Traditional and presentday topics in real analysis
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„FOLIA MATHEMATICA” SERIES EDITORS Marek Śmietański, Piotr Fulmański, Cezary Obczyński

PROJECT EDITORS Cezary Obczyński, Małgorzata Terepeta

COVER DESIGN Mariusz Kubiński

PHOTOS Małgorzata Terepeta

© Copyright by University of Łódź, Łódź 2013

Publication reviewed Printed directly from camera‐ready materials provided to the Łódź University Press by Chair of Real Functions First Edition. W.06372.13.0.I

ISBN 978-83-7525-971-1

Łódź University Press 90-131 Łódź, ul. Lindleya 8 www.wydawnictwo.uni.lodz.pl e-mail: [email protected] tel. (42) 665 58 63, faks (42) 665 58 62 Print and setting: Quick Druk

Dedicated to ´ Professor Jan Stanisław Lipinski on the occasion of his 90th birthday

Contents

1

A Modest Review of a Great Deal of Work . . . . . . . . . . . . . . . . . . . Paul D. Humke

11

2

´ Jan Lipinski – our teacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Władysław Wilczy´nski

27

3

´ The legacy of Professor Jan Stanisław Lipinski for the sake of ´ (1969 - 1998) . . . . . . the development of mathematics in Gdansk Lech Górniewicz, Zbigniew Grande

4

´ The Scientific Family of Professor Jan Lipinski ............... Małgorzata Filipczak, Małgorzata Terepeta

5

Existence theorems on convolution of functions, distributions and ultradistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

29

35

39

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Convergence of sequences of measurable functions . . . . . . . . . . . . Elz˙ bieta Wagner-Bojakowska, Władysław Wilczy´nski

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Ideal convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rafał Filipów, Tomasz Natkaniec, Piotr Szuca

69

8

On uniform convergence and some related types of convergence Robert Drozdowski, Jacek J˛edrzejewski, Stanisław Kowalczyk, Agata Sochaczewska

93

7

8

Contents

9

Points of quasicontinuity and of similar generalizations of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Ján Borsík

10 On Extension Problem, Decomposing and Covering of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Zbigniew Grande, Mariola Marciniak 11 On Baire generalized topological spaces and some problems connected with discrete dynamical systems . . . . . . . . . . . . . . . . . . . 151 Anna Loranty, Ryszard J. Pawlak 12 On rings of Darboux-like functions. From questions about the existence to discrete dynamical systems . . . . . . . . . . . . . . . . . . . . . . 173 Ewa Korczak-Kubiak, Helena Pawlak, Ryszard J. Pawlak 13 I-approximate differentiation of real functions . . . . . . . . . . . . . . . 195 Ewa Łazarow 14 Lineability, algebrability and strong algebrability of some sets in RR or CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Artur Bartoszewicz, Marek Bienias, Szymon Gła¸b 15 On approximately continuous integrals (a survey) . . . . . . . . . . . . 233 Valentin A. Skvortsov, Tatiana Sworowska, Piotr Sworowski 16 Axial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Marcin Szyszkowski 17 Measurability of multifunctions with the (J) property . . . . . . . . . 265 Graz˙ yna Kwieci´nska 18 Bilinear mappings – selected properties and problems . . . . . . . . . 281 Marek Balcerzak, Filip Strobin, Artur Wachowicz 19 Stability Aspects of the Jensen-Hosszú Equation . . . . . . . . . . . . . . 307 Zygfryd Kominek 20 Properties of the σ - ideal of microscopic sets . . . . . . . . . . . . . . . . . 325 Graz˙ yna Horbaczewska, Aleksandra Karasi´nska, Elz˙ bieta Wagner-Bojakowska

Contents

9

21 Topological and algebraic aspects of subsums of series . . . . . . . . 345 Artur Bartoszewicz, Małgorzata Filipczak, Franciszek Prus-Wi´sniowski 22 On ψ-density topologies on the real line and on the plane . . . . . . 367 Małgorzata Filipczak, Małgorzata Terepeta 23 Density type topologies generated by functions. f -density as a generalization of hsi-density and ψ-density . . . . . . . . . . . . . . . . . . . 389 Tomasz Filipczak 24 Density type topologies generated by functions. Properties of f -density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Małgorzata Filipczak, Tomasz Filipczak 25 On the abstract density topologies generated by lower and almost lower density operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Jacek Hejduk, Renata Wiertelak 26 Path continuity connected with the notion of density . . . . . . . . . . 449 Stanisław Kowalczyk, Katarzyna Nowakowska 27 Decompositions of permutations of N with respect to divergent permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Roman Wituła List of denotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Chapter 1

A Modest Review of a Great Deal of Work

PAUL D. HUMKE

It is both an honor and a pleasure and, I might say, somewhat daunting to write a paper describing the mathematical career of a mathematician who has made such major contributions to both the scientific literature and to his chosen profession. I met Jan Lipi´nski in 1978 at the International Congress in Helsinki. He came to a session at which I presented a poster and was very encouraging. I recall vividly; it meant a great deal to the beginner that I was. Two days later we met again at a dinner we’d arranged for several real analysts attending the conference. Jan was there as was my wife Bonnie, Andy and Judy Bruckner, Mik Laczkovich, Krishna Garg, Ladislav Mišík, and Dan and Mudite Waterman. All of us have remained in close contact throughout our lives, and all have certainly enriched my own life immeasurably both scientifically and personally. In 1987 a group of us here in the United States applied for a series of grants to invite several European real analysts to a Special Real Analysis Session at the Annual Joint Meeting of the A.M.S. in San Antonio, Texas. Jan was among those invited, as was one of Jan’s former students, Władek Wilczy´nski. After the meeting Jan and I flew back to Minnesota where he stayed with me and my family in Minnesota for a week. We talked a great deal of mathematics during that week and got to know each other pretty well. Jan enchanted our children with stories and “tricks;” his separating finger trick is still a favorite of our eldest son, Eric who has demonstrated it to all of his nieces and nephews and now to his own children.

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Paul D. Humke

Jan was a first student of Zygmunt Zahorski and much of Jan’s mathematical work reflects the style and delicacy of the Zahorski school. Although Zahorski wrote his dissertation under the direction of Tadeusz Waz˙ ewski, he had early worked with both Mazurkiewicz and Banach and his professional life reflects a devotion to working on hard problems and developing deep understanding of the intricacies of sets and functions. An insightful reflection on Zahorski’s life and work was written by Władek Wilczy´nski, see [62], to whom I am indebted for introducing me to Zahorski during the Warsaw International Congress in 1983. Several of Jan’s earliest papers solve problems left unresolved in papers published by Zahorski, but he quickly branched out, successfully attacking problems in allied areas of real analysis and set theoretic topology. The forty seven papers reflected in this review reveal a sparkling energy and creative spirit that characterize their author. Jan has fathered eight mathematical children, twenty nine mathematical grand children and thirty eight mathematical great grandchildren. It is a wonderfully diverse and dedicated family of professional mathematicians who continue to push back the frontiers of real analysis. In the sections that follow I have chosen four categories within which to discuss Jan’s scientific work and most of that work is at least touched upon in the sequel. Any division of a body of intellectual work is artificial and, in some sense can detract from the overall vision of the whole. Still, I found it helpful to make some categorization and I hope it is not distractive to the reader. Inevitably, there is overlap between the sections, but frequently the overlap reflects Jan’s new ways of looking at old ideas. We’ll begin where Jan himself began in the early 1950’s by looking at Jan’s contributions to understanding derivatives.

1.1 Derivatives The papers I’ve categorized as Jan’s derivative body of work1 is substantial and his interest in derivatives is clearly career long. And a solid portion of this work involves the hierarchy of classes of Fσ sets and of Baire 1 functions introduced by Zahorski in his celebrated 1950 paper, Sur la première dérivée, [64] . I’ll first give the briefest of descriptions of these classes; a characteristically elegant and complete treatment can be found as Chapter 6 of Andy Bruckner’s book, Differentiation of Real Functions, [2]. I’ll use that treatment here, para1

See [39], [34], [44], [32], [28], [27], [24], [23], [19], [16], [17], [14].

1. A Modest Review of a Great Deal of Work

13

phrasing and giving enough background within which to place some of Jan’s contributions. Definition 1.1. Let 0/ 6= E ∈ Fσ . Then E is said to belong to class M0 if every point of E is a bilateral accumulation point of E; M1 if every point of E is a bilateral condensation point of E; M2 if every one-sided neighborhood of each point of E intersects E in a set of positive measure; M3 if for each x ∈ E and each sequence of closed intervals, {In } converging to x but not containing x such that λ (In ∩ E) = 0 for each n, we have limn→∞ λ (In )/dist(x, In ) = 0; M4 if there exists a sequence of closed sets, {Kn } and a sequence of positive S numbers ηn such that E = Kn and for each x ∈ Kn and each c > 0 there is a number ε = ε(x, c) such that if h and k satisfy hk > 0, h/k < c, |h+k| < ε, then λ (E ∩ (x + h, x + h + k)) > ηn . |k| M5 if every point of E is a point of density of E. These classes of sets give rise to corresponding classes of Baire 1 functions in a most natural way. Definition 1.2. A function f ∈ Mi if every associated set of f is in class Mi for i = 0, 1, . . . , 5. By associated sets of f we mean the sets of the form {x : f (x) < α} or {x : f (x) > α}. Let DB1 be the class of Darboux Baire 1 functions and Cap - the class of the approximately continuous functions. Two foundational results concerning the Zahorski Classes (see [2], Theorem 1.3, Corollary 2.4 and Theorem 2.5) are the following: Theorem 1.3. Cap = M5 ( M4 ( M3 ( M2 ( M1 = M0 = DB1 . In the next theorem, ∆ 0 denotes the set of derivatives and b∆ 0 denotes the set of bounded derivatives. Theorem 1.4. ∆ 0 ( M3 and b∆ 0 ( M4 .

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Paul D. Humke

David Preiss in [58] and David and Maria Tartaglia in [57] also had a good deal to say about this story and a relatively complete list of related papers can be found by searching for papers citing Zahorski, [64]. Zahorski defined the classes described in the paragraph above and among many other results, showed that ∆ 0 ( M2 and b∆ 0 ( M3 . Several questions remained, however and Jan answered one of these in [14] where he proves the following theorem.2 Theorem 1.5. There is a set E ∈ M3 such that E is not an associated set for any finite derivative. The condition required for a set to belong to the class M4 is rather complicated, but is not dissimilar to the original formulation of the M3 condition. The latter was simplified to that given above, and Zahorski asked whether a similar simplification could be made for M4 by taking the η to be dependent only on x and not on n. In [16] Jan showed that this is not possible via the following theorem. Theorem 1.6. Un condition nécessaire qu’un ensemble linéaire E soit identique á l’ensemble des pointes en lesquels une fonction dérivée, bornée en module, prend une valeur finie donnée est que E soit un Gδ contenant tous les points d’accumulation en mesure. Several other Zahorski type derivative results can be found the following Section 1.2. But now I’ll turn to derivative results of different types. In 1957 Jan published a paper in Colloquium Mathematicum, [17] in which he studies monotone jump functions. Definition 1.7. 1. If f : R → R, then D∞ ( f 0 ) = {x : f 0 (x) = ∞}. 2. If f : R → R is bounded and non-decreasing, then f is called a jump function provided a. ∑R f (x + 0) − f (x − 0) < ∞, and b. f (b − 0) − f (a + 0) = ∑a 0 for infinitely many n ∈ N, then for every increasing sequence of integers, {in }, λ (lim sup Enin ) = 1.

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Theorem 1.28. For every fixed increasing sequence of integers, {in } and each ε > 0 there is a sequence of measurable sets, {En } such that 1. ∑ λ (En ) = +∞ S 2. λ ( Enin ) < ε, and 3. λ (lim sup Enin ) = 0. There are three additional papers I’ll make some remarks about here and the first of these, [46] concerns transfinite limits either of functions. Here Jan considers families of mappings, F = { f : E → Y } where E is a fixed set and Y is a metric space. Definition 1.29. Such a family is closed with respect to transfinite convergence if whenever { fα : α < ω1 } ⊂ F converges pointwise to a function f , then f ∈ F. Under the assumption of CH, Jan then shows several classical families of mappings to be closed in this sense. These include the bounded functions, increasing functions, differentiable functions and several others. The second, [26] is a contribution to the general topic of the algebra of continuous functions, C[a, b] = { f : [a, b] → [a, b]}. Given f ∈ C[a, b], define S( f ) = {g ∈ C[a, b] : f ◦ g = g ◦ f }. Jan proves the following. Theorem 1.30. For every f ∈ C[a, b], S( f ) is infinite. Moreover, if f is strictly increasing, then the cardinality of S( f ) is 2ℵo . Theorem 1.31. There is a function f : [a, b] → [a, b] with two simple discontinuities for which S( f ) = 0. / Finally, in [40] Jan and Tibor Šalát define a general Banach Indicatrix function and study its measurability. More specifically, let X and Y be arbitrary sets and f : X → Y . Definition 1.32. The Banach Indicatrix of f is τ f : Y → N ∪ {∞} defined as ( card( f −1 (y)), if f −1 (y) is finite, τ f (y) = ∞, otherwise. Among the theorems proved are the following. Theorem 1.33. If f : R → R is monotone, then τ f is in Baire class 2. Theorem 1.34. If f : R → R is a Baire function, then τ f is Lebesgue measurable.

1. A Modest Review of a Great Deal of Work

21

1.4 Generalized Continuity Jan has written more than ten papers5 concerning various notions of generalized continuity. Too, this topic proved a rich source for his active collaboration with other real analysts, and his interest in notions of generalized continuity can be seen as threading a good portion of his research career, beginning with [33] in 1968 and extending through [48] in 1993. As with other sections of this paper, the setting is not always the real line R, but sometimes a general topological space; I’ll try to keep the more general notions somewhat separate from those specifically related to those of the real line, R, but, of course, this is not always completely possible. In [33], Jan began an investigation of the relationship between functions that are continuous and functions that preserve connectedness. This relationship is important for a variety of reasons not the least of which is that derivatives of functions f : R → R preserve connectedness, but need not be continuous. Here are the definitions and theorems Jan proves in 1971. Definition 1.35. Suppose that X is a topological space and f : X → R. Then, 1. f is said to have property (G) if there exists a dense set Y ⊂ R, such that the set f −1 (y) is closed for each y ∈ Y . 2. f is said to have property (D) if it maps connected sets onto connected sets. Theorem 1.36. If X is locally connected, then f is continuous if and only if f possesses both properties (G) and (D). Theorem 1.37. If, for every open subspace A ⊂ X, every f : A → R and possessing the properties (G) and (D) is continuous, then X is locally connected. A great deal has been done in this area since 1971 and I will mention several more of Jan’s contributions in the next few paragraphs, but before listing those I would like to insert some purely real analysis references. Mike Evans and I published a general audience paper on the subject in the Monthly in 2009, [6] and a more technical paper with a reasonable bibliography in [7]. A wonderful introduction to the subject can be found in Andy Bruckner’s classic, [2] while a paper illustrating the significance of the study is Jan Malý’s paper, [53]. The list of real analysts who have written on this topic is both broad and long, yet much still needs to be discovered. 5

See [4], [5], [8], [9], [11], [18], [20], [33], [48], [51].

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Three of the last five papers Jan published in the area of generalized continuity were coauthored with Janina Ewert, [9], [8], [11], one was coauthored with Tibor Šalát, [51] and the final paper he wrote by himself, [48]. In these papers, written over a period spanning two decades, Jan and his coauthors study the relationship between points of continuity, points of quasicontinuity and points of cliquishness. Definitions follow. For quasicontinuity, let X and Y be general topological spaces and let x ∈ X. Definition 1.38. A function f : X → Y is said to be quasicontinuous at x if for every pair of neighborhoods, U of x and V of f (x) there is a nonempty open set W ⊂ U such that f (W ) ⊂ V . For cliquish, let X be a topological spaces, Y be a metric space with metric, ρ and let x ∈ X. Definition 1.39. A function f : X → Y is said to be cliquish at x if for every ε > 0 and neighborhood U of x there is a nonempty open set W ⊂ U such that whenever w1 , w2 ∈ W , then ρ( f (w1 ), f (w2 )) < ε. Definition 1.40. The sets of points of continuity, quasicontinuity and cliquishness are denoted C( f ), E( f ), and A( f ) respectively. It is clear from the definitions that C( f ) ⊂ E( f ) ⊂ A( f ). Two results from the Lipi´nski-Šalát paper, [51] relate the nature of the individual sets E( f ) and A( f ). Theorem 1.41. Let X be a topological space and Y be a metric space. Then, for an arbitrary function f : X → Y A( f ) is a closed subset of X. If both X and Y are metric spaces, they have a full characterization, namely: Theorem 1.42. If X and Y are metric spaces, then there exists a function f : X → Y with A( f ) = A if and only if A is closed. Further, if X = Rn is a Euclidean space and Y = R, then a similar characterization is proved for E( f ), namely: Theorem 1.43. If X = Rn and Y = R, then a set E ⊂ X is the set of quasicontinuity points for a function f : X → Y if and only if int(E)\E is a set of the first Baire category. In [11], Ewert and Lipi´nski build on these results using the fact that A( f )\C( f ) is of the first Baire category to prove the following theorem.

1. A Modest Review of a Great Deal of Work

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Theorem 1.44. Suppose X = Rn and Y = R (or alternately that X and Y are real normed linear spaces and Y is a Baire space). Then whenever C, E, and A are sets with 1. C ⊂ E ⊂ A = A, and 2. A\C is of the first Baire category, then there is a function f : X → Y such that C = C( f ), E = E( f ), and A = A( f ). The next two papers in this program, also by Ewert and Lipi´nski, [9], [8] are more technical, but continue the investigation of the relationship between the sets C( f ), E( f ) and A( f ). Too, they tend to assume more specific conditions on the underlying spaces X and Y . But they are interesting and reveal insight into several classical theorems. I’ll give you a taste. Theorem 1.45. Let X be a topological space which is the union of two disjoint dense subsets, and let Y be a metric space with at least one accumulation point. Then for each decreasing sequence {Wn : n = 1, 2, . . . } of open subsets of X and each E satisfying the inclusions C=

∞ \ n=1

Wn ⊂ E ⊂

∞ \

Wn = A

n=1

there is a function f : X → Y such that C = C( f ), E = E( f ), and A = A( f ). The applications presented are using Y = R with the usual topology and X = R with the density topology. In this instance “quasicontinuity” is referred to as “density-quasicontinuity” etc. They first note the following theorem. Theorem 1.46. A function f : R → R is measurable if and only if f is densitycliquish. Using Theorem 1.45 and the fact that “approximate continuity” is equivalent to “density-continuity” they give a simple proof of Denjoy’s Theorem. Denjoy’s Theorem A function is Lebesgue measurable if and only if it is approximately continuous almost everywhere.

References [1] A. M. Bruckner and C. Goffman, Approximate differentiation, Real Anal. Exchange 6(1) (1980/81), 9–65.

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[2] A. M. Bruckner, Differentiation of real functions, volume 5 of CRM Monograph Series, American Mathematical Society, Providence, RI, second edition, 1994. [3] J. Ceder, On Darboux points of real functions, Period. Math. Hungar. 11(1) (1980), 69–80. [4] M. C. Chakrabarty and J. S. Lipi´nski, On points of absolute continuity. III, Colloq. Math. 22 (1971), 281–284. [5] M. Czajka-Zgirska and J. S. Lipi´nski, On the continuity for connected functions, Math. Slovaca 31(4) (1981), 341–345. [6] M. J. Evans, P. D. Humke, Revisiting a century-old characterization of Baire class one, Darboux functions, Amer. Math. Monthly 116(5) (2009), 451–455. [7] M. J. Evans, P. D. Humke, Collections of Darboux-like, Baire one functions of two variables, J. Appl. Anal. 16(1) (2010), 135–149. [8] J. Ewert, J. S. Lipi´nski, On points of continuity, quasicontinuity and cliquishness of maps, Topology, theory and applications (Eger, 1983), volume 41 of Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam 1985, 269–281. [9] J. Ewert, J. S. Lipi´nski, On relations between continuity, quasi-continuity and cliquishness of maps, General topology and its relations to modern analysis and algebra, VI (Prague, 1986), volume 16 of Res. Exp. Math., Heldermann, Berlin 1988, 177–185. [10] J. Ewert, J. S. Lipi´nski, Weakly connected multivalued maps, Rad. Mat. 5(2) (1989), 193–200. [11] J. Ewert, J. S. Lipi´nski, On points of continuity, quasicontinuity and cliquishness of real functions, Real Anal. Exchange 8(2) (1982/83), 473–478. [12] K. M. Garg, Monotonicity, continuity and levels of Darboux functions, Colloq. Math 28 (1973), 91–103, 162. [13] Z. Grande, J. S. Lipi´nski, Un exemple d’une fonction sup-mesurable qui n’est pas mesurable, Colloq. Math. 39(1) (1978), 77–79. [14] J. S. Lipi´nski, Une propriété des ensembles { f 0 (x) > a}, Fund. Math. 42 (1955), 339–342. [15] J. S. Lipi´nski, Sur les ensembles { f (x) > a}, où f (x) sont des fonctions intégrables au sens de Riemann, Fund. Math. 43 (1956), 202–229. [16] J. S. Lipi´nski, Sur certains problèmes de Choquet et de Zahorski concernant les fonctions dérivées, Fund. Math. 44 (1957), 94–102. [17] J. S. Lipi´nski, Sur la dérivée d’une fonction de sauts, Colloq. Math. 4 (1957), 197–205. [18] J. S. Lipi´nski, Sur l’uniformisation des fonctions continues, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 1019–1021, LXXXV. [19] J. S. Lipi´nski, Sur les ensembles { f 0 (x) > a}, Fund. Math. 45 (1958), 254–260. [20] J. S. Lipi´nski, Sur les fonctions approximativement continues, Colloq. Math. 5 (1958), 172–175. [21] J. S. Lipi´nski, Sur une intégrale, Colloq. Math. 7 (1959), 67–74. [22] J. S. Lipi´nski, Sur un problème de E. Marczewski concernant les fonctions périodiques, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 695–697. [23] J. S. Lipi´nski, Mesure et dérivée, Colloq. Math. 8 (1961), 83–88. [24] J. S. Lipi´nski, Une simple démonstration du théorème sur la dérivée d’une fonction de sauts, Colloq. Math. 8 (1961), 251–255. [25] J. S. Lipi´nski, Sets of points of convergence to infinity of a sequence of continuous functions, Fund. Math. 51 (1962/1963), 35–43.

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[26] J. S. Lipi´nski, Sur la composition commutative des fonctions, Colloq. Math. 10 (1963), 271–276. [27] J. S. Lipi´nski, Sur la discontinuité approximative et la dérivée approximative, Colloq. Math. 10 (1963), 103–109. [28] J. S. Lipi´nski, Sur quelques problèmes de S. Marcus relatifs à la dérivée d’une fonction monotone, Rev. Math. Pures Appl. (Bucarest) 8 (1963), 449–454. [29] J. S. Lipi´nski, On a kind of dispersion of sets, Colloq. Math. 12 (1964) 249–251. [30] J. S. Lipi´nski, On periodic extensions of functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 12 (1964), 373–375. [31] J. S. Lipi´nski, On periodic extensions of functions, Colloq. Math. 13 (1964), 65–71. [32] J. S. Lipi´nski, Sur les dérivées de Pompeiu, Rev. Roumaine Math. Pures Appl. 10 (1965), 447–451. [33] J. S. Lipi´nski, Une remarque sur la continuité et al connexité, Colloq. Math. 19 (1968), 251–253. [34] J. S. Lipi´nski, On derivatives of singular functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 625–628. [35] J. S. Lipi´nski, On measurability of functions of two variables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 131–135. [36] J. S. Lipi´nski, On transfinite sequences of approximately continuous functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 817–821. [37] J. S. Lipi´nski, On level sets of Darboux functions, Fund. Math. 86 (1974), 193–199. [38] J. S. Lipi´nski, On a problem of Bruckner and Ceder concerning the sum of Darboux functions, Proc. Amer. Math. Soc. 62(1) (1977) 1976, 57–61. [39] J. S. Lipi´nski, A remark about Laczkovich’s theorem on functions whose sections are derivatives, Period. Math. Hungar. 18(4) (1987), 319–323. ˇ [40] J. S. Lipi´nski, T. Šalát, On the generalized Banach indicatrix, Mat. Casopis Sloven. Akad. Vied 22 (1972), 219–225. [41] J. S. Lipi´nski, Convergence to infinity of a sequence of continuous functions, Dokl. Akad. Nauk SSSR 140 (1961), 752–754. [42] J. S. Lipi´nski, On continuous periodic extensions of functions, Proceedings of the Real Analysis Summer Symposium (Syracuse, N.Y., 1981), volume 7 (1981/82), 129–134. [43] J. S. Lipi´nski, Two nonmeasurable functions of two variables with measurable sections, Rev. Roumaine Math. Pures Appl. 35(1) (1990), 49–52. ˇ [44] Jan Stanisław Lipi´nski, Sur la classe M02 , Casopis Pˇest. Mat. 93 (1968), 222–226, 227. [45] J. S. Lipi´nski, Über eine Frage von Herr L. Mišik, Czechoslovak Math. J. 21 (96) (1971), 234–235. ˇ [46] J. S. Lipi´nski, On transfinite sequences of mappings, Casopis Pˇest. Mat. 101(2) (1976), 153–158. [47] J. S. Lipi´nski, On Darboux points, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(11) (1979), 1978, 869–873. [48] J. S. Lipi´nski, On some extensions of almost continuous functions and of connectivity functions, Tatra Mt. Math. Publ. 2 (1993), 15–18. [49] J. S. Lipi´nski, On sequences and transfinite sequences of closed sets, Słup. Prace Mat. Przyr. Mat. Fiz. 9A (1994), 23–28. [50] J. S. Lipi´nski, Zygmunt Zahorski (1914–1998), Wiadom. Mat. 36 (2000), 73–83.

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Paul D. Humke

[51] J. S. Lipi´nski, T. Šalát, On the points of quasicontinuity and cliquishness of functions, Czechoslovak Math. J. 21 (96) (1971), 484–489. [52] A. J. Lohwater, The boundary behavior of derivatives of univalent functions, Math. Z. 119 (1971), 115–120. [53] J. Malý, The Darboux property for gradients, Real Anal. Exchange 22(1) (1996/97), 167–173. [54] S. Marcus, Sur les dérivées dont les zéros forment un ensemble frontière partout dense, Rend. Circ. Mat. Palermo (2) 12 (1963), 5–40. [55] S. Markus, Points of discontinuity and points at which the derivative is infinite, Rev. Math. Pures Appl. 7 (1962), 309–318. [56] E. Marczewski, Remarks on sets of measure zero and the derivability of monotonic functions, Prace Mat. 1 (1955), 141–144. [57] D. Preiss, M. Tartaglia, On characterizing derivatives, Proc. Amer. Math. Soc. 123(8) (1995), 2417–2420. [58] D. Preiss, Level sets of derivatives, Trans. Amer. Math. Soc. 272(1) (1982), 161–184. [59] W. Sierpi´nski, Sur l’ensemble de valeurs de qu’une fonction continue prend une infinité non dénombrable de fois, Fund. Math. 8 (1926), 370–373. [60] J. Szmelter, T. Sulikowski, J. S. Lipi´nski, Bending of a rectangular plate clamped at one edge, Arch. Mech. Stos. 13 (1961), 63–75. [61] G. Tolstov, La méthode de Perron pour l’intégrale de Denjoy, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 149–168. [62] W. Wilczy´nski, Zygmunt Zahorski [1914–1998]—an obituary, Real Anal. Exchange 23(2) (1997/98), 359–361. [63] Z. Zahorski, Sur la classe de Baire des dérivées approximatives d’une fonction quelconque, Ann. Soc. Polon. Math. 21 (1949) 1948, 306–323. [64] Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1–54.

PAUL H UMKE Department of Mathematics, St. Olaf College Northfield, Minnesota 45701 E-mail: [email protected]

Chapter 2

´ Jan Lipinski – our teacher

´ WŁADYSŁAW WILCZYNSKI

Jan Stanisław Lipi´nski was born Nov. 24, 1923 in Przemy´sl (now southeastern Poland). During the Second World War he lived in Warsaw, where he survived the outbreak of the Warsaw uprising. In 1945 he obtained (as an extern) the certificate of maturity and worked as a teacher in Zgierz -– a small town several kilometers from Łód´z. From 1946 to 1952 he studied mathematics at the University of Łód´z. In 1949 he started research under professor Zygmunt Zahorski, who was the chief of the Chair of Mathematics II. Professor Lipi´nski has obtained his Ph.D. in 1958 and habilitation in 1960, both from University of Łód´z. He became a full professor in 1967. Fifty years later University of Łód´z has organized for him a 50th Anniversary Celebration. In 2013 University of Łód´z awarded the medal Universitatis Lodziensis Amico to Professor Jan Lipi´nski. I met Jan Lipi´nski as a student of the third course in 1965. Then I wrote my master thesis on rectifiable plane continua in 1968 and the supervisor was Professor Lipi´nski. After that I was employed at the University of Łód´z, Department of Mathematics, Physics and Chemistry, Chair of Mathematics II. I expected to learn a lot from Zygmunt Zahorski and from Jan Lipi´nski (working also in the Chair of Math. II). Unfortunately, thanks to the knotted history of University of Łód´z Professor Lipi´nski left our University going to Gda´nsk in 1969 (his work in Gda´nsk is described in the article of Lech Górniewicz and Zbigniew Grande), Professor Zahorski went to Gliwice and continued work on Silesian Technical University. The group of young mathematicians interested in real analysis at the University of Łód´z became scientific orphans. Luck-

28

Władysław Wilczy´nski

ily, we could participate from time to time in the seminar hold by Professor Lipi´nski in Gda´nsk. The second lucky circumstance was that at the Technical ´ atkowski, University of Łód´z there was Professor Tadeusz Swi ˛ also eminent in real analysis. In the seventieth it was a great common effort of Mirosław Filipczak, Jacek J˛edrzejewski and me to continue studies in real analysis in Łód´z. Our students and students of our students are still working on it. Professor Jan Lipi´nski was still with us: he participated in numerous conferences, wrote a lot of opinions for doctorates, habilitations and professorships. Thanks to his contacts with eminent mathematicians in real analysis throughout the world we had the opportunity to deepen our knowledge and to learn new trends in our field of interest. Paul Humke in his article described almost all scientific achievements of Professor Jan Lipi´nski. In the text he didn’t mention papers [30] and [31]. The results there, although this is a lateral branch of scientific creation of Professor Lipi´nski, belong to my favourite in the whole mathematics. Professor Lipi´nski proved that there exists a sequence (quickly tending to infinity) of real numbers such that each bounded real function defined on the set of terms of this sequence can be extended to a continuous periodic function defined on the whole real line, and he even strengthened this result proving that instead of the sequence of numbers one can find the sequence of intervals (obviously, with necessary changes concerning the function defined on the union of intervals) having similar property. I believe this result should have a considerable place in THE BOOK refered to by Pal Erdös. ´ W ŁADYSŁAW W ILCZY NSKI Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 3

´ The legacy of Professor Jan Stanisław Lipinski for the sake of the development of mathematics in ´ (1969 - 1998) Gdansk

LECH GÓRNIEWICZ, ZBIGNIEW GRANDE

In the post-war years, mathematics in Gda´nsk was concentrated at the Technical University. The professors who pioneered the development of mathematics there were Eustachy Tarnawski and Wacław Pawelski, and a little later they were joined by Piotr Besala and Marian Kwapisz. Then at the beginning of the 1960’s, the development of mathematics slowly began at the School of Teacher Education. The first titular professor at WSP (School of Teacher Education) was Ryszard Bitner, and by the mid-1960’s an intensive effort was undertaken to establish a university. Great achievements in this effort were due to Professor Janusz Sokołowski, then Rector of WSP, and later Rector and Honorary Rector of the University of Gda´nsk. In order to establish a firm foundation for the new university, the authorities in charge identified certain key disciplines and created significant incentives to attract strong researchers in those disciplines to work at WSP. One of those key disciplines was mathematics and in the years from 1964-1970 many strong researchers were recruited for the newly emerging center at Gda´nsk. Among them were Professors Kazimierz G˛eba, Andrzej Granas, Andrzej Włodzimierz Mostowski and Jan Stanisław Lipi´nski. Professor Lipi´nski had been a professor at the University of Łód´z until he joined the School of Education in Gda´nsk as a professor on 1 May, 1969. Fourteen months later, on 1 July, 1970, three universities, the School of Education, the School of Economics and the Education Teachers College merged to form the newly minted Gda´nsk University. It was here where Professor Lipi´nski spent the remainder of his rich professional career until he retired in 1998.

30

Lech Górniewicz, Zbigniew Grande

In this paper we focus on the scientific, educational and organizational aspects of the legacy of Professor Jan Stanisław Lipi´nski during his years in Gda´nsk from 1969 until 1998. The period of his career at the University of Łód´z before 1969 was also remarkably fruitful, but those contributions are the subject of a separate chapter. Immediately after his employment and in his capacity as head of the Department of Numerical Methods, Professor Jan Stanisław Lipi´nski began fostering the development of mathematics in Gda´nsk. But, in 1970 with the establishment of the University of Gda´nsk, it became clear that the faculty of mathematics would need to be radically reorganized. This reorganization resulted in the establishment of the Institute of Mathematics and the clear choice for its first director was Professor Jan Stanisław Lipi´nski who led the Institute from 1 July 1970 to 30 September 1972. This was an extraordinarily busy and exciting period, marked by the establishment of dynamic new faculties of mathematics, physics and chemistry. Later these faculties were reorganized into two divisions, namely, the Faculty of Mathematics, Physics and Computer Science and the Faculty of Chemistry. It is impossible to overestimate the role of Professor Lipi´nski played in this process. His leadership and energy were so integral to the developmental process that it is difficult to separate the achievements of the Institute from those of Professor Lipi´nski. Calling on his experience at the University of Łód´z, he was able to anticipate and circumvent many of the problems inherent with a fast growing institution and his patience and experience proved invaluable in solving those substantive issues that did arise. Among his many other duties, he was named head of the Department of Probability Theory on 1 October 1970 and continued to serve in that capacity through the transformation of that department to the Department of Real Functions and Probability Theory in 1973 up until his retirement on 30 September 1991. An outstanding specialist in the theory of real functions and the theory of measure and integration himself, he instilled his intellectual excitement and interest in a large active group of young mathematicians in Gda´nsk. Within the workings of the Institute, he delivered a weekly research seminar that was attended by mathematics students and researchers from Łód´z, Cz˛estochowa and Słupsk as well as those from Gda´nsk. The remarkably rich mathematical environment and high level of activity at Gda´nsk directly resulted in four of the doctorates supervised by Professor Lipi´nski: • Mieczysław Gutowski, On the derivative of functions with infinite jump, 1973,

3. The legacy of Professor Jan Stanisław Lipi´nski - Gda´nsk (1969 - 1998)

31

• Zbigniew Grande, Measurability of functions defined on product spaces, 1974, • Stanisław Wojtan, Convergence of sequences of open functions, 1974, • Grzegorz Krzykowski, Homeomorphisms preserving Zahorski classes, 1983. Moreover, although these four PhDs were clearly inspired by the high level of research fostered by Professor Lipi´nski, they only represent a small portion of the effect of that seminar on the mathematical community. For example, the Ph.D. thesis of Stanisław Wojtan contains the first correct proof of an early result of A. M. Bruckner in which Professor Lipi´nski discovered a significant error. While in Slovakia, Professor Lipi´nski in collaboration with Professor L. Misik initiated a broad study of the measurability of functions of two and several variables. These joint results quickly generated a flurry of activity by a good many mathematicians, both Polish and foreign. Among these are R. O. Davies, J. Dravecky, M. Laczkovich, and Z. Grande. Professor Z. Grande, a pupil of Professor Lipi´nski supervised three doctoral theses in this subject ´ ezak at the University of area. The first, in 1982 was written by Włodzimierz Sl˛ Pozna´n. The second thesis, entitled "Measurability of real functions defined on the product of metric spaces" was written by Graz˙ yna Kwieci´nska in 1984 at the University of Gda´nsk, and in 2003, Professor Grande supervised the Ph.D. thesis of Katarzyna Chmielewska at the Technical University of Łód´z. The school of scientific inquiry led by Professor Lipi´nski has included a widely aclaimed list of senior research real analysts. Among these are Professors: Zbigniew Grande, Janina Ewert, Tomasz Natkaniec, Graz˙ yna Kwieci´nska, Aleksander Maliszewski and a host of others all inspired by the love of real functions and measure theory they experienced in Professor Lipi´nski’s seminar at the University of Gda´nsk. Not only did his practiced ability and penchant to inspire and train budding mathematicians have a major influence on the development of mathematics in Gda´nsk, but also in Bydgoszcz and Słupsk. Of particular note, was his influence and contributions to the Ph.D. thesis work of Tadeusz Lipski and Zygmunt Wojtowicz. In the main, Professor Jan Stanisław Lipi´nski’s scientific work focused on real function theory and the theory of measure and integration. Throughout his entire tenure in Gda´nsk, he taught his classes and conducted his seminars on these subjects. His scientific output encompasses more than 50 scientific papers, and he has written on such diverse topics as derivatives of real functions, continuous and approximately continuous functions, function sequences and series, approximate derivative, functions enjoying the Bolzano-Darboux property, and properties of sets associated with such functions. One highlight of Professor Lipi´nski’s professional career is connected with the problem of

32

Lech Górniewicz, Zbigniew Grande

Caratheodory superposition measurability. In a series of papers and in collaboration with Canadian mathematician, K. Garg and Czech mathematician, D. Preiss he studied global and local properties of different classes of functions. Then, in 1978, and in collaboration with Zbigniew Grande, he constructed a non-measurable function that is measurable in the Caratheodory superposition sense. This example caused quite a stir in the mathematics community and is sufficiently important that it gained the permanent name as the "Polish monster". The most extensive work of Professor Lipi´nski’s is the article, "Elements of the general theory of measure and integration," IKN, Math workbook 1, 1977. Professor Lipi´nski has served on a number of Ph.D. and habilitation committees as well having served as a professional referee for several candidates for Professorships. Professor Lipi´nski has taught and lectured extensively at the invitation of universities in the United States, Canada, and the United Kingdom and he has served as a plenary speaker at several international and national conferences devoted to the subject matter in which he both loved and excelled. His professional colleagues abroad are and his adherents in Poland include the leaders of mature schools of real analysis, especially notable are Władysław Wilczy´nski in Łód´z, Zbigniew Grande in Bydgoszcz and Tomasz Natkaniec in Gda´nsk. Professor Lipi´nski’s scientific achievements are widely recognized in our country and abroad and are widely cited in the professional literature. It is beyond question that in honoring Professor Jan Stanisław Lipi´nski we are honoring a world renowned mathematician who has made important contributions to the internationally esteemed Polish school of real functions initiated by Wacław Sierpi´nski, Zygmunt Zahorski and Roman Sikorski. Professor Jan Stanisław Lipi´nski was a great teacher. He regarded teaching as a very important part of his activities. Students took a lot of interest in his lectures and they also highly praised them. He was very kind and understanding for students. He conducted- practically every two years- new MA seminars. The work prepared under his supervision is always characterized by being scientifically flawless. He supervised over 70 Masters. An important initiative of Professor Lipi´nski was to open the mathematical class in Nicolaus Copernicus Secondary School in Gda´nsk. The existence of this class almost immediately increased the number of participants of Mathematical Olympiads. Many graduates of this class studied mathematics at the Gda´nsk University and had the best students. Consequently some of them are well known researches working in Mathematical Institute of Gda´nsk University It is worth mentioning Professor’s all-embracing university-related activities and the awards he was bestowed with at the University of Gdansk. In

3. The legacy of Professor Jan Stanisław Lipi´nski - Gda´nsk (1969 - 1998)

33

1977 he obtained the title of full professor. He has worked in numerous departmental and university committees. He was an active member of the Polish Mathematical Society and the Gda´nsk Scientific Society. Between 1979-1983 he was President of Gda´nsk Department of the Polish Mathematical Society. He worked actively on the agenda of Polish Academy of Sciences and the Ministry of National Education. He was several times winner of the Minister of National Education and the Rector of the University of Gda´nsk awards. He was awarded with the Knight’s Cross of the Polish Order of Polonia Restituta and the Medal of the National Education Commission. To celebrate the 80th birthday anniversary of Professor Lipi´nski at Faculty of Mathematics, Physics and Computer Science at the University of Gdansk there took place on 4 December 2003, a celebrative academic session. University of Gda´nsk, together with the Polish Mathematical Society published on this occasion a commemorative booklet devoted to the life and legacy of Professor Jan Stanisław Lipi´nski. The contribution of Professor Jan Stanisław Lipi´nski to the organization of mathematics as a discipline of study and to ensuring a high level of science cannot be overestimated. Thanks to Professor, the Institute of Mathematics at the University of Gdansk has an important and eminent Real Function and Probability Theory Department. Professor Jan Stanisław Lipi´nski belongs to the eliterian group of mathematicians from Gda´nsk, who marked his contribution to the development of this discipline and field of study. The mentioned contribution is now a historical event in itself. Acknowledgement: The authors would like to thank Mrs Halina Łyz˙ wi´nska for her kind assistance in obtaining materials. This article was written on the basis of academic materials which we obtained from Mrs Halina Łyz˙ wi´nska and also thanks to the long collaboration with Professor Jan Stanisław Lipi´nski of organizational nature (first author) and of scientific nature (second author). L ECH G ÓRNIEWICZ Institute of Mathematics, Kazimierz Wielki University Plac Weyssenhoffa 11, 85-072 Bydgoszcz, Poland E-mail: [email protected]

Z BIGNIEW G RANDE Institute of Mathematics, Kazimierz Wielki University Plac Weyssenhoffa 11, 85-072 Bydgoszcz, Poland E-mail: [email protected]

Chapter 4

´ The Scientific Family of Professor Jan Lipinski

MAŁGORZATA FILIPCZAK, MAŁGORZATA TEREPETA

The scientific family of Professor Jan Lipi´nski consists of 77 members: 8 scientific children (Professor’s PhD students), 29 scientific grandchildren (PhD students of Professor’s PhD students) and 40 scientific great-grandchildren (PhD students of Professor’s PhD students of Professor’s PhD students). Below we present the list of all members of this family.

´ Jan Stanisław Lipinski (Łód´z University 1958) 1. Franciszek Mirosław Filipczak (Łód´z University 1965) – Ryszard Pawlak (Łód´z University 1978) Andrzej Rychlewicz (Łód´z University 1992) Mohammed Imad Mutaib (Łód´z University 1993) ´ atek Boz˙ ena Swi ˛ (Łód´z University 1997) Dariusz Doliwa (Łód´z University 1999) Mariola Marciniak (Łód´z University 2000) Aneta Tomaszewska (Łód´z University 2002) Joanna Kucner (Łód´z University of Technology 2003) Ewa Korczak-Kubiak (Łód´z University 2009)

36

Małgorzata Filipczak, Małgorzata Terepeta

´ – Bo˙zena Szkopinska (Łód´z University 1978) – Czesław Krawczyk (Łód´z University 1978) ´ – Jan Omiecinski (Łód´z University 1982) – Zygmunt Wojtowicz (Łód´z University 1982) – Eliza Wajch (Łód´z University 1987) – Bogusław Kaczmarski (Łód´z University 1996)

2. Leonarda Filipczak (Łód´z University 1969) ´ 3. Władysław Wilczynski (Łód´z University 1970) – Jerzy Niewiarowski (Łód´z University 1976) – Genowefa Rzepecka (Łód´z University 1977) – Wiesława Poreda (Łód´z University 1977) – El˙zbieta Wagner-Bojakowska (Łód´z University 1978) Graz˙ yna Horbaczewska (Łód´z University 1995) Małgorzata Terepeta (Łód´z University 1997) Aleksandra Karasi´nska (Łód´z University 2007) – Ewa Łazarow (Łód´z University 1979) Katarzyna Flak (Łód´z University 2001) Agnieszka J˛edrzejewska-Vizvary (Łód´z University 2002) Agnieszka Kubi´s-Lipowska (Łód´z University 1997) Katarzyna Nowakowska (Łód´z University of Technology 2012) – Wacława Tempczyk (Łód´z University 1979) ´ – Stanisław Wronski (Łód´z University 1980) – Inga J˛edrzejewska (Łód´z University 1982) – Wojciech Wojdowski (Łód´z University of Technology 1983) – Marek Balcerzak (Łód´z University 1984) Dorota Rogowska (Łód´z University 1996) Joanna Peredko (Łód´z University 1999)

4. The Scientific Family of Professor Jan Lipi´nski

37

Elz˙ bieta Kotlicka (Łód´z University of Technology 2003) Joanna Rzepecka (Łód´z University of Technology 2004) Artur Wachowicz (Łód´z University of Technology 2004) Katarzyna Dems (Łód´z University of Technology 2005) Monika Potyrała (Łód´z University of Technology 2006) Monika Lindner (Łód´z University of Technology 2007) Marek Małolepszy (Łód´z University of Technology 2007) Szymon Głab ˛ (IMPAN 2007) – Jan Jastrz˛ebski (Łód´z University 1984) – Jacek Hejduk (Łód´z University 1985) Sebastian Lindner (Łód´z University 2003) Anna Loranty (Łód´z University 2005) Renata Wiertelak (Łód´z University 2010) Magdalena Górajska (Łód´z University 2012) – Mariusz Strze´sniewski (Łód´z University 1986) – Tomasz Filipczak (Łód´z University 1987) – Małgorzata Filipczak (Łód´z University 1990) – Agnieszka Niedziałkowska (Łód´z University of Technology 2008) – Rafał Zdu´nczyk (Łód´z University 2012)

4. Janusz Jaskuła (Łód´z University 1971) ´ 1973) 5. Mieczysław Gutowski (University of Gdansk ´ 1974) 6. Zbigniew Grande (University of Gdansk ´ ezak (UAM - Pozna´n 1982) – Włodzimierz Sl˛ ´ – Gra˙zyna Kwiecinska (University of Gda´nsk 1983) – Tomasz Natkaniec (Łód´z University 1985) Krzysztof Banaszewski (Łód´z University 1996) Dariusz Banaszewski (Łód´z University 1997)

38

Małgorzata Filipczak, Małgorzata Terepeta

Joanna Wesołowska (University of Gda´nsk 2001) Piotr Szuca (University of Gda´nsk 2004) Grzegorz Matusik (University of Gda´nsk 2012) – Aleksander Maliszewski (Łód´z University 1990) Jolanta Jałocha (Kosman) (Łód´z University of Technology 2003) Paulina Szczuka (Łód´z University of Technology 2006) Agnieszka Łukasiewicz (Łód´z University of Technology 2007) Marcin Kowalewski (Łód´z University of Technology 2010) ´ – Ewa Stronska (Łód´z University 1990) – Stanisław Kowalczyk (Łód´z University 1994) – Katarzyna Chmielewska (Łód´z University of Technology 2003)

´ 1974) 7. Stanisław Wojtan (University of Gdansk ´ 1983) 8. Grzegorz Krzykowski (University of Gdansk

M AŁGORZATA F ILIPCZAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

M AŁGORZATA T EREPETA Center of Mathematics and Physics, Łód´z University of Technology al. Politechniki 11, 90-924 Łód´z, Poland E-mail: [email protected]

Chapter 5

Existence theorems on convolution of functions, distributions and ultradistributions

´ ´ ANDRZEJ KAMINSKI, SVETLANA MINCHEVA-KAMINSKA

2010 Mathematics Subject Classification: Primary: 44A35, Secondary: 46F05, 46F10. Key words and phrases: convolution of functions, convolution of (tempered) distributions, convolution of Beurling (tempered) ultradistributions, compatible sets, polynomially compatible sets, M-compatible sets.

5.1 Introduction There are known sufficient conditions for existence of convolution in various spaces of functions and generalized functions. They are often given in the form of suitable assumptions concerning the growth of generalized functions. There exist also conditions of another type, formulated without any restriction on the growth, but expressed in terms of supports of generalized functions and called compatibility conditions (see e.g. [1], p. 124-127). We discuss this notion in some spaces of functions, distributions and ultradistributions and present theorems on existence of the convolution assuming compatibility of supports of the considered functions or generalized functions. As a matter of fact, we will present conditions of compatibility which are not only sufficient for existence of convolution in the considered spaces of generalized functions, but also necessary in some sense (namely, in the sense of S.Yu. Prishtshepionok who posed in 1977 certain problems concerning the convolution in D0 and in S 0 ); for suitable results we refer to [10], [11], [15].

40

Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

We will show certain new situations, in which the convolution of generalized functions exists, delivering interesting cases of compatibility of supports. In particular, we show that there is a variety of sets in Rd , that we call spiral, such that functions, distributions or ultradistributions having supports contained in such sets are convolvable in the corresponding spaces, in spite of the fact that the supports are unbounded in each direction of Rd .

5.2 Notation We use the standard multi-dimensional notation in Rd and Nd0 as well as the standard notation concerning the known spaces of real- or complex-valued functions on Rd : Lr (Rd ) for r ∈ [1, ∞] (with the norm denoted by k · kr ), 1 (Rd ), C(Rd ), C ∞ (Rd ), E(Rd ), D(Rd ), S(Rd ) as well as the spaces D 0 (Rd ) Lloc of distributions and S 0 (Rd ) of tempered distributions on Rd with the respective topologies (cf. [25], [1]). For a given set E ⊆ Rd and a function φ on Rd , we will use the following convenient notation: E @ Rd if E is a compact subset of Rd and E 4 := {(x, y) ∈ R2d : x + y ∈ E};

φ 4 (x, y) := φ (x + y),

x, y ∈ Rd .

In sections 5.6 and 5.7, we will consider the space D0(Mp ) (Rd ) of Beurling ultradistributions and the space S 0(Mp ) (Rd ) of Beurling tempered ultradistributions for a given sequence (Mp ) of positive numbers satisfying the following three conditions: (M.1)

Mp2 ≤ Mp−1 Mp+1

(M.2)

Mp ≤ A H p Mq Mp−q

(M.3)

∑∞p=q+1 Mp−1 Mp−1 ≤ A q Mq Mp−1

for p ∈ N; for p, q ∈ N, 0 ≤ q ≤ p; for q ∈ N,

where A > 0 and H > 0 are certain constants. It will be convenient to extend the sequence (Mp ) (for p ∈ N0 ) to its multidimensional version (Mk ) (for k ∈ Nd0 ) in the following way: Mk := Mκ1 +...+κd

for k = (κ1 , . . . , κd ) ∈ Nd0 .

By the associated function for the sequence (Mp ) we will mean the function M : [0, ∞) → [0, ∞) given by M(t) := sup log+ (t p /Mp ) for t > 0, where p∈N0

log+ := max{logt, 0} for t > 0 and M(0) := 0.

5. Existence theorems on convolution

41

5.3 Compatibile sets We formulate certain equivalent forms of the known condition, connected with the existence of the convolution of distributions, for given closed sets in Rd : Proposition 5.1 (see e.g. [6], p. 383). Let A, B ⊆ Rd be arbitrary closed sets. The following conditions are equivalent: (Σ ) (Σ 0 ) (Σ 00 )

(A × B) ∩ K 4 @ R2d for every K @ Rd ; A ∩ (K − B) @ Rd for every K @ Rd ; (K − A) ∩ B @ Rd for every K @ Rd .

The meaning of the conditions for d = 1 can be seen on Fig. 5.1 (for X = Y = R).

Fig. 5.1

In general, without the assumption that the sets A and B are closed, we have the following equivalence: Proposition 5.2 (see [11]). Let A, B ⊆ Rd be arbitrary sets. The following conditions are equivalent: (Σb )

(A × B) ∩ K ∆ is bounded in R2d for every bounded set K in Rd ;

(Σb0 )

A ∩ (K − B) is bounded in Rd for every bounded set K in Rd ;

(Σb00 )

(K − A) ∩ B is bounded in Rd for every bounded set K in Rd ;

if xn ∈ A and yn ∈ B for n ∈ N, then |xn | + |yn | → ∞ as n → ∞ implies |xn + yn | → ∞ as n → ∞. If A, B are closed, then each of the above conditions is equivalent to any of conditions (Σ )-(Σ 00 ).

(M)

42

Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

Definition 5.3 (see [17], [1], [11]). Sets A, B ⊆ Rd are called compatible if any of equivalent conditions (Σb )-(M) is satisfied. There are two well known particular cases of compatible sets A, B in R1 : 1◦ at least one of the sets A, B is bounded (see the lower part of Fig. 5.2); 2◦ both sets A, B are bounded from the same side: both from the left or both from the right (see the upper part of Fig. 5.2).

Fig. 5.2

Case 1◦ extends clearly to Rd for d > 1 and case 2◦ can be described in Rd in the following form: A, B ⊂ Rd are (or are contained in) suitable cones with vertices at 0 such that A is an open convex cone and B ⊂ A∗ , where A∗ means the cone dual to A (see [2], pp. 4-6; [1], pp. 129-130; [29], pp. 63-64). For d = 2, case 2◦ is illustrated on Fig. 5.3.

Fig. 5.3

5. Existence theorems on convolution

43

Fig. 5.4

That condition (Σb0 ) is satisfied in this case can be seen from Fig. 5.3 and 5.4 (to simplify presentation we show on Fig. 5.4 the set A ∩ (c − B) only for B := A, with A as on Fig. 5.3, and for a specific c ∈ R2 , but a general case is easily seen). There exist, however, another case of compatible sets in R1 , not so well known as 1◦ and 2◦ : 3◦ both sets A, B in R1 are unbounded from both sides: unbounded both from the left and from the right.

Fig. 5.5

The set presented in each of the three parts of Fig. 5.5, let us denote it by the common symbol A, is a union of countably many intervals of length 1 situated in three different ways on R1 . The set A in the two lower parts of Fig. 5.5 is compatible with itself and the set A in the upper part of Fig. 5.5 is not compatible with itself (for details see [7], [8] and [15]).

44

Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

Fig. 5.6

Fig. 5.7

Case 3◦ can be extended to Rd in various manners. An interesting extension are sets which can be described as infinite spirals or helices with a suitable way of developing their coils in Rd . Notice that compatible sets which are unbounded in each direction of Rd can be obtained in this way (see [15]). On Fig. 5.6 and 5.8 particular examples of such a spiral set A in R2 which is compatible with itself are shown. On Fig. 5.7 and 5.9 (analogously to Fig. 5.4) we show only the set A ∩ (c − A) for specific vectors c ∈ R2 .

5. Existence theorems on convolution

45

Fig. 5.8

Fig. 5.9

Compatibility of supports of two tempered distributions, elements of the subspace S 0 (Rd ) of the space D0 (Rd ), does not guarantee that their convolution in D0 is again a tempered distribution (see section 5.5). Similarly, compatibility of supports of two tempered ultradistributions, elements of the subspace S 0(Mp ) (Rd ) of the space D0(Mp ) (Rd ), does not guarantee that their convolution in D0(Mp ) is again a tempered ultradistribution (see sections 5.6 and 5.7). This is a consequence of Theorem 5.9 formulated in the next section. Therefore the notion of compatibility requires suitable modifications in the spaces S 0 (Rd ) of tempered distributions and S 0(Mp ) (Rd ) of tempered ultradistributions. In [7], [8], the following modifications of compatibility, corresponding to the mentioned spaces were introduced.

46

Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

Definition 5.4 (see [7], [8], [10]). Two sets A, B ⊆ Rd are polynomially compatible if there is a positive polynomial p on [0, ∞) such that the following implication holds: x ∈ A, y ∈ B ⇒ |x| + |y| ≤ p (|x + y|) or, equivalently, if there are C > 0 and k ∈ N0 such that x ∈ A, y ∈ B ⇒ |x| + |y| ≤ C (1 + |x + y|)k . In [14], a certain modification of compatibility condition, corresponding to the space S 0(Mp ) (Rd ), was given via the associated function M for the sequence (Mp ). We present it here in a slightly relaxed form: Definition 5.5 (cf. [14]). Two sets A, B ⊆ Rd are M-compatible, if there is a constant a > 0 such that M(|x|) + M(|y|) ≤ M(a|x + y|) + a for all x ∈ A and y ∈ B.

1 5.4 Existence of convolution in Lloc

Definition 5.6. Let F and G be Lebesgue measurable functions on Rd . For a given x ∈ Rd we define (F ∗ G)(x) :=

Z Rd

F(x − t)G(t) dt,

(5.1)

saying that (F ∗ G)(x) exists, whenever the function under the integral sign in (5.1) is Lebesgue integrable as a function of t for the fixed x. If (F ∗G)(x) exists for all (almost all) x ∈ Rd , we say that the convolution F ∗ G exists everywhere (almost everywhere) on Rd . 1 (Rd ). We say that Definition 5.7 (see [15]). Let F and G be functions in Lloc 1 the convolution F ∗ G exists in Lloc if F ∗ G exists almost everywhere on Rd 1 (Rd ). and |F| ∗ |G| ∈ Lloc

Let us recall the known particular case (Young’s theorem) of the existence 1 of functions from suitable subspaces L p (Rd ) and of the convolution in Lloc q d 1 L (R ) of the space Lloc (Rd ):

5. Existence theorems on convolution

47

If F ∈ L p (Rd ) and G ∈ Lq (Rd ) for p, q ∈ [1, ∞) such that 1/p + 1/q ≥ 1, then the convolution F ∗ G exists almost everywhere in Rd and F ∗ G ∈ Lr (Rd ), where r = (1/p + 1/q − 1)−1 . In particular, if F, G ∈ L1 (Rd ), then the convolution F ∗ G exists almost everywhere in Rd and F ∗ G ∈ L1 (Rd ). 1 of two locally integrable functions may not exist, The convolution in Lloc but it exists if the supports of the functions are compatible (see section 5.3): 1 (Rd ) and the supports of the Theorem 5.8 (see e.g. [1], p. 124). If F, G ∈ Lloc functions F and G are (contained in) compatible sets, then the convolution 1 . F ∗ G exists in Lloc 1 (similarly, However the existence of the convolution of functions in Lloc the existence of the convolution of distributions or of ultradistributions in the respective spaces, see sections 5.5 and 5.6) does not guarantee any restriction of growth of the convolution. For instance, the convolution of two measurable 1 but their convolution may be a slowly increasing functions may exist in Lloc function of arbitrarily fast increase. As a matter of fact, the following much stronger result was proved in [7] (see also [8] and [15]):

Theorem 5.9 (see [7], [8], [15]). Let F ∈ C(Rd ) be an arbitrary continuous nonnegative function (of an arbitrary increase). Then there exists a nonnegative smooth function φ ∈ C ∞ (Rd ) such that its support is compatible with itself and the convolution ϕ ∗ ϕ satisfies the inequality (φ ∗ φ )(x) > F(x) for each x ∈ Rd . Moreover, lim|x|→∞ φ (x) = 0. Due to its strong formulation the above theorem can serve as a universal counter-example in considerations concerning the existence and the growth of the convolution of functions, distributions and ultradistributions.

5.5 Existence of convolution in D0 and in S 0 There are several general definitions of the convolution of distributions in D0 given consecutively by C. Chevalley [3], L. Schwartz [26], R. Shiraishi [27], V. S. Vladimirov [28], [29], P. Dierolf - J. Voigt [4], A. Kami´nski [9] and S. Mincheva-Kami´nska [19], [20] (see also [32], [5], [30], [31], [21], [18] and [15]). These general definitions allow one to define the convolution f ∗ g in D0 for arbitrary distributions f , g ∈ D0 (Rd ) and to determine, for each pair ( f , g) of distributions, whether the convolution f ∗g exists in D0 (then f ∗g ∈ D0 (Rd )) or not. Most of the mentioned definitions are equivalent (for details see e.g.

48

Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

[9]). We will recall only one of them, the sequential definition of Vladimirov [28], [29], based on the notion of strong approximate unit. Definition 5.10 (see [28], [29], [4], [9]). A sequence (ηn ) of elements of D(Rd ) is said to be a strong approximate unit on Rd if for every K @ Rd there exists an n0 ∈ N such that ηn (x) = 1 for x ∈ K and n ≥ n0 (hence ηn → 1 in E(Rd )) and, in addition, (k)

sup k ηn k∞ < ∞ for every k ∈ Nd0 . n∈N

We denote the set of all strong approximate units on Rd by U(Rd ). Definition 5.11 (see [28], [29], [4], [9]). For given f , g ∈ D0 (Rd ) the convolution f ∗ g in D0 is defined by h f ∗ g, ϕi := lim h f ⊗ g, ηn ϕ 4 i, n→∞

ϕ ∈ D(Rd ),

whenever the above limit exists for every strong approximate unit (ηn ) ∈ U(R2d ) and ϕ ∈ D(Rd ). We say then that the convolution f ∗ g exists in D0 . Recall that the space S 0 (Rd ) of tempered distributions is a subspace of the space D0 (Rd ) of distributions. Analogously to and independently of the above Definition 5.11, one may define the convolution of tempered distributions in S 0 in various ways (see e.g. [27], [4], [9] and other references given earlier). We present below only one of several equivalent definitions of the convolution in S 0 , namely the respective counterpart of the above sequential definition of Vladimirov (cf. [28], [29]): Definition 5.12 (see [4], [9]). For given f , g ∈ S 0 (Rd ) we define the convolution f ∗ g in S 0 by h f ∗ g, ψi := lim h f ⊗ g, ηn ψ 4 i, n→∞

ψ ∈ S(Rd ),

whenever the above limit exists for every strong approximate unit (ηn ) ∈ U(R2d ) and ψ ∈ S(Rd ). We say then that the convolution f ∗ g exists in S 0 . For existence of the convolution in D0 of two distributions f , g ∈ D0 (Rd ) the condition, introduced in section 5.4, of compatibility of their supports is sufficient. Namely Theorem 5.13 (see [1]; cf. [6], [28], [29]). Let f , g ∈ D0 (Rd ) be distributions. If the supports of f and g are (contained in) compatible sets in Rd , then f ∗ g exists in D0 and f ∗ g ∈ D0 (Rd ).

5. Existence theorems on convolution

49

In [27], R. Shiraishi posed the problem whether the assumption that the convolution f ∗ g of two tempered distributions f , g ∈ S 0 (Rd ) (⊂ D0 (Rd )) exists in D0 implies that the convolution f ∗ g exists in S 0 , in particular, whether the existence in D0 of the convolution f ∗ g of tempered distributions f , g ∈ S 0 (Rd ) implies that the convolution f ∗ g not only belongs to D0 (Rd ), but is even a member of S 0 (Rd ). The negative answer to the problem of Shiraishi follows directly from Theorem 5.9 recalled in section 5.4, proved in [7], [8] (see also [15]), and from the paper [4]. In [4], an example of two tempered measures f , g is given, concentrated on a countable set in R1 , such that the convolution f ∗ g exists in D0 , but f ∗ g ∈ / S 0 (R1 ). Theorem 5.9 is much stronger than the result from [4] and stands for a counter-example concerning the convolution in various spaces of functions and generalized functions. In particular, it follows from Theorem 5.9 that the counterpart of Theorem 5.13 for tempered distributions is not true under the assumption of compatibility of their supports. However, if one replaces this assumption by polynomial compatibility of supports of given tempered distributions, the result concerning their convolution in S 0 is analogous: Theorem 5.14 (see [7], [8], [10]). Let f , g ∈ S 0 (Rd ) be tempered distributions. If the supports of f and g are (contained in) polynomially compatible sets in Rd , then the convolution f ∗ g exists in S 0 and f ∗ g ∈ S 0 (Rd ).

5.6 Definition of spaces D0(Mp ) and S 0(Mp ) We recall the definition of Beurling spaces of ultradifferentiable functions for a fixed numerical sequence (Mp ) satisfying conditions (M.1)-(M.3), formulated in section 5.2, which will be assumed to the end of this and the next section. We start with defining, for given h > 0 and regular compact subset K of (M ) (M ) (M ) d R (see [16]), the spaces EK,hp (Rd ) and DK,hp (Rd ). The space EK,hp (Rd ) is defined to consist of all functions ϕ from E (Rd ) such that | ϕ (k) (x) | 0 and s ∈ [1, ∞], as the space of all functions ϕ from C∞ (Rd ) satisfying the inequality: ) ( kϕ (k) ks d : k ∈ N0 < ∞, kϕks,h := sup hk Mk and then we define (M p )

DL s

(M )

(Rd ) := proj lim DLs ,hp (Rd ). h→0

(M )

B˙ (Mp ) (Rd ),

By we denote the completion of D(Mp ) (Rd ) in DL∞ p (Rd ). For more details concerning all the above spaces we refer to [16], [23], [24], [12], [13] and [2]. (M ),m In addition, for a fixed m > 0, we denote by S2 p (Rd ) the space of all smooth functions ϕ such that  σm,2 (ϕ) := 

∑ α,β ∈Nd0

m2(α+β ) Mα2 Mβ2

1/2 2 β (α) hxi ϕ (x) dx < ∞,

Z Rd

where hxi := (1 + |x|2 )1/2 for x ∈ Rd , equipped with the topology induced by the above norm σm,2 , Then we define (M p ),m

S (Mp ) (Rd ) := proj lim S2

(Rd ).

m→∞

For more details concerning the above spaces we refer to [22], [14] and [2]). (M )

Remark 5.15. Notice that the basic spaces E (Mp ) (Rd ), D(Mp ) (Rd ), DLs p (Rd ) for s ∈ [1, ∞], B˙ (Mp ) (Rd ) and S (Mp ) (Rd ) contain sufficiently many functions (in case of the space D(Mp ) (Rd ), this is a consequence of the Denjoy-Carleman theorem). In particular, there exists a function η ∈ D(Mp ) (Rd ) such that η = 1 in some neighbourhood of 0.

5. Existence theorems on convolution

51

The strong dual of D(Mp ) (Rd ), denoted by D0(Mp ) (Rd ), is called the space of Beurling ultradistributions. (M ) Notice that D(Mp ) (Rd ) is dense in DLs p (Rd ) for s ∈ [1, ∞) as well as in B˙ (Mp ) (Rd ); moreover, the respective inclusion mappings are continuous. (M ) Hence the strong duals of DLs p (Rd ) (s ∈ [1, ∞)) and B˙ (Mp ) (Rd ) are subspaces of the space D0(Mp ) (Rd ) of all Beurling ultradistributions. We denote them tra(M ) (M ) ditionally by D0 Lt p (Rd ), where t := s/(s − 1) ∈ (1, ∞], and D0 L1 p (Rd ), respectively (see [16], [2]). The space S 0(Mp ) (Rd ) of all Beurling tempered ultradistributions is meant as the strong dual of the space S (Mp ) (Rd ) defined above (see [22], [14], [2]). Since D(Mp ) (Rd ) is dense in S (Mp ) (Rd ) and the inclusion mapping is continuous, so S 0(Mp ) (Rd ) can be embedded into the space D0(Mp ) (Rd ). For other properties of the space S 0(Mp ) we refer to [22], [14] and [2]. For given ultradistributions f , g ∈ D0(Mp ) (Rd ) by their tensor product f ⊗ g we mean an ultradistribution in D0(Mp ) (R2d ) defined in a standard way.

5.7 Existence of convolution in D0(Mp ) and in S 0(Mp ) There are various general definitions of the convolutions in D0(Mp ) of Beurling ultradistributions (see [12]) and in S 0(Mp ) of Beurling tempered ultradistributions (see [14]). They are counterparts of the known general definitions of the convolutions in D0 and in S 0 (see section 5). That the mentioned definitions of the convolution in D0(Mp ) of Beurling ultradistributions are equivalent and that the corresponding definitions of the convolution in S 0(Mp ) of Beurling tempered ultradistributions are equivalent was proved in [12] and [14], respectively (see also [2]). We will recall here only these definitions of the convolution in D0(Mp ) and in S 0(Mp ) which correspond to Vladimirov’s definition of the convolution in D0 and in S 0 , respectively. The definitions are based on the notions of strong D(Mp ) -approximate unit and strong S (Mp ) -approximate unit. Definition 5.16 (see [12], [13]). A sequence (ηn ) of elements of D(Mp ) (Rd ) is said to be a strong D(Mp ) -approximate unit on Rd if for every K @ Rd there exists an n0 ∈ N such that ηn (x) = 1 for x ∈ K and n ≥ n0 (hence ηn → 1 in E (Mp ) (Rd )) and, in addition, if there exists a positive constant h such that   k h (k) k ηn k∞ < ∞. sup sup n∈N k∈Nd Mk 0

52

Andrzej Kami´nski, Svetlana Mincheva-Kami´nska

We denote the set of all strong D(Mp ) -approximate units on Rd by U

(M p )

(Rd ).

Definition 5.17 (see [13], [14]). If in the above definition the assumption ηn ∈ D(Mp ) (Rd ) for n ∈ N is replaced by ηn ∈ S (Mp ) (Rd ) for n ∈ N and the remaining assumptions are preserved, then the sequence (ηn ) is called a strong S (Mp ) approximate unit. We denote the set of all strong S (Mp ) -approximate units on (M ) Rd by Us p (Rd ). Vladimirov’s version of the definition of the convolution in D0(Mp ) of Berling ultradistributions has the following form: Definition 5.18 (see [12], [13]). For given Beurling ultradistributions f , g ∈ D0(Mp ) (Rd ) the convolution f ∗ g in D0(Mp ) is defined by h f ∗ g, ϕi := lim h f ⊗ g, ηn ϕ 4 i, n→∞

ϕ ∈ D(Mp ) (Rd ),

whenever the above limit exists for every strong approximate unit (ηn ) ∈ (M ) U p (R2d ) and ϕ ∈ D(Mp ) (Rd ). We say then that the convolution f ∗ g exists in D0 (Mp ). Analogously, the convolution in S 0(Mp ) of Beurling tempered ultradistributions can be defined as follows: Definition 5.19 (see [13], [14]). For given two Beurling tempered ultradistributions f , g ∈ S 0(Mp ) (Rd ) we define the convolution f ∗ g in S 0(Mp ) by h f ∗ g, ψi := lim h f ⊗ g, ηn ψ 4 i, n→∞

ψ ∈ S (Mp ) (Rd ),

whenever the above limit exists for every strong approximate unit (ηn ) ∈ (M ) Us p (R2d ) and ψ ∈ S (Mp ) (Rd ). We say then that the convolution f ∗ g exists in S 0 (Mp ). The following analogue of Theorem 5.13 is true for the convolution in D0(Mp ) : Theorem 5.20 (see [13]). Let f , g ∈ D0(Mp ) (Rd ) be Beurling ultradistributions. If the supports of f and g are (contained in) compatible sets in Rd , then the convolution f ∗ g exists in D0(Mp ) and f ∗ g ∈ D0(Mp ) (Rd ). There is also a counterpart of Theorem 5.14 for the convolution in S 0(Mp ) of Beurling tempered ultradistributions. Namely, we have

5. Existence theorems on convolution

53

Theorem 5.21 (see [13]). Let f , g ∈ S 0(Mp ) (Rd ) be Beurling tempered ultradistributions. If the supports of f and g are (contained in) M-compatible sets in Rd , then the convolution f ∗ g exists in S 0(Mp ) and f ∗ g ∈ S 0(Mp ) (Rd ). Acknowledgements. We would like to express our gratitude to Mr. Klaudiusz Majchrowski for preparing the figures illustrating certain topics discussed in the article.

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[18] S. Mincheva-Kami´nska, Sequential approach to integrable distributions, Novi Sad J. Math. 41 (2011), 123–131. [19] S. Mincheva-Kami´nska, Equivalence of sequential definitions of the convolution of distributions, Rend. Sem. Mat. Univ. Politec. Torino 69 (2011), 367–376. [20] S. Mincheva-Kami´nska, Equivalent conditions for integrability and convolvability of distributions and tempered distributions, submitted. [21] N. Ortner, On convolvability conditions for distributions, Monatsh. Math. 160 (2010), 313–335. [22] S. Pilipovi´c, Tempered ultradistributions, Boll. Un. Mat. Ital. (7) 2-B (1988), 235–251. [23] S. Pilipovi´c, On the convolution in the space of Beurling ultradistributions, Comm. Math. Univ. St. Paul. 40 (1991), 15–27. [24] S. Pilipovi´c, Characterizations of bounded sets in the space of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191–1206. [25] L. Schwartz, Théorie des distributions, Vol. 1-2, Hermann, Paris, 1950-51; nouvelle édition 1966. [26] L. Schwartz, Définitions integrale de la convolution de deux distributions, in: Séminaire Schwartz, Année 1953-54. Expose n◦ 22, Secr. math. Fac. Sci., Paris, 1954. [27] R. Shiraishi, On the definition of convolution for distributions, J. Sci. Hiroshima Univ. Ser. A 23 (1959), 19–32. [28] V. S. Vladimirov, Equations of Mathematical Physics, Nauka, Moscow 1967 (in Russian); English edition: Marcel Dekker, New York, 1971. [29] V. S. Vladimirov, Methods of the Theory of Generalized Functions, Taylor & Francis, London-New York, 2002. [30] P. Wagner, Zur Faltung von Distributionen, Math. Ann. 276 (1987), 467–485. [31] R. Wawak, Improper integrals of distributions, Studia Math. 86 (1987), 205–220. [32] K. Yoshinaga, H. Ogata, On the convolutions, J. Sci. Hiroshima Univ. Ser. A 22 (1958), 15–24.

´ A NDRZEJ K AMI NSKI Institute of Mathematics, University of Rzeszów ul. Rejtana 16A, 35-959 Rzeszów, Poland E-mail: [email protected]

´ S VETLANA M INCHEVA -K AMI NSKA Institute of Mathematics, University of Rzeszów ul. Rejtana 16A, 35-959 Rzeszów, Poland E-mail: [email protected]

Chapter 6

Convergence of sequences of measurable functions

˙ ´ ELZBIETA WAGNER-BOJAKOWSKA, WŁADYSŁAW WILCZYNSKI

2010 Mathematics Subject Classification: 28A20, 28A05, 54C50. Key words and phrases: σ -algebra, σ -ideal, convergence I-a.e., convergence with respect to the σ -ideal I.

6.1 Introduction Let S be a σ -algebra of subsets of R and I – a proper σ -ideal included in S. We shall say that some property holds I-a.e. (I-almost everywhere) if and only if the set of points for which this property does not hold belongs to I. In this chapter we shall consider extended real functions defined I-a.e. and finite I-a.e. on [0, 1]. The σ -algebra will be usually the family L of Lebesgue measurable sets or B – the family of sets having the Baire property. The σ ideal associated with L will be the σ -ideal N of null sets in R and σ -ideal associated with B will be the σ -ideal M of first category sets (meager sets). The σ -algebra Bor of Borel sets will not be used since the Lebesgue measure restricted to Bor is not complete. We shall say that two I-a.e. finite extended real-valued functions f , g defined I-a.e. on [0, 1] are equivalent if and only if f (x) = g(x) I-a.e. on [0, 1], i.e. {x ∈ [0, 1] : f (x) 6= g(x)} ∈ I.

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In this case we shall write f ∼I g. Let FI be the family of all extended real-valued functions defined I-a.e. on [0, 1], finite I-a.e. and measurable with respect to S. Denote by FI / ∼I the quotient space. If [ f ] ∈ FI / ∼I , then there exists an S-measurable real function g defined and finite everywhere on [0, 1] such that g ∼I f . Indeed, it is sufficient to put g(x) = 0 if | f (x)| = +∞ or f (x) is not defined and g(x) = f (x) at remaining points. Therefore in the sequel we shall usually assume that all functions under considerations are finite-valued and defined everywhere on [0, 1]. We shall consider different types of convergence of sequences of elements of FI / ∼I using the symbols of functions rather than of equivalence classes. Also the limit of the sequence will be written as a function (although it will be always determined up to the equivalence). So we shall write limn→∞ fn = f I-a.e. rather than limn→∞ [ fn ] = [ f ].

6.2 Convergence almost everywhere Suppose now that S = L and I = N , so we shall deal with Lebesgue measurable functions. Usually we shall write a.e. instead of N -a.e. If { fn }n∈N is a sequence of real-valued Lebesgue measurable functions defined on [0, 1] convergent a.e. to a function f (not necessarily a.e. finite), then it is well known that f is also Lebesgue measurable. Recall the theorem of D. Egorov (see, for example [2], p. 184 or [8], p. 143). Theorem 6.1. If { fn }n∈N is a sequence of real-valued Lebesgue measurable functions convergent a.e. to a.e. finite function f , then for each ε > 0 there exists a set Eε ∈ L, Eε ⊂ [0, 1] such that λ (Eε ) < ε and the sequence { fn }n∈N converges uniformly to f on [0, 1] \ Eε . Remark 6.2. Obviously we can take Eε as an open set. However, if fn (x) = xn for x ∈ [0, 1], then it is not possible to find a set E0 such that λ (E0 ) = 0 and the convergence is uniform on [0, 1] \ E0 . So the theorem of Egorov cannot be improved in this direction. It can be reformulated in the following way: Theorem 6.3. If { fn }n∈N is a sequence of real-valued Lebesgue measurable functions convergent a.e. to a.e. finite function f , then there exists a sequence S {Am }m∈N of sets from L such that λ ([0, 1] \ ∞ m=1 Am ) = 0 and the convergence is uniform on each Am .

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It is evident that Theorem 6.1 implies Theorem 6.3. Suppose now that the conclusion of Theorem 6.3 holds and take ε > 0. There exists m0 ∈ N such Sm0 S 0 that λ ([0, 1] \ m m=1 Am ) < ε and the convergence on m=1 Am is uniform, so Theorem 6.3 implies Theorem 6.1. Remark 6.4. Observe that the theorem is not true without the assumption of measurability. Indeed, let {Em }m∈N be a sequence of pairwise disjoint subsets S of [0, 1] such that λ ∗ (Em ) = 1 for each m ∈ N and ∞ m=1 Em = [0, 1]. Put f n = S∞ χ i=n+1 Ei . It is easy to see that fn (x) → 0 for each x ∈ [0, 1]. If A ⊂ [0, 1] is n→∞

a measurable set such that { fn }n∈N converges uniformly on A to zero, then S Sm A⊂ m i=1 Ei for some m ∈ N and λ (A) = 0, since λ∗ ( i=1 Ei ) = 0 (where λ∗ is an inner Lebesgue measure).

6.3 Convergence I-a.e. We shall generalize the notion of convergence in the following way: Suppose that (X, S) is a measurable space, i.e. X is a non-empty set and S is a σ -algebra of subsets of X. Suppose also that I ⊂ S is a proper σ -ideal of sets. Definition 6.5 (see [10] or [12]). We shall say that a pair (S, I) fulfills the condition (E) if for every set D ∈ S \ I and for each double sequence {B j,n } j,n∈N of subsets belonging to S and satisfying the conditions: B j,n ⊂ B j,n+1 for each S j, n ∈ N, ∞ n=1 B j,n = D for each j ∈ N there exists an increasing sequence { j p } p∈N of positive integers and a sequence {n p } p∈N of positive integers such T that ∞p=1 B j p ,n p ∈ / I. Definition 6.6. We shall say that a pair (S, I) fulfills the countable chain condition (ccc) if every pairwise disjoint family of sets from S \ I is at most denumerable. Definition 6.7. We shall say that a sequence { fn }n∈N of real-valued S-measurable functions defined on X converges to a real-valued function f defined on X in the sense of Egorov if there exists a sequence {Am }m∈N of sets from S S such that X \ ∞ m=1 Am ∈ I and { f n }n∈N converges to f uniformly on each Am , m ∈ N. Theorem 6.8 (see [12]). Suppose that the pair (S, I) fulfills ccc. Then the convergence I-a.e. of a sequence { fn }n∈N of a real-valued S-measurable functions defined on X to a real-valued function f defined on X implies the convergence of { fn }n∈N to f in the sense of Egorov if and only if the pair (S, I) fulfills the condition (E).

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Elz˙ bieta Wagner-Bojakowska, Władysław Wilczy´nski

Obviously the following theorem is true. Theorem 6.9. If a sequence { fn }n∈N of real-valued functions defined on X converges to a real-valued function f defined on X in the sense of Egorov, then { fn }n∈N converges I-a.e. to f . Remark 6.10. Observe that in the above theorem the assumption of S-measurability of fn , n ∈ N, is not necessary. Now we shall present related kinds of convergence of sequences of functions. Definition 6.11 (compare [8], p. 141 or [7]). We shall say that a sequence { fn }n∈N of real-valued S–measurable functions defined on X converges to a real-valued function f defined on X in the sense of Taylor if there exists a non-decreasing sequence {tn }n∈N tending to +∞ such that the sequence {tn · ( fn − f )}n∈N converges I-a.e. to zero. Definition 6.12 (compare [8], p. 141). We shall say that a sequence { fn }n∈N of real-valued S–measurable functions defined on X converges to a real-valued function f defined on X with the convergence regulator if there exists a nonnegative extended real-valued function g defined on X and a sequence of positive numbers {αn }n∈N convergent to zero such that | fn (x) − f (x)| ≤ αn · g(x) for each n ∈ N and x ∈ X. Definition 6.13 (compare [12]). We shall say that a sequence { fn }n∈N of realvalued S–measurable functions defined on X converges to a real-valued function f defined on X in the sense of Yoneda if there exists a non-negative extended real-valued S-measurable function d defined on X such that for each ε > 0 there exists a positive integer n(ε) such that | fn (x) − f (x)| < ε · d(x) for each n > n(ε) and x ∈ X. Theorem 6.14. Let { fn }n∈N be a sequence of real-valued S-measurable functions defined on X and let f be a real-valued function defined on X. The following statements are equivalent: (a) { fn }n∈N converges to f in the sense of Egorov, (b) { fn }n∈N converges to f in the sense of Taylor, (c) { fn }n∈N converges to f with the convergence regulator, (d) { fn }n∈N converges to f in the sense of Yoneda. Proof. (a) =⇒ (b). Let {Am }m∈N be a sequence of sets from S such that S X\ ∞ m=1 Am ∈ I and for each m ∈ N the convergence is uniform on Am . For

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each m ∈ N choose an increasing sequence {nm,r }r∈N of positive integers such 1 for each x ∈ Am and each n ≥ nm,r . Let {nm }m∈N be an that | fn (x) − f (x)| < r+1 increasing sequence of positive integers such that limr→∞ nnm,rr = +∞ for each m ∈ N (for example nr = r − max(n1r , n2r , . . . , nrr )). Put ( 1 for 1 ≤ n < n1 , tn = √ r for nr−1 ≤ n < nr , (r = 2, 3, . . . ), If x ∈ ∞ m=1 Am , then x ∈ Am for some m ∈ N. Then there exists r0 such that nr ≥ nm,r for r ≥ r0 . If n ≥ nr0 , then there exists r ≥ r0 such that nr ≤ n < nr+1 . 1 = t12 , so tn · | fn (x) − f (x)| < t1n . Hence for such n we have | fn (x) − f (x)| < r+1 Sn Finally {tn · ( fn − f )}n∈N converges to zero on ∞ m=1 Am , which means I-a.e. on X. (b) =⇒ (c) Put g(x) = supn tn | fn (x) − f (x)| for x ∈ X and αn = t1n for n ∈ N. Obviously g is S-measurable. By virtue of (b) g is I-a.e. finite and the inequality | fn (x) − f (x)| ≤ t1n · g(x) is obvious. (c) =⇒ (d) Put d(x) = g(x). Take ε > 0. There exists n(ε) such that αn < ε for n > n(ε). Then, obviously, | fn (x) − f (x)| < ε · d(x) for n > n(ε) and x ∈ X. S (d) =⇒ (a) Put Am = {x ∈ X : d(x) ≤ m}. Then we have X \ ∞ m=1 Am ∈ I and { fn }n∈N converges to f uniformly on each Am . t u S

Theorem 6.15. The pair (L, N ) fulfills the condition (E). Proof. Take D ∈ L \ N and such that λ (D) < ∞. Let {B j,n } j,n∈N be a double sequence fulfilling both conditions. Put j p = p for each p ∈ N and choose n p T such that λ (D \ B p,n p ) < 31p λ (D). Then λ ( ∞p=1 B p,n p ) > 21 λ (D) > 0. t u Remark 6.16. If we choose n p such that λ (D \ B p,n p ) < 2εp for p ∈ N, where ε T is fixed, then we obtain λ ( ∞p=1 B p,n p ) > λ (D) − ε. This choice is used when proving Egorov’s theorem. Theorem 6.17. The pair (B, M) does not fulfill the condition (E). Proof. Let Q be the set of all rational numbers in [0, 1]. Put D = [0, 1] \ Q and S j−1 i 1 i 1 let B j,n = [0, 1] \ (Q ∪ A j,n ), where A j,n = [0, 1] ∩ i=1 ( j − n , j + n ), for n ∈ N, j ∈ N \ {1}. If { j p } p∈N is an arbitrary increasing sequence of positive integers T and {n p } p∈N is an arbitrary sequence of positive integers, then ∞p=1 B j p ,n p = S S [0, 1] \ (Q ∪ ∞p=1 A j p ,n p ) ∈ M, because ∞p=1 A j p ,n p is an open set dense in [0, 1]. t u

Elz˙ bieta Wagner-Bojakowska, Władysław Wilczy´nski

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6.4 Convergence in measure Definition 6.18. We shall say that a sequence { fn }n∈N of real-valued measurable functions defined on [0, 1] converges in measure to a real-valued measurable function f defined on [0, 1] if λ ({x ∈ [0, 1] : | fn (x) − f (x)| > ε}) → 0 n→∞

for each ε > 0.

Recall well known facts: Theorem 6.19. If a sequence { fn }n∈N of real-valued measurable functions defined on [0, 1] converges a.e. to a real-valued function f defined on [0, 1], then { fn }n∈N converges to f in measure. Remark 6.20. It is essential that λ ([0, 1]) < ∞. In fact, the theorem holds in all finite measure spaces. If we take a sequence { fn }n∈N of real-valued functions defined on [0, ∞) by the formula fn = χ[n,+∞) , n ∈ N, then fn (x) → 0 for n→∞

each x ∈ [0, ∞) and simultaneously { fn }n∈N does not converge to zero in measure. The assumption of measurability is also essential. Indeed, the sequence of functions from Remark 6.4 converges to zero everywhere without converging in measure. Theorem 6.21 (F. Riesz, compare [4], Theorem 11.26). If a sequence { fn }n∈N of real-valued measurable functions defined on [0, 1] converges in measure to a real-valued measurable function f defined on [0, 1], then there exists an increasing sequence {nm }m∈N of positive integers such that { fnm }m∈N converges a.e. to f . The above theorem has a very nice version which will be useful for us: Theorem 6.22. A sequence { fn }n∈N of real-valued measurable functions defined on [0, 1] converges in measure to a real-valued measurable function f defined on [0, 1] if and only if for each subsequence { fnm }m∈N of { fn }n∈N there exists a subsequence { fnm p } p∈N which converges a.e. to f . Proof. Necessity follows immediately from the theorem of Riesz and from the obvious fact that convergence in measure is preserved by subsequences. To prove the sufficiency suppose that { fn }n∈N does not converge to f in measure. Then there exist a pair of positive numbers ε and δ and an increasing sequence {nm }m∈N of positive integers such that λ ({x ∈ [0, 1] : | fnm (x) − f (x)| > ε}) > δ for each m ∈ N. If { fnm p } p∈N is a subsequence of { fnm }m∈N convergent a.e. to f , then by virtue of Theorem 6.19 { fnm p } p∈N converges to f

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in measure. Hence lim p→∞ λ ({x ∈ [0, 1] : | fnm p (x) − f (x)| > ε}) = 0, a contradiction.  In the last theorem there is proved that the convergence in measure can be defined without measure, only using the notion of set of measure zero. It leads us to the notion of convergence with respect to the σ -ideal I, which is obtained by changing the convergence a.e. (except on a set of measure zero) with the convergence I-a.e., i.e. except on a set belonging to I. t u

6.5 Convergence with respect to the σ -ideal Suppose again that (X, S) is a measurable space and I ⊂ S is a σ -ideal. Definition 6.23 (see [10]). We shall say that a sequence { fn }n∈N of real-valued S-measurable functions defined on X converges to a real-valued S-measurable function f defined on X with respect to the σ -ideal I if for each subsequence { fnm }m∈N of { fn }n∈N there exists a subsequence { fnm p } p∈N which converges I

I-a.e. to f . We shall use the denotation fn → f . n→∞ I

Remark 6.24. If fn → f I-a.e., then fn → f . Obviously, the limit function n→∞ n→∞ with respect to I is determined up to equivalent functions. It is not difficult to see that the following conditions are fulfilled (we shall formulate all conditions in terms of functions rather than of elements of the quotient space): I

(L1) If fn = f for each n ∈ N, then fn → f ; n→∞

I

I

(L2) If fn → f , then fnm → f for each increasing sequence {nm }m∈N of posn→∞ n→∞ itive integers. (L3) If the sequence { fn }n∈N does not converge to f with respect to I, then there exists a subsequence { fnm }m∈N , no subsequence of which converges to f with respect to I. So the family of real-valued S-measurable functions defined on X (or, more precisely, the quotient spaceFI / ∼I ) equipped with the convergence with respect to I is an L∗ space (see [3], p. 90). In such a space one can define the closure operation assuming that f belongs to cl(A) if and only if there exists I

a sequence { fn }n∈N of functions from A such that fn → f (or, equivalently, n→∞

there exists a sequence { fn }n∈N of functions from A such that fn → f I−a.e.). n→∞

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This operation has always the following properties: cl(0) / = 0, / A ⊂ cl(A), cl(A ∪ B) = cl(A) ∪ cl(B) for each A, B but the condition cl (cl(A)) = cl(A) need not hold. To assure this last equality it is necessary and sufficient that the following condition holds (see [3], p. 90): (L4) If { f j } j∈N , { f j,n } j,n∈N consist of real-valued S-measurable functions defined on X and f is a real-valued S-measurable function defined on X such I I that f j → f , f j,n → f j for each j ∈ N, then there exist two sequences j→∞

n→∞

I

{ j p } p∈N , {n p } p∈N of positive integers such that f j p ,n p → f . p→∞

After a moment of reflection, choosing the subsequence from the column of limit functions and subsequences of the rows of functions, we can observe that the above condition is equivalent to the following one: (L4)’ If { f j } j∈N , { f j,n } j,n∈N and f are as above and f j → f I-a.e. and f j,n → j→∞

n→∞

f j I-a.e. for each j ∈ N, then there exist two sequences { j p } p∈N and {n p } p∈N of positive integers such that f j p ,n p → f I-a.e. p→∞

Remark 6.25. Observe that the condition (E) is equivalent to the condition T (E’) which requires that ∞p=1 B j p ,n p ∈ / I holds for j p = p for p ∈ N. Theorem 6.26 ([10], Theorem 1). Suppose that the pair (S, I) fulfills ccc. Then FI / ∼I is equipped with the Frechet topology generated by the convergence with respect to I (i.e. the closure operator fulfills all axioms of Kuratowski) if and only if the pair (S, I) fulfills the condition (E). Below, we shall present several characterizations of the convergence I-a.e. and the convergence with respect to the σ -ideal I. Theorem 6.27. Suppose that the pair (S, I) fulfills ccc. Then the sequence { fn }n∈N of real-valued S-measurable functions defined on X converges I-a.e. to a real-valued S-measurable function f defined on X if and only if for each ε > 0 the sequence {hεn }n∈N of functions defined in the following way: hεn = χE(n,ε) , where E(n, ε) = {x ∈ X : | fn (x) − f (x)| > ε}, converges to zero with respect to σ -ideal I. Theorem 6.28 (compare Lemma 4 in [10]). Suppose that the pair (S, I) fulfills ccc. Then the sequence { fn }n∈N of real-valued S-measurable functions defined on X converges to zero with respect to I if and only if the following conditions are fulfilled:

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1. for each increasing sequence {nm }m∈N of positive integers, for each set D ∈ S \ I and for each ε > 0 there exists a subsequence {nm p } p∈N and a set B ⊂ D, B ∈ S \ I such that lim sup p fnm p (x) < ε for each x ∈ B. 2. for each increasing sequence {nm }m∈N of positive integers, for each set D ∈ S \ I and for each ε > 0 there exists a subsequence {nm p } p∈N and a set B ⊂ D, B ∈ S \ I such that lim inf p fnm p (x) > −ε for each x ∈ B. Let { fn }n∈N be a sequence of real-valued S-measurable functions defined on X and f – a real-valued S-measurable function defined on X. Put En (α) =

∞ [

{x ∈ X : | fi (x) − f (x)| > α}.

i=n

Definition 6.29 (see [11]). We shall say that the sequence { fn }n∈N satisfies the T vanishing restriction with respect to f if and only if ∞ n=1 En (α) ∈ I for all α > 0. Put φn (x) = sup{| fi (x) − f (x)| : i ≥ n, i ∈ N} for n ∈ N. Theorem 6.30 ([11], Theorem 2). If { fn }n∈N is a sequence of real-valued Smeasurable functions defined on X and f – a real-valued S-measurable function defined on X, then the following conditions are equivalent: (i) the sequence { fn }n∈N converges to f I-a.e. on X; (ii) the sequence { fn }n∈N satisfies the vanishing restriction with respect to f ; (iii) the sequence {φn }n∈N converges to zero with respect to I. Remark 6.31. As the pair (L, N ) fulfills ccc, it follows from Theorem 6.15 and Theorem 6.26 that the convergence in measure in a finite measure space yields the topology in FN / ∼ N . This topology is metrizable (see for example [4], p. 182-183). Theorems 6.17 and 6.26 imply that the convergence in category (i.e. with respect to M) does not generate a topology. Remark 6.32. If (X, S) is a measurable space and I ⊂ S is a maximal σ -ideal, then the countable chain condition is fulfilled (every disjoint family in S \ I can have at most one element). We shall prove that the condition (E) also holds. Let D ∈ S \ I and let {B j,n } j,n∈N be a double sequence of S-measurable sets S such that B j,n ⊂ B j,n+1 for j, n ∈ N and ∞ n=1 B j,n = D for j ∈ N. Put j p = p for every natural p and choose n p in such a way that B p,n p ∈ / I (it is possible T for every p). Then ∞p=1 B p,n p ∈ / I, because X\

∞ \ p=1

B p,n p =

∞ [

(X \ B p,n p ) ∈ I.

p=1

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Remark 6.33. Let X be an arbitrary uncountable set, S = 2X and I = {0}. / Then the convergence with respect to I is simply the pointwise convergence. It is well known that this kind of convergence does not yield a topology in this case. However the condition (E) is fulfilled. Indeed, let D ∈ S \ I and let {B j,n } j,n∈N be a double sequence of sets B j,n ⊂ B j,n+1 for each j, n ∈ N, S∞ to put j p = p and to choose n=1 B j,n = D for each j ∈ N. Let x0 ∈ D. It suffices T T n p in such a way that x0 ∈ B p,n p . Hence x0 ∈ ∞p=1 B p,n p , so ∞p=1 B p,n p ∈ / I. Observe that the pair (S, I) does not fulfill the countable chain condition. So this condition in Theorem 6.26 is important. Below we shall give some examples showing that the convergence in measure (with respect to N ) differs essentially from the convergence in category (with respect to M). It is easy to construct a sequence of measurable and having the property of Baire functions, which is convergent in measure but not in category (or conversely). Indeed, let A, B ⊂ [0, 1] be a pair of sets such that A is of the first category, B is of Lebesgue measure zero and A ∪ B = [0, 1] (see [5], p. 5). If fn = (−1)n χA for every n, then the sequence { fn }n∈N converges Malmost everywhere (so also in category) to the function f ≡ 0, but { fn }n∈N does not converge in measure. If we put gn = (−1)n χB for n ∈ N, then we obtain a sequence convergent almost everywhere to g ≡ 0 but not convergent in category. Obviously, every fn and gn is measurable and has the Baire property. It is a little more difficult to construct a sequence of continuous functions which is convergent in measure but not in category (or conversely), (compare [9], p. 310-311). Let A ⊂ [0, 1] be a closed nowhere dense set of positive measure and let {(an , bn )}n∈N be a sequence of components of [0, 1] \ A. Put  S  (−1)n for x ∈ A ∪ ∞  i=n+1 (ai , bi ),   Sn 0 for x ∈ i=1 [ai + (bi − ai )2−n , bi − (bi − ai )2−n ], fn (x) =  linear on the intervals [ai , ai + (bi − ai )2−n ] and     [bi − (bi − ai )2−n , bi ] for i = 1, . . . , n. It is not difficult to see that the sequence { fn (x)}n∈N converges to zero for every x ∈ / A, so { fn }n∈N converges in category to f ≡ 0. Simultaneously { fn }n∈N does not converge in measure, because this sequence does not fulfill the Cauchy condition in measure. Indeed, for ε < 2 we have the inclusion {x : | fn (x) − fn+1 (x)| > ε} ⊃ A for each n ∈ N. Let now

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  1       0   

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i 1 1 i − + , 2 2 , 2(n+1) i h n+1 2(n+1) i h i=1 n 1 1 1 − (n+1) for x ∈ 0, n+1 2 ∪ n+1 + (n+1)2 , 1 h i S i 1 i+1 1 ∪ n−1 gn (x) = i=1 n+1 + (n+1)2 , n+1 − (n+1)2 ,  h i   i 1 i 1  linear on the intervals − , − ,  2 2 n+1  (n+1)  i n+1 2(n+1) h   1 1 i i  n+1 + 2(n+1)2 , n+1 + (n+1)2 , i = 1, 2, . . . , n. for x ∈

Sn

h

i n+1

It is not difficult to see that the sequence {gn }n∈N converges in measure to g ≡ 0. Indeed, the Lebesgue measure of the set {x : gn (x) 6= 0} is equal to 2n/(n + 1)2 . We shall prove that {gn }n∈N does not converge in category. Let {gmn }n∈N be an arbitrary subsequence of {gn }n∈N . We shall show that the set {x : lim supn gmn (x) = 1 and lim infn gmn = 0} is residual in [0, 1]. Denote An = int ({x : gn (x) = 0}) and Bn = int ({x : gn (x) = 1}) for n ∈ N. T S∞ S∞ Then {x : lim supn gmn = 1} ⊃ ∞ n=k+1 Bmn . The set n=k+1 Bmn is open T∞ Sk=1 and dense in [0, 1]. Hence k=1 ∞ B is a residual set. Similarly one can n=k+1 T∞mn S∞ prove that {x : lim infn gmn (x) = 0} ⊃ k=1 n=k+1 Amn is a residual set. From the fact that {gmn }n∈N was an arbitrary subsequence we conclude that {gn }n∈N does not converge in category.

6.6 Small translations of sets It is well known (see, for example [1], p. 901-902) that if A ⊂ [0, 1], A ∈ L, λ (A) > 0, then limx→0 λ ((A − x) ∩ A ∩ (A + x)) = λ (A). Obviously, also limx→0 λ ((A + x) ∩ A) = λ (A). It means that if {xn }n∈N converges to zero, then the sequence {χA+xn }n∈N of characteristic functions converges in measure to χA . This statement cannot be improved. Theorem 6.34 (see [13], Theorem 1). There exists a set A ⊂ [0, 1], A ∈ L, λ (A) > 0 and a sequence {xn }n∈N convergent to 0 such that the sequence {χA+xn }n∈N does not converge almost everywhere to χA . The set A has been constructed in such a way that λ (A ∩ [a, b]) > 0 and λ ([a, b] \ A) > 0 for each [a, b] ⊂ [0, 1]. Obviously, if A ∈ N , then the sequence {χA+xn }n∈N converges to χA a.e. Theorem 6.35 (see [13], Theorem 2). If A ⊂ [0, 1], A ∈ B, and {xn }n∈N is an arbitrary sequence convergent to 0, then the sequence {χA+xn }n∈N converges to χA M-almost everywhere.

Elz˙ bieta Wagner-Bojakowska, Władysław Wilczy´nski

66

The proof is based upon the representation A = G4P, where G is open and P ∈ M. Similarly one can prove a slightly more general theorem: Theorem 6.36 (see [6], Theorem 1). If A ⊂ [0, 1], A ∈ B and { fn }n∈N is a sequence of continuous strictly increasing functions convergent uniformly to the identity function, then the sequence {χ fn (A) }n∈N converges to χA M-almost everywhere. The situation is more complicated in the case of measurable sets: Theorem 6.37 (see [6], Thoerem 2). Let { fn }n∈N be a sequence of continuous increasing functions convergent uniformly to the identity function. Then limn→∞ λ ∗ (A4 fn (A)) = 0 for each A ⊂ [0, 1], A ∈ L if and only if for the sequences of terms {gn }n∈N and {hn }n∈N from the Lebesgue decomposition of fn (gn is absolutely continuous and hn is singular for n ∈ N, both are nondecreasing) the following conditions are fulfilled: 1. limn→∞ hn (1) = 0 (i.e. {hn }n∈N converges uniformly to 0), 2. the sequence {gn }n∈N consists of uniformly absolutely continuous functions (i.e. for each ε > 0 there exists δ > 0 such that for each n ∈ N and for each finite collection [a1 , b1 ], [a2 , b2 ], . . . , [ak , bk ] of nonoverlapping intervals contained in [0, 1] if ∑ki=1 (bi − ai ) < δ , then ∑ki=1 (gn (bi ) − gn (ai )) < ε.

References [1] N. Bary, Trigometric series, Moscow 1961 (in Russian). [2] A. Bruckner, J. Bruckner, B. Thomson, Real Analysis, Prentice-Hall 1997. [3] R. Engelking, General topology, PWN – Polish Scientific Publishers, Warszawa 1977. [4] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, BerlinHeidelberg-New York 1965. [5] J. C. Oxtoby, Measure and category, Springer-Verlag, Berlin-Heidelberg-New York 1980. [6] G. Rzepecka, W. Wilczy´nski, On the transformations of measurable sets and sets with the Baire property, Real Anal. Exchange 20(1) (1994/95), 178–182. [7] S. J. Taylor, An alternative form of Egoroff’s theorem, Fund. Math. 48 (1960), 169–174. [8] B. Vulikh, A brief course in the theory of functions of a real variable, Mir Publisher Moscow 1976. [9] E. Wagner, Convergence in category, Estratto dal Rend. Acad. Sci. Fis. Mat. Soc. Naz. Sci. Lettere e Arti in Napoli, Serie IV 45 (1978), 303–312. [10] E. Wagner, Sequences of measurable functions, Fund. Math. 112 (1981), 89–102.

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[11] E. Wagner-Bojakowska, Remarks on convergence of sequences of measurable functions, Acta Univ. Lodziensis, Folia Mathematica 4 (1991), 173–179. [12] E. Wagner, W. Wilczy´nski, Convergence almost everywhere of sequences of measurable functions, Colloq. Math. 45(1) (1981), 119–124. [13] A. Kharazishvili, W. Wilczy´nski, On translations of measurable sets and sets having the Baire property, Bulletin of the Academy of Sciences of Georgia 145(1) (1992), 43–46 (In Russian, English and Georgian summary). [14] K. Yoneda, On control function of a.e. convergence, Math. Japonicae 20 (1975), 101–105.

˙ E L ZBIETA WAGNER -B OJAKOWSKA Faculty of Mathematics and Computer Sciences, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

´ W ŁADYSŁAW W ILCZY NSKI Faculty of Mathematics and Computer Sciences, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 7

Ideal convergence

RAFAŁ FILIPÓW, TOMASZ NATKANIEC, PIOTR SZUCA

2010 Mathematics Subject Classification: 40A35, 40A30, 26A03, 54A20, 28A05, 05D10, 26A15, 26A21. Key words and phrases: ideal, dual filter, Borel ideal, Katetov order, ideal convergence, Bolzano-Weierstrass property, ideal pointwise convergence, ideal equal convergence, ideal discrete convergence, ideal Baire class, Lunina 7-tuple.

The notion of the ideal convergence is dual (equivalent) to the notion of the filter convergence introduced by Cartan in 1937 ([10]). The notion of the filter convergence is a generalization of the classical notion of convergence of a sequence and it has been an important tool in general topology and functional analysis since 1940 (when Bourbaki’s book [8] appeared). Nowadays many authors prefer to use an equivalent dual notion of the ideal convergence (see e.g. frequently quoted work [32]). In this paper we survey ideal convergence of sequences of reals and functions. We focus on three aspects of ideal convergence that are connected with the following well-known results. 1. Every bounded sequence of reals has the convergent infinite subsequence. (Section 7.2). 2. The limit of a convergent sequence of continuous functions need not be continuous. (Section 7.3). 3. The set of points where a sequence of continuous functions is convergent forms an Fσ δ set. (Section 7.4).

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7.1 Preliminaries An ideal on N is a family of subsets of N closed under taking finite unions and subsets of its elements. We can speak about ideals on any other infinite countable set by identifying this set with N via a fixed bijection. If not explicitly said, we assume that an ideal is proper (6= P(N)) and contains all finite sets. By FIN we denote the ideal of all finite subsets of N. A filter on N is a family of subsets of N closed under taking finite intersections and supersets of its elements. For an ideal I we define I ? = {A : N \ A ∈ I} and call it the dual filter to I; and for a filter F we define F ? = {A : N \ A ∈ F } and call it the dual ideal to F. The filter FIN? is called the Fréchet filter. Let I be an ideal on N. Let xn ∈ R (n ∈ N) and x ∈ R. We say that the sequence (xn ) is I-convergent to x if {n ∈ N : |xn − x| ≥ ε} ∈ I for every ε > 0. We write I − lim xn = x in this case. If I = FIN, then Iconvergence is equivalent to the classical convergence. By identifying subsets of N with their characteristic functions, we equip P(N) with the product topology of {0, 1}N . It is known that P(N) with this topology is a compact Polish space without isolated points (it is homeomorphic to the Cantor set). An ideal I is an Fσ ideal (analytic ideal, respectively) if I is an Fσ subset of P(N) (if it is a continuous image of a Gδ subset of P(N), respectively). A map φ : P (N) → [0, ∞] is a submeasure if φ (0) / = 0, φ is monotone (i.e. φ (A) ≤ φ (B) whether A ⊆ B) and φ is subadditive (i.e. φ (A ∪ B) ≤ φ (A) + φ (B)). We will assume also that φ (N) > 0. For a submeasure φ we define Z (φ ) = {A ⊆ N : φ (A) = 0} and Fin(φ ) = {A ⊆ N : φ (A) < ∞}. For any A ⊆ N we define kAkφ = lim φ (A \ {0, 1, . . . , n − 1}) . n→∞

k·kφ is also a submeasure on N. A submeasure φ is lower semicontinuous (lsc, in short) if for all A ⊆ N we have φ (A) = limn→∞ φ (A ∩ {0, 1, . . . , n − 1}). For example, FIN = FIN(φ ) for φ (A) = card(A). Theorem 7.1 ([37]). An ideal I is Fσ if and only if there exists an lsc submeasure φ such that I = Fin(φ ).

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Example 7.2 ([37]). Let f : N → [0, ∞) be such that ∑n f (n) = ∞. We define I f = {A ⊆ N : ∑n∈A f (n) < ∞}. An ideal I is a summable ideal if I = I f for some f . Every summable ideal is an Fσ ideal (I f = FIN(φ ), φ (A) = ∑i∈A f (i)).   For an lsc submeasure φ we define Exh (φ ) = Z k·kφ . An ideal I is a P-ideal if for every family {An : n ∈ N} ⊆ I there exists A ∈ I such that An \ A is finite for all n. Theorem 7.3 ([43]). An ideal I is an analytic P-ideal if and only if there exists an lsc submeasure φ such that I = Exh (φ ). Example 7.4 ([28]). Let f : N → [0, ∞) be a function such that ∑n f (n) = ∞ and limn f (n)/(∑i≤n f (i)) = 0 We define EU f = Z d f , for d f (A) = lim sup n→∞

∑i∈A∩{0,1,...,n−1} f (i) . ∑i 0 there exists a partition of N on finitely many sets A0 , A1 , . . . , An−1 such that φ (Ai ) ≤ ε for each i = 0, 1, . . . , n − 1. In [15] the authors showed that the ideal Z (φ ) is nonatomic for every strongly nonatomic submeasure φ . And they also showed that the converse does not hold. Theorem 7.12 ([1]). If a submeasure φ is strongly nonatomic, then Z (φ ) does not have BW property. It is not possible to prove the converse of Theorem 7.12. The counterexample is the submeasure defined by φ (A) = 0 if u(A) = 0 and φ (A) = 1 otherwise, for u being the upper Banach density ([1]). However, Theorem 7.11 can be reversed for φ being the lim sup of lsc submeasures ([1]). For an analytic P-ideal one can prove the following equivalence. Theorem 7.13 ([19]). An analytic P-ideal I = Exh (φ ) has BW property if and only if the submeasure k·kφ is not strongly nonatomic.

7.2.2 Splitting families A family S ⊆ P(N) is an I-splitting family if for every A ∈ / I there is S ∈ S such that A ∩ S ∈ / I and A \ S ∈ / I. Theorem 7.14 ([19]). An ideal I has BW property if and only if does not exist a countable I-splitting family. Let s(I) denote the smallest cardinality of an I-splitting family. For I = FIN we write s = s(FIN) and it is called the splitting number (for more about s, see e.g. [5] or [46]). The above theorem shows that s(I) = ℵ0 for every ideal without BW property. It is known that s = c if we assume Martin’s Axiom. In [18] it was shown that if we assume Martin’s Axiom then s(I) = c for every Fσ ideal, and for every analytic P-ideal with BW property.

7.2.3 Katˇetov order and extendability to Fσ ideals Let I, J be ideals. We write I ≤K J if there exists a function h : N → N such that h−1 [A] ∈ J for all A ∈ I. The relation ≤K is called Katˇetov order and was

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introduced in [29] and used in [30] for study of filter convergence of sequences of functions. By CONV we denote the ideal of all subsets of Q ∩ [0, 1] which have only finitely many cluster points. Theorem 7.15 ([38]). An ideal I has FinBW property if and only if CONV 6≤K I. Let I and J be ideals. We say that: • J extends an ideal I if I ⊆ J ; • I contains an isomorphic copy of an ideal J if there is a bijection h : N → N such that h−1 [A] ∈ I for each A ∈ J . It is known that I contains an isomorphic copy of the ideal CONV iff CONV ≤K I ([2]). In [27] the author asked if for a Borel ideal I, CONV 6≤K I ⇐⇒ I can be extended to a proper Fσ ideal. Using Theorem 7.15 this question can be reformulated in the following way. Problem 7.16 ([27]). Let I be a Borel ideal. Are the following conditions equivalent? (1) I has FinBW property. (2) I can be extended to a proper Fσ ideal. In [19] the authors proved that the implication (2) ⇒ (1) holds for every ideal. Since every analytic P-ideal is Borel (in fact Fσ δ ) so the following theorem gives a partial answer to the Hrušák’s question (as far as we know, this question in its general version is still open). Theorem 7.17 ([19], [2]). Let I be an analytic P-ideal. The following conditions are equivalent: • • • •

I has FinBW property; I can be extended to a proper Fσ ideal; CONV 6≤K I; I does not contain an isomorphic copy of the ideal CONV. An ideal I is called maximal if there is no proper ideal J extending I.

Theorem 7.18 ([19]). Assume the Continuum Hypothesis. Let I be an analytic P-ideal. The following conditions are equivalent:

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• I has FinBW property; • I can be extended to a maximal P-ideal. The dual filter to a maximal ideal is called an ultrafilter. Recall that the set ˇ of all ultrafilters defined on N with an appropriate topology is the Cech-Stone compactification β N of the set of natural numbers N. Dual filters to maximal ˇ P-ideals are called P-points in the Cech-Stone compactification β N realm and it is known that its existence is independent from the axioms of ZFC ([41]).

7.2.4 Combinatorics The Ramsey theorem is one of the best known theorems of combinatorics. This theorem has many generalizations. For example Frankl, Graham and Rödl provided its iterated density version for the submeasure d—i.e. upper asymptotic density—defined in Example 7.5. Recall that by [N]2 we mean a family of all two-element subsets of N, i.e. [N]2 = {{x, y} : x, y ∈ N, x 6= y}. In this section we will use the word “coloring” instead of “partition”. Theorem 7.19 ([24]). For every coloring [N]2 = C0 ∪ C1 ∪ . . . ∪ Cr there exist δ = δ (r) > 0 and i ≤ r such that     d x ∈ N : d y ∈ N : d ({z ∈ N : {x, y} , {x, z} , {y, z} ∈ Ci }) ≥ δ ≥ δ ≥ δ . An analogous result is true for every analytic P-ideal. Theorem 7.20 ([22]). Let I = Exh (φ ) be an analytic P-ideal. Then for every coloring [N]2 = C1 ∪C2 ∪ . . . ∪Cr there exist δ = δ (r) and i ≤ r with

 

n o



x∈N:

y ∈ N : k{z ∈ N : {x, y}, {x, z}, {y, z} ∈ Ci }kφ ≥ δ ≥ δ

≥ δ. φ

φ

We have the following stronger version of the above result for ideals with Bolzano-Weierstrass property. Theorem 7.21 ([20]). Let I = Exh (φ ) be an analytic P-ideal with BW property. Then there exist δ = δ (φ ) such that for every finite coloring [N]2 = C1 ∪ C2 ∪ . . . ∪ Cr there exist i ≤ r and A ⊆ N with kAkφ ≥ δ such that for every x ∈ A

n o

k{z {x, {x, {y, }k y ∈ A : ∈ A : y} , z} , z} ∈ C ≥ δ

≥ δ.

i φ φ

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Corollary 7.22. An analytic P-ideal has the Bolzano-Weierstrass property if and only if the constant δ in Theorem 7.20 can be found independently on the number of colors r. Proof. The implication “⇒” follows from Theorem 7.21. Now we show the implication “⇐”. Let I = Exh (φ ) be an analytic P-ideal without BW property and suppose that there is δ > 0 such that for every coloring [N]2 = C1 ∪ C2 ∪ . . . ∪Cr there is i ≤ r with

 

n o



≥ δ.

x∈N: k{z }k ≥ δ y ∈ N : ∈ N : {x, y}, {x, z}, {y, z} ∈ C ≥ δ

i φ

φ

φ

Since I does not have BW property so by Theorem 7.13, there is a partition N = A1 ∪ · · · ∪ AN such that kAi kφ ≤ δ /2 for every i ≤ N. For k, l ≤ N let  Ck,l = {x0 , x1 } ∈ [N]2 : ∃i ∈ {0, 1} (xi ∈ Ak and x1−i ∈ Al ) . Since [N]2 = k,l≤N Ck,l , there are k0 , l0 ≤ N with

 

n o



x∈N:

≥ δ. }k k{z y ∈ N : ∈ N : {x, y}, {x, z}, {y, z} ∈ C ≥ δ ≥ δ

k ,l 0 0 φ

S

φ

φ

It is easy to check that if {x, y}, {x, z}, {y, z} ∈ Ck0 ,l0 then Ak0 ∩ Al0 6= 0. / Since sets A1 , . . . , AN are pairwise disjoint, so k0 = l0 . Thus {z ∈ N : {x, y}, {x, z}, {y, z} ∈ Ck0 ,l0 } ⊆ Ak0 so k{z ∈ N : {x, y}, {x, z}, {y, z} ∈ Ck0 ,l0 }kφ ≤ kAk0 kφ ≤ δ /2. Then n o y ∈ N : k{z ∈ N : {x, y}, {x, z}, {y, z} ∈ Ck0 ,l0 }kφ ≥ δ = 0/ so

 

n o



x∈N:

= 0, k{z }k ≥ δ y ∈ N : ∈ N : {x, y}, {x, z}, {y, z} ∈ C ≥ δ

k ,l 0 0 φ

φ φ

a contradiction.

t u

Another well-known theorem from infinite combinatorics is the Schur theorem which says that for every coloring of the set of natural numbers N =

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C0 ∪ . . . ∪Cr , there exist i ≤ r and x, y, z ∈ Ci with x + y = z. This theorem also has many generalizations. Bergelson and Hindman provided density version of the Schur theorem for the submeasure d. Theorem 7.23 ([4]). For every coloring N = C0 ∪ C1 ∪ . . . ∪ Cr there exist δ = δ (r) > 0 and i ≤ r such that   d x ∈ N : d ({y ∈ N : x, y, x + y ∈ Ci }) ≥ δ ≥ δ . We will say that a submeasure φ is invariant under translations if φ (A + t) = φ (A) for each A ⊆ N and t ∈ N (where A + t = {a + t : a ∈ A}). In [22] it was shown that it is possible to generalize Theorem 7.23 on any submeasure of the form k·kφ , for k·kφ being invariant under translations (in particular, d is of this form). Theorem 7.24 ([22]). Let I = Exh (φ ) be an analytic P-ideal with k·kφ invariant under translations. Then for every coloring N = C1 ∪ C2 ∪ . . . ∪ Cr there exists δ = δ (r) and i ≤ r with

n o

x ∈ N : k{y ∈ N : x, y, x + y ∈ Ci }kφ ≥ δ ≥ δ . φ

Theorem 7.25 ([22]). Let I = Exh (φ ) be an analytic P-ideal such that k·kφ is invariant under translations. The ideal I has the BW property if and only if there exists δ > 0 such that for every r ∈ N and every coloring N = C1 ∪C2 ∪ . . . ∪Cr there is i ≤ r with

n o

x ∈ N : k{y ∈ N : x, y, x + y ∈ Ci }kφ ≥ δ ≥ δ . φ

Note that the constant δ in Theorems 7.23 and 7.24 depends on the number of colors r. Theorem 7.25 yields the following corollary. Corollary 7.26. An analytic P-ideal I = Exh (φ ) with k·kφ invariant under translations has the Bolzano-Weierstrass property if and only if the constant δ in Theorem 7.24 can be found independently on the number of colors r. In this context it seems to be interesting to find out in which density theorems (see e.g. [24], Th. 3.1, 5.2, 6.1) the condition “the constant δ does not depend on the number of colors” characterizes non strongly nonatomic submeasures.

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7.3 Ideal convergence of sequences of functions Let Φ be a fixed kind of convergence of sequences of real-valued functions (e.g. pointwise convergence, equal convergence or discrete convergence, or their ideal counterpart). For a family F of real-valued functions defined on X there is the smallest family B Φ (F) of all real-valued functions defined on X which contains F and which is closed under the process of taking Φ-limits of sequences. This family is called the Baire system with respect to Φ generated by F. One method of generating B Φ (F) from F is by iteration of Φ-limits: • B0Φ (F) = F; • BαΦ (F) = LIMΦ

 Φ (F) for α > 0, B β N {k ∈ N : (n, k) ∈ A} is finite. Theorem 7.29 ([13], [33], [2]). Let X be an uncountable Polish space and I be an analytic ideal. The following conditions are equivalent: 1. B1 (X) = B1I (X); 2. I does not contain an isomorphic copy of the ideal FIN × FIN; 3. FIN × FIN 6≤K I. It is not possible to generalize Theorem 7.29 on the class of all ideals. This is a consequence of the fact that a maximal ideal is a P-ideal if and only if it does not contain an isomorphic copy of the ideal FIN × FIN. Thus, using Example 7.27 for I being a maximal P-ideal, we get χI ∈ B1I (X) \ B1 (X). However, the following weak version of Theorem 7.29 holds for any ideal I. Theorem 7.30 ([40]). For any ideal I and an uncountable Polish space X the following conditions are equivalent: In this section we use only Fσ -separability of I and I ? . In Section 7.4.2 we introduce the definition of the rank of an ideal I ; the rank of I is equal to 1 if and only if I and I ? can be Fσ -separated.

1

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1. B1 (X) = B1I (X) ∩ Bor(X)2 ; 2. I does not contain an isomorphic copy of the ideal FIN × FIN. Recall also that B1I (X) ⊆ Bor(X) for every analytic I and a Polish space X ([13]).

7.3.2 Ideal Baire classes Let X be a topological space and α be a countable ordinal. For an ideal I and a topological space X we define I-Baire classes: • B0I (X) = C(X); • BαI (X) = I − LIM

S

 I (X) , for α > 0, B β N; • discretely convergent to f (d-lim fn = f ) if for every x ∈ X there is N with fn (x) = f (x) for every n > N. 2

Here Bor(X) denotes the class of all Borel functions on X.

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The notions of discrete and equal convergence were introduced by Császár and Laczkovich in [11]. It is known that if ( fn ) is uniformly convergent to f then ( fn ) is equally convergent to f ; and if ( fn ) is equally convergent to f then ( fn ) is pointwise convergent to f ; and if ( fn ) is discretely convergent to f then ( fn ) is equally convergent to f . For a family of functions E ⊆ RX by the symbol d-LIM (E) (respectively: e-LIM (E)) we denote the family of all discrete limits (equal limits, respectively) of discretely convergent (equally-convergent, respectively) sequences of functions from the family E. Analogously to the definition of Baire classes (with respect to pointwise (d) convergence) one can define discrete Baire classes Bα (X) and equal Baire (e) classes Bα (X) ([11]). (d)

(e)

• B0 (X) = B0 (X)= C(X);  S (d) (d) • Bα (X) = d-LIM β 0; S  (e) (e) • Bα (X) = e-LIM β 0. The ideal versions of discrete and equal convergence and Baire classes were introduced in [23] in the following manner. Let I be an ideal on N. A sequence ( fn ) is: • equally? I-convergent to f (I − e? -lim fn = f ) if there exists a sequence of positive reals (εn ) such that limn εn = 0 and for every x ∈ X the set {n ∈ N : | fn (x) − f (x)| ≥ εn } ∈ I;3 • discretely I-convergent to f (I − d-lim fn = f ) if for every x ∈ X the set {n ∈ N : fn (x) 6= f (x)} ∈ I. For a family E ⊆ RX by the symbol I − d-LIM (E) (I − e? -LIM (E), respectively) we denote the family of all discrete I-limits (equal I-limits, respectively) of all discretely I-convergent (equally? I-convergent, respectively) sequences of functions belonging to E. And finally, ideal discrete and equal? Baire classes are defined in the following way. (I−d)

• B0

(I−e? )

(X) = B0

(X)= C(X);  S (I−d) (X) , for every α > 0; • = I − d-LIM β 0, I − lim xn = −∞ if {n ∈ N : xn > M} ∈ I for any M < 0, I − lim xn = inf {α : {n : xn > α} ∈ I} , I − lim xn = sup {α : {n : xn < α} ∈ I} . It is easy to see that {EI1 (f), . . . , EI7 (f)} is a partition of X. Moreover, we have E i (f) = EIi (f) for i = 1, 2, . . . , 7 and I = FIN.

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Let F ⊆ RX be a family of real-valued functions defined on a set X. Let E i ⊆ X (i = 1, . . . , 7). The sequence (E 1 , . . . , E 7 ) is called an I-Lunina’s 7tuple for F if there is a sequence f = ( fn ), fn ∈ F (n ∈ N) such that E i = EIi (f) for i = 1, . . . , 7. The family of all I-Lunina’s 7-tuples for F is denoted by ΛI (F). Let I, J be ideals. We write I ≤RK J if there exists a function h : N → N such that A ∈ I ⇐⇒ h−1 [A] ∈ J (the relation ≤RK is called the Rudin-Keisler order). Proposition 7.40 ([7]). Let X be a set. Let I, J be ideals. Let F be a family of real-valued functions defined on X. If I ≤RK J then ΛI (F) ⊆ ΛJ (F). Since FIN ≤RK I for every ideal I with the Baire property ([45]) so we get the following corollary. Corollary 7.41 ([7]). Let X be a set. Let I be an ideal with the Baire property and F be a family of real-valued functions defined on X. Then Λ (F) ⊆ ΛI (F). For ideals without the Baire property it is possible that ΛI (F) 6= Λ (F) even for F = C(X), which is shown by Example 7.42. Example 7.42. Let f = (n · hn ) (where hn are defined as in Example 7.27). Then EI1 (f) = I, and for a maximal ideal I, EI1 (f) does not have the Baire property. Hence by Theorem 7.37, EI1 (f) 6= E 1 (g) for any sequence g of continuous functions. Thus ΛI (F) 6= Λ (F).

7.4.1 Fσ ideals Theorem 7.43 ([7]). Let X be a metric space. Let I be an Fσ ideal. Then ΛI (C(X)) = Λ (C(X)). Theorem 7.44 ([7]). Let X be a metric space which contains a subspace homeomorphic to the Cantor space. If Λ (C(X)) = ΛI (C(X)) then I is an Fσ ideal. Theorem 7.45 ([6]). Let A be a σ -additive and (finitely) multiplicative family of subsets of a set X. Let Ac = {X \ M : M ∈ A} and FA be the family of A-measurable functions. Let I be an Fσ ideal. If {E 1 , . . . , E 7 } ∈ ΛI (FA ) then 1. E 1 , E 2 , E 3 ∈ (Ac )σ δ , 2. E 2 ∪ E 5 ∪ E 7 and E 3 ∪ E 6 ∪ E 7 are Aδ in X.

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Corollary 7.46 ([6]). Let X be a separable metric space. Let I be an Fσ ideal. Then ΛI (Bα (X)) = Λ (Bα (X)) for every countable ordinal α. Corollary 7.47 ([6]). Let I be an Fσ ideal and A be a σ -algebra of subsets of X. Then ΛI (FA )) = Λ (FA ). Theorem 7.48 ([39]). Let X be a dense in itself separable metric Baire space. Let I be an Fσ ideal. Then ΛI (QC(X)) = Λ (QC(X)).

7.4.2 Borel ideals and continuous functions Let F be a family of real-valued functions defined on a set X. Let I be an ideal on N. We define the following families of subsets of X: EI1 (F) = {EI1 (f) : f ∈ F }, EI2 (F) = {EI2 (f) : f ∈ F }, EI3 (F) = {EI3 (f) : f ∈ F }. The sequence (E 1 , E 2 , E 3 ) ∈ X 3 is called an I-Lipi´nski’s triple for F if there is a sequence f = ( fn ), fn ∈ F (n ∈ N) such that E i = EIi (f) for i = 1, 2, 3. The family of all I-Lipi´nski’s triples for F is denoted by ΛI3 (F). We write Λ 3 (F) 3 (F). instead of ΛFIN Theorem 7.49 ([40]). Let X be a metric space, F = C(X) and I be an ideal. S If I ∈ Π 0α \ β 4 (at least for Polish spaces or zero-dimensional spaces)? The following theorem gives some partial answer to the problem. Proposition 7.55 ([40]). Let X be a separable, zero-dimensional metric space, S F = C(X) and let I ∈ Π 0α \ β ) will denote a topological space and (Y, ρ) will denote a metric space. We start from definitions of some types of convergence of nets of functions f : X −→ Y . The best known types of convergence are pointwise and uniform convergence.

94 Robert Drozdowski, Jacek J˛edrzejewski, Stanisław Kowalczyk, Agata Sochaczewska

 Definition 8.1. We say that a net f j : j ∈ J of functions defined on a topological space X with values in a metric space (Y, ρ) is pointwise convergent to a function f : X −→ Y if ∀x∈X ∀ε>0 ∃ j0 ∈J ∀ j≥ j0 (ρ( f j (x), f (x)) < ε) .  Definition 8.2. We say that a net f j : j ∈ J of functions defined on a topological space X with values in a metric space (Y, ρ) is uniformly convergent to a function f : X −→ Y if ∀ε>0 ∃ j0 ∈J ∀ j≥ j0 ∀x∈X (ρ ( f j (x), f (x)) < ε) . Relations between these two kinds of convergence are well known, we will use them without any further remarks. Let us make a review of those kinds of convergence which will be considered. The first one is so called quasi-uniform convergence. It is sometimes called Arzelá convergence after the name of the author, who introduced this kind of convergence into mathematical life.  Definition 8.3. ([1], [4], [12]) A pointwise convergent net j j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is called quasi-uniformly convergent to a function f : X −→ Y if   ∀ε>0 ∀ j∈J ∃k j ∃l1 ,...,lk j ≥ j ∀t∈X min ρ ( fli (t), f (t)) : i ∈ {1, . . . , k j } < ε . (8.1) Of course everybody can see that this kind of convergence is weaker than uniform convergence but stronger than pointwise convergence. In the literature there was considered another kind of convergence which is also a bit weaker than uniform convergence and stronger than pointwise convergence. It is called almost-uniform convergence. For any point x in a topological space X by Bx we will denote the class of all open neighbourhoods of the point x.  Definition 8.4. ([5], [13]). A pointwise convergent net f j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is called almost-uniformly convergent to a function f : X −→ Y if ∀x∈X ∀ε>0 ∀ j∈J ∃ jx ≥ j ∃Ux ∈Bx ∀t∈Ux (ρ ( f jx (t), f (t)) < ε) .

(8.2)

In further part of the article we will make use of the following denotations. The symbol C(X,Y ) will denote the class of all continuous functions defined in a topological space X with values in a metric space Y . Similarly, the symbol

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F(X,Y ) will denote the class of all functions defined in a topological space X with values in a metric space Y . Moreover, let R+ = (0, ∞).  Definition 8.5. ([14], [2], [8], [10], [11]). A net f j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is said to be convergent to a function f : X −→ Y in the sense of Whitney (shortly W-convergent) if for each function ϕ from C(X, R+ ) there exists j0 ∈ J such that ρ ( f j (x), f (x)) < ϕ(x) for each x ∈ X and for each j ∈ J such that j ≥ j0 . This kind of convergence is even stronger than uniform one, since each constant function is continuous. Quasi-uniform convergence and convergence in the sense of Whitney can be combined, hence we can define:  Definition 8.6. ([6]) A net f j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is said to be convergent to a function f : X −→ Y in the sense of Arzelá-Whitney (shortly AW-convergent) if this net is pointwise convergent to the function f and for every ϕ ∈ C(X, R+ ), every j0 in J there exists a finite subset J1 of J such that j ≥ j0 for each j ∈ J1 and  min ρ ( f j (x), f (x)) : j ∈ J1 < ϕ(x) if x ∈ X. Similarly, combining Whitney convergence with almost-uniform convergence we are able to define another kind of convergence.  Definition 8.7. A net f j : j ∈ J of functions defined in a topological space with values in a metric space (Y, ρ) is said to be almost-Whitney convergent to a function f : X −→ Y if this net is pointwise convergent to function f and for every x ∈ X, ϕ ∈ C(X, R+ ), and for every j in J there exist jx ∈ J and a neighbourhood Ux of x such that ρ ( f j (t), f (t)) < ϕ(x) for each t ∈ Ux . We have defined 7 kinds of convergence of nets of functions. In further parts of the work we will consider some properties of these types of convergence.

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8.2 Relations between considered types of convergence Since positive constant function ϕε : X −→ R+ , ϕε (x) = ε for all x in X belongs to C(X, R+ ), it is easy to observe.  Remark 8.8. If a net f j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is W-convergent to f : X −→ Y then it is uniformly convergent to f .  Remark 8.9. If a net f j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is almost-Whitney convergent to f : X −→ Y then it is almost-uniformly convergent to f .  Remark 8.10. If a net f j : j ∈ J of functions defined in a topological space X with values in a metric space (Y, ρ) is AW-convergent to f : X −→ Y then it is quasi-uniformly convergent to f . Immediately from definitions of quasi-uniform convergence and almostuniform convergence, we infer that if a net { f j : j ∈ J} of functions from F(X,Y ) is quasi-uniformly convergent or almost-uniformly convergent to f : X −→ Y then this net is pointwise convergent to f . Moreover, it is obvious that each uniformly convergent net of functions { f j : j ∈ J} from F(X,Y ) is quasi-uniformly convergent, almost-uniformly convergent and pointwise convergent. Similarly, it is easy to see that each W-convergent net of functions from F(X,Y ) is AW-convergent and almost-Whitney convergent. Theorem 8.11. Let (X, >) be a topological space and let (Y, ρ) be a metric space. If a net f j : j ∈ J of functions from F(X,Y ) is almost-uniformly convergent to f : X −→ Y then it is almost-Whitney convergent to f .  Proof. Let a net f j : j ∈ J be almost-uniformly convergent to a function f : X −→ Y . Fix x ∈ X, j ∈ J and ϕ ∈ C(X, R+ ). Then ε = 21 ϕ(x) > 0. By continuity of ϕ, there exists a neighbourhood U1 of x such that ϕ(t) > ε for  all t ∈ U1 . Since f j : j ∈ J is almost-uniformly convergent to f , there exist jx ≥ j and a neighbourhood U2 of x such that ρ ( f jx (t), f (t)) < ε for all t ∈ U2 . Let U = U1 ∩U2 . Then U is a neighbourhood of x and ρ( f jx (t), f (t)) < ε < ϕ(t) if t ∈ U.   Since f j : j ∈ J is pointwise convergent to f , it proves that f j : j ∈ J is almost-Whitney convergent to f . t u

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Corollary 8.12.  Let (X, >) be a topological space and let (Y, ρ) be a metric space. If a net f j : j ∈ J of functions from F(X,Y ) is uniformly convergent to f : X −→ Y then it is almost-Whitney convergent to f . Corollary 8.13. For every topological space (X, T ) and every metric space (Y, ρ) almost-uniform convergence and almost Whitney convergence are equivalent in F(X,Y ).  Remark 8.14. If a net f j : j ∈ J of functions from F(X,Y ) is pointwise convergent to f : X → Y and it has a subnet which is W-convergent then it is AW-convergent to f . The diagram shows the relations among the considered kinds of convergence. Arzelá-Whitney convergence 

Whitney convergence @ R @

-

quasi-uniform convergence @ R @

 -

uniform convergence

almost-Whitney  convergence

-

pointwise convergence

@ R @ - almost-uniform



convergence

No other implication can be stated as a few examples show. Example 8.15. Pointwise convergence implies neither almost-uniform convergence nor quasi-uniform convergence. Let X = [0, 1] and Y = R and fn (x) = xn if x ∈ [0, 1]. Then the sequence ( fn )∞ n=1 is pointwise convergent to the function  0, if x ∈ [0, 1), f (x) = 1, if x = 1, but this sequence is not almost-uniformly convergent to f . Similarly, it is not quasi-uniformly convergent to f . Example 8.16. Almost-uniform convergence does not imply quasi-uniform convergence. Let X = [0, 1) and Y = R and fn (x) = xn if x ∈ [0, 1). Then the sequence ( fn )∞ n=1 is almost-uniformly convergent to the zero function, but this sequence is not quasi-uniformly convergent to f .

98 Robert Drozdowski, Jacek J˛edrzejewski, Stanisław Kowalczyk, Agata Sochaczewska

Example 8.17. Neither AW-convergence nor quasi-uniform convergence imply almost uniform convergence. Let X = [−1, 1] and Y = R and if n is even positive integer then    0, if x ∈ [−1, 0] ∪ n1 , 1 , fn (x) = 1, if x ∈ 0, 1n , and if n is odd positive integer then    0, if x ∈ −1, −n1 ∪ [0, 1] , fn (x) = 1, if x ∈ − 1n , 0 . Then the sequence ( fn )∞ n=1 is AW-convergent (and also quasi-uniformly convergent) to the zero function in [−1, 1], but this sequence is not almostuniformly convergent to f . Further examples state that neither quasi-uniform convergence nor almostuniform convergence imply uniform convergence. Example 8.18. None of the uniform convergence, almost-Whitney convergence, quasi-uniform convergence implies AW-convergence. Let fn : (0, 1) −→ [0, 1] be defined as follows: fn (x) =

1 n

if x ∈ (0, 1).

Then the sequence ( fn )∞ n=1 is not AW-convergent to zero function in (0, 1), but it is uniformly convergent, almost Whitney and quasi-uniformly convergent. Example 8.19. Almost Whitney convergence implies neither uniform convergence nor AW-convergence. Let fn (x) = xn and f (x) = 0, x ∈ (0, 1). Then { fn : n ∈ N} is almost Whitney convergent to f but it is neither uniformly convergent nor AW-convergent. Hence it is not W-convergent.

8.3 Continuity of limit of nets of continuous functions Everyone does know that if a net of continuous functions is uniformly convergent to a function f , then f is continuous as well and that pointwise convergence is too weak to get the continuity of the limit function. The following facts concerning quasi-uniform convergence are well known.

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Theorem 8.20. ([13]). If a net of continuous functions defined in a topological space X with values in a metric space (Y, ρ) is quasi-uniformly convergent to a function f : X −→ Y then f itself is continuous.  Proof. Let f j : j ∈ J be any net of functions from C(X,Y ). Let ε be a fixed positive number and x0 be a fixed point from X. There exists an index j0 ∈ J such that ρ ( f j (x0 ), f (x0 ))
lim sup f (t)} and

T l( f )

= {x : f (x) < lim inf f (t)}.

t→x t→x The quintuplet (S( f ), Sl ( f ),C( f ), T ( f ), T l ( f )) was characterized by T. Natka-

niec in 1983. Theorem 9.4 ([49]). Let S, Sl ,C, T, T l be subsets of R. Then S = S( f ), Sl = Sl ( f ), C = C( f ), T = T ( f ) and T l = T l ( f ) for some function f : R → R if and only if S ∩ Sl = C, C is dense in the set int(S) ∪ int(Sl ), C is a Gδ -set, T ⊂ S \C, T l ⊂ Sl \C and the set T ∪ T l is countable. For a real function f : R → R let us define the qualitative upper limit at the point x as

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q-lim sup f (t) = inf{y ∈ R : {t ∈ R : f (t) < y} is residual at x}. t→x

Similarly let us define the qualitative lower limit of f at x. Denote Cq ( f ) = {x ∈ R : q-lim sup f (t) = f (x) = q-lim inf f (t)}, t→x

t→x

Sq ( f ) = {x ∈ R : q-lim sup f (t) ≤ f (x)}, t→x

Tq ( f ) = {x ∈ R : q-lim sup f (t) < f (x)}, t→x

Sql ( f ) = {x ∈ R : q-lim inf f (t) ≥ f (x)} and t→x

Tql ( f ) = {x ∈ R : q-lim inf f (t) > f (x)}. t→x

The triplet (Cq ( f ), Sq ( f ), Tq ( f )) was characterized by Z. Grande in 1985 in [32] and the quintuplet (Cq ( f ), Sq ( f ), Sql ( f ), Tq ( f ), Tql ( f )) by T. Natkaniec. In the proof it is assumed that every subset of R of cardinality less than continuum is of first category. So, if we assume Continuum Hypothesis or Martin’s Axiom, then we have Theorem 9.5 ([50]). . Assume CH or MA. For every sets S, Sl ,C, T, T l in R the following condintions are equivalent: (i) S ∩ Sl = C, T ∪ T1 ∈ B, T ⊂ S \C, T l ⊂ Sl \C, the sets S \C and Sl \C do not contain sets of second category having Baire property and there exists a Gδ -set D such that C = D \ (T ∩ T l ), (ii) there is a function f : R → R such that S = Sq ( f ), Sl = Sql ( f ), C = Cq ( f ), T = Tq ( f ) and T l = Tql ( f ).

9.4 Quasicontinuity and cliquishness Recall that a function f : X → Y (X and Y are topological spaces) is said to be quasicontinuous at a point x if for each neighbourhood U of x and each neighbourhood V of f (x) there is an open nonempty set G ⊂ U such that f (G) ⊂ V ([41]). A function f : X → Y (X is a topological space and (Y, d) is a metric space) is said to be cliquish at a point x ∈ X if for each neighbourhood U of x and each ε > 0 there is an open nonempty set G ⊂ U such that d( f (y), f (z)) < ε for each y, z ∈ G ([53]). Denote by Q( f ) the set of all quasicontinuity points of f and by A( f ) the set of all cliquishness points of f . The sets Q( f ) and A( f ) were characterized for the first time by J. S. Lipi´nski and T. Šalát in 1970. They showed that A( f ) is always closed and gave the following characterizations:

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Theorem 9.6 ([46]). Let (X, d) be a metric space without isolated points and let (Y, p) be a metric space containing some one-to-one Cauchy sequence. Let A be a subset of X. Then A = A( f ) for some f : X → Y if and only if A is closed. Theorem 9.7 ([46]). Let (X, d) be a complete metric space dense in itself and let (Y, p) be a metric space possesing at least one accumulation point. Let A be a subset of X. Then A = Q( f ) for some f : X → Y if and only if the set int (cl(A)) \ A is of first category (in the sense of Baire). If Y is a metric space, then evidently C( f ) ⊂ Q( f ) ⊂ A( f ). A. Neubrunnová in 1974 showed (see [54]) that the set A( f ) \C( f ) is of first category. J. Ewert and J. S. Lipi´nski investigated the triplet (C( f ), Q( f ), A( f )). For the sets C, Q and A denote (*) C ⊂ Q ⊂ A, C is a Gδ -set, A is closed and A \C is of first category. Therefore (*) is a necessary condition for the triplet (C( f ), Q( f ), A( f )). In [23] they showed that if X is a Baire real normed space and Y is a normed space then (C, Q, A) = (C( f ), Q( f ), A( f )) for some function f : X → Y if and only if we have (*). In [22] they showed that if (*) implies the equality (C, Q, A) = (C( f ), Q( f ), A( f )) for some function f : X → Y then for each closed set A in X there is a decreasing sequence (Un )n of open sets such that T A = n cl(Un ) and for each closed nowhere dense subset F ⊂ X there is a continuous function g : X \ F → R such that the oscillation ωg (x) > 0 for each point x ∈ F. Further, in [24], they showed that the necessary condition on X is T not only A = n cl(Un ) but even the sets Un are the same as for C( f ), therefore (**) there is a decreasing sequence (Wn )n of open subsets of X such that T T n Wn = C ⊂ Q ⊂ A = n cl(Wn ). By [24], (**) implies (*). By [9], (*) does not imply (**), however, if X is perfectly normal, then (*) and (**) are equivalent. Further generalizations of conditions on a space X are investigated in [9]. Theorem 9.8 ([9]). Let X be a Baire resolvable perfectly normal locally connected space (or let X be a Baire pseudometrizable space without isolated points). Let (Y, p) be a metric space containing a subspace isometric with R. Let C, Q and A be subsets of X. Then the following conditions are equivalent: (i) there is a function f : X → Y such that C = C( f ), Q = Q( f ) and A = A( f ), (ii) C ⊂ Q ⊂ A, C is a Gδ -set, A is closed and A \C is of first category, (iii) there is a decreasing sequence (Wn )n of open subsets of X such that T T n Wn = C ⊂ Q ⊂ A = n cl(Wn ).

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If Q = X then the assumption “X is resolvable” can be omitted. So, we have Theorem 9.9 ([9]). Let X be a Baire perfectly normal locally connected space or X be a Baire pseudometrizable space. Then the set M is the set of all discontinuity points of some quasicontinuous function f : X → R if and only if M is an Fσ -set of first category. In [58], the question to characterize the sets of discontinuity points of quasicontinuous functions f : X → R (X is a topological space) is posed. It was solved in [28] for X = R2 . Theorem 9.9 is not true if X is normal is replaced with X is T1 completely regular, as the Niemytzki plane shows ([9]). Other characterization (for spaces not Baire only) we can find in [48]. Theorem 9.10 ([48]). Let for a Fréchet-Urysohn space X at least one of the following conditions holds: (i) X (ii) X (iii) X (iv) X (v) X

is a hereditarily separable perfectly normal; is hereditarily quasi-separable perfectly normal; is a regular space with a countable net; is a paracompact with a σ -locally finite net; is metrizable.

Then a set M is the set of all discontinuity points of some quasicontinuous function f : X → R if and only if M is an Fσ -set of first category. It is interesting that the pairs (C( f ), A( f )) and (Q( f ), A( f )) can be characterized under very general conditions, while for the pair (C( f ), Q( f )) I known the same conditions on X and Y as for the triplet (C( f ), Q( f ), A( f )) only. Theorem 9.11 ([9]). Let X be a resolvable topological space and let (Y, p) be a metric space with at least one accumulation point. Let Q and A be subsets of X. Then there is a function f : X → Y such that Q = Q( f ) and A = A( f ) if and T only if there is a decreasing sequence (Wn )n of open sets such that n Wn ⊂ T Q ⊂ n cl(Wn ) = A. Theorem 9.12 ([9]). Let X be a resolvable topological space and let (Y, p) be a metric space with at least one accumulation point. Let C and A be subsets of X. Then there is a function f : X → Y such that C = C( f ) and A = A( f ) if and T only if there is a decreasing sequence (Wn )n of open sets such that C = n Wn T and A = n cl(Wn ). If A = X then the assumption of resolvability of X can be omitted and we obtain a characterization of the set of discontinuity points of cliquish functions.

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Theorem 9.13 ([9]). Let X be a topological space and let Y be a metric space with at least one accumulation point. Then a set M is the set of all discontinuity points of some cliquish function f : X → Y if and only if M is an Fσ set of first category.

9.5 Bilateral quasicontinuity and cliquishness A function f : R → R is said to be left (right) hand sided quasicontinuous at a point x ∈ R if for every δ > 0 and for every open neighbourhood of f (x) there exists an open nonempty set G ⊂ (x − δ , x) (G ⊂ (x, x + δ )) such that f (G) ⊂ V [33]. A function f is bilaterally quasicontinuous at x if it is simultaneously left and right hand quasicontinuous at x. Denote by Q− ( f ), Q+ ( f ) and BQ( f ) the set of all left hand side quasicontinuity points, right hand side quasicontinuity points and bilateral quasicontinuity points of f . In this case we can find a characterization of the sixtuple (C( f ), BQ( f ), Q− ( f ), Q+ ( f ), Q( f ), A( f )). Theorem 9.14 ([6]). Let C, D, D1 , D2 , Q and A be subsets of R. Then C = C( f ), D = BQ( f ), D1 = Q− ( f ), D2 = Q+ ( f ), Q = Q( f ) and A = A( f ) for some f : R → R if and only if C ⊂ D = D1 ∩ D2 ⊂ D1 ∪ D2 = Q ⊂ A, C is a Gδ -set, A is closed, A \C is of first category and Q \ D is countable. Let Y be a topological space. If (X, d) is a metric space, we can generalize the bilateral quasicontinuity as follows: a function f : X → Y is Squasicontinuous at x if for every neighbourhood V of f (x) and every y ∈ X, y 6= x, there exists an open nonempty set G ⊂ S(y, d(x, y)) such that f (G) ⊂ V . Denote the set of all S-quasicontinuity points of f by QS( f ). If X is a topological space, another definition of bilateral quasicontinuity is possible, too. We will say that a function f : X → Y is B-quasicontinuous at x if for every neighbourhood V of f (x) and for every open connected set A such that x ∈ cl(A) there exists an open nonempty set G ⊂ A such that f (G) ⊂ V . Denote the set of all B-quasicontinuity points of f by QB( f ). Evidently, if X = R, the notions of bilateral quasicontinuity, B-quasicontinuity and S-quasicontinuity coincide. The characterizations of QB( f ) and QS( f ) are similar. Theorem 9.15 ([3]). Let X be a locally connected perfectly normal almost resolvable topological space. Let B be a set in X. Then B = QB( f ) for some function f : X → R if and only if the set cl(B) \ B is of first category.

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Theorem 9.16 ([3]). Let (X, d) be a metric space without isolated points such that cl (S(x, δ )) = {y ∈ X : d(x, y) ≤ δ } for each x ∈ X and each δ > 0. Let S be a subset of X. Then S = QS( f ) for some f : X → R if and only if the set cl(S) \ S is of first category. Similarly, we can define an one sided and bilateral cliquishness. A function f : R → Y ((Y, d) is a metric space) is said to be left-side (right-side) cliquish at x ∈ R if for each δ > 0 and ε > 0 there is nonempty open set G ⊂ (x − δ , x) (G ⊂ (x, x + δ )) such that d( f (y), f (z)) < ε for each y, z ∈ G. A function f is bilaterally cliquish at x if it is both right-side and left-side cliquish at x [26]. Denote by A+ ( f ), A− ( f ) and BA( f ) the sets of all points at which f is rightside cliquish, left-side cliquish and bilaterally cliquish, respectively. For a set M ⊂ R denote by L+ (M) (L− (M)) the set of all right-sided (left-sided) cluster points of M. Theorem 9.17 ([7]). Let A, B,C, D be subsets of R. Then A = A( f ), B = A+ ( f ), C = A− ( f ) and D = BA( f ) for some function f : R → R if and only if L+ (A) ⊂ B, L− (A) ⊂ C, A = B ∪C, D = B ∩C and the set A \ D is countable. If X is a topological space (and (Y, d) a metric one) we say that a function f : X → Y is B-cliquish at x if for each ε > 0 and for each open set V with x ∈ cl(V ) there is a nonempty open set G ⊂ V such that diam f (G) < ε. Denote by AB( f ) the set of all B-cliquishness points of f . We have AB( f ) ⊂ A( f ) and the set A( f ) \ AB( f ) is nowhere dense. Theorem 9.18 ([7]). Let X be a resolvable space and let M be a subset of X. T Then M = AB( f ) for some f : X → Y if and only if M = n Mn where Mn are open and such that int (cl(Mn+1 )) ⊂ Mn .

9.6 Upper and lower quasicontinuity A function f : X → R is said to be upper (lower) quasicontinuous at x if for each ε > 0 and for each neighbourhood U of x there is a nonempty open set G ⊂ U such that f (y) < f (x) + ε ( f (y) > f (x) − ε) for each y ∈ G ([25]). Denote by E( f ) the set of all points of both upper and lower quasicontinuity of f . In [31] it is shown that if a function f : X → R is upper and lower quasicontinuous at each point then it is cliquish. However, the inclusion E( f ) ⊂ A( f ) does not hold. Nevertheless, the set E( f ) \ A( f ) is nowhere dense ([10]). According to [21], the set E( f ) is the countable intersection of semi-open sets. A set M is

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semi-open ([44]) (or quasi-open, see [53]) if M ⊂ cl (int(M)). For X = R we have a characterization of the quadruplet (C( f ), Q( f ), E( f ), A( f )). Theorem 9.19 ([10]). Let C, Q, E and A be subsets of R. Then C = C( f ), Q = Q( f ), E = E( f ) and A = A( f ) for some f : R → R if and only if C ⊂ Q ⊂ A∩E, C is a Gδ -set, A is closed, A \C is of first category and E \ A is nowhere dense. This characterization is not true for functions f : R2 → R. In fact, for X = R2 this characterization is not true for the triplet (C( f ), Q( f ), E( f )). The remaining triplets can be characterized for Baire metric spaces without isolated points. The triplet (C( f ), Q( f ), A( f )) is characterized in Theorem 9.8. Remaining two cases: Theorem 9.20 ([8]). Let X be a Baire metric space without isolated points. Let C, E and A be subsets of X. Then C = C( f ), E = E( f ) and A = A( f ) for some function f : X → R if and only if C ⊂ A ∩ E, C is a Gδ -set, A is closed, A \C is of first category and E \ A is nowhere dense. Theorem 9.21 ([8]). Let X be a Baire metric space without isolated points. Let Q, E and A be subsets of X. Then Q = Q( f ), E = E( f ) and A = A( f ) for some function f : X → R if and only if Q ⊂ A ∩ E, A is closed, A \ Q is of first category and E \ A is nowhere dense.

9.7 Strong quasicontinuity The set Q( f ) of points of quasicontinuity of a real function f : R → R, in general, need not be Lebesque measurable. If a function f : R → R is Lebesgue measurable then the set Q( f ) is measurable [42]. Similarly, although the set Q( f ) \ C( f ) is of first category, it need not be measurable or of measure zero [12]. Even, there is a Darboux function such that the measure of Q( f ) \ C( f ) is positive [43]. Remind that λ (λ ∗ ) denote the Lebesgue measure (outer Lebesgue measure) ∗ the upper outer density in R. Denote by du (A, x) = lim supr→0+ λ (A∩(x−r,x+r)) 2r ∗ of A ⊂ R at a point x ∈ R; similarly dl (A, x) = lim infr→0+ λ (A∩(x−r,x+r)) is the 2r lower outer density at x. Denote Td = {A ⊂ R : A is measurable and for every x ∈ A we have dl (A, x) = 1}. Td is a topology called the density topology. Z. Grande in [29] has defined some ”stronger“ quasicontinuities. A function f : R → R has the property A(x) at a point x if there is an open set U such that du (U, x) > 0 and the restriction f  (U ∪ {x}) is continuous at

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x. A function f has property B(x) at x if for every η > 0 we have du (int({t : | f (t) − f (x)| < η}), x) > 0. A function f is strongly quasicontinuous at x if for every η > 0 and every U ∈ Td containing x there is a nonempty open set G such that U ∩ G 6= 0/ and | f (t) − f (x)| < η for all t ∈ U ∩ G. Denote by QA ( f ) the set of all points with property A(x), by QB ( f ) the set of all points with property B(x) and by Qs ( f ) the set of all strong quasicontinuity points of f . Obviously C( f ) ⊂ QA ( f ) ⊂ QB ( f ) ⊂ Qs ( f ) ⊂ Q( f ). The set Qs ( f ) \C( f ) need not be of measure zero ([29]), however, the set QB ( f ) \ C( f ) is of measure zero ([30]). Moreover, the sets QA ( f ) and QB ( f ) have Baire propery, however, they need not be borelian. Futher, he gave a characterization of the set QA ( f ). Theorem 9.22 ([30]). Let A ⊂ R. Then A = QA ( f ) for some f : R → R if and S T only if A = m n Am,n , where Am,n be such that there are open sets Gn such that for each m, n ∈ N we have du (int(Am,n ), x) ≥ 1/m for each x ∈ A, Am,n+1 ⊂ Am,n , Am,n ⊂ Am+1,n , Gn+1 ⊂ Gn , Gn ⊂ Am,n and du (Gn , x) ≥ 1/m for all x ∈ Am,n . Also, there exist characterizations of the pairs (C( f ), QA ( f )) and (C( f ), QB ( f )). Theorem 9.23 ([5]). Let A and C be subsets of R. Then C = C( f ) and A = QA ( f ) for some function f : R → R if and only if there exist open sets Gn such T that C = n Gn ⊂ A, Gn+1 ⊂ Gn and inf{du (Gn , x) : n ∈ N} > 0 for each x ∈ A. Theorem 9.24 ([5]). Let B and C be subsets of R. Then C = C( f ) and B = QB ( f ) for some function f : R → R if and only if there exist open sets Gn such T that C = n Gn ⊂ B, Gn+1 ⊂ Gn and du (Gn , x) > 0 for each x ∈ B.

9.8 Simple continuity Let X and Y be topological spaces. A set A is simply open if it is the union of open set and a nowhere dense set. A function f : X → Y is said to be simply continuous if the inverse image f −1 (V ) is a simply open set in X for each open set V in Y ([1]). Evidently, each quasicontinuous function is simply continuous. A suitable pointwise definition of simple continuity is given in [13]. We say that f : X → Y is simply continuous at a point x ∈ X if for each open neighbourhood V of f (x) and for each neighbourhood U of x, the set f −1 (V ) \ int f −1 (V ) is not dense in U. Denote by N( f ) the set of all simple continuity points of f . Then f if simply continuous if and only if N( f ) = X and moreover Q( f ) ⊂ N( f ) ([13]).

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Theorem 9.25 ([13]). Let X be a perfectly normal resolvable space. Let Y be a metric space with at least one accumulation point. Further, let moreover, X be a Baire space and Y be separable (or let Y be totally bounded). Let N ⊂ X. Then N = N( f ) for some function f : X → Y if and only if cl (int(N)) ⊂ N and the set cl(N) \ N is of first category.

9.9 Closed graph A function f : X → Y has a closed graph if the set Gr( f ) = {(x, f (x)) : x ∈ X} T is a closed subset of X ×Y . For a function f denote C( f , x) = {cl ( f (U)) : U is a neighbourhood of x}. We say that a function f : X → Y has a closed graph at x if C( f , x) = { f (x)}. Denote by H( f ) the set of all closedness graph points of f . Then f has closed graph if and only if H( f ) = X ([35], [55]). Then characterizations of the set H( f ) and the pair (C( f ), H( f )) are following. Theorem 9.26 ([4]). Let X be an almost resolvable topological space. Let H be a subset of X. Then H = H( f ) for some function f : X → R if and only if H is a Gδ -set. Theorem 9.27 ([4]). Let X be a Baire almost resolvable perfectly normal topological space. Let C and H be subsets of X. Then C = C( f ) and H = H( f ) for some function f : X → R if and only if C ⊂ H, C and H are Gδ -sets, C is open in H and int (H \C) = 0. / There are examples that any condition on X cannot be omitted. By [19], the set of discontinuity points of a function with the closed graph is of first category and closed: and if moreover X is a Baire space, then it is even nowhere dense. Nevertheless, the set H( f ) \C( f ) can be residual and not closed (even for a function f : R → R).

9.10 Generalized topology Generalized continuities in the above section usually are not continuous (in a some suitable topology). Nevertheless, many of them are “continuous” in some weaker “topology”. Let us recall some notions. Let X be a nonempty set and P(X) the power set of X. We call a class g ⊂ P(X) a generalized topology (briefly GT, see [16]), if 0/ ∈ g and the arbitrary union of elements of g belongs

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to g. A GT g is strong if X ∈ g. A set X with a GT g is called a generalized topological space (briefly, GTS) and is denoted by (X, g). For x ∈ X we denote g(x) = {A ∈ g : x ∈ A}. By [17], if (X, g) and (Y, h) are GTS’s, then a mapping f : X → Y is called (g, h)-continuous, if f −1 (V ) ∈ g for each V ∈ h. A function f : X → Y is (g, h)continuous at x ∈ X if for each V ∈ h( f (x)) there is U ∈ g(x) such that f (U) ⊂ V . By [17], a function f is (g, h)-continuous if it is such at each point. Denote by C(g,h) ( f ) the family of all (g, h)-continuity points of f . In generally, the set C(g,h) ( f ) can be arbitrary. However, if (Y, d) is a metric space then this set is the countable intersection of sets from g. From now, we will assume that (Y, d) is a metric space. We will use the notion g-continuity for (g, d)-continuity and Cg ( f ) for continuity points C(g,d) ( f ). By [18], H ⊂ P(X) is a hereditary class, if B ⊂ A ∈ H implies B ∈ H. Theorem 9.28 ([11]). Let g be a GT on X and let Y be a metric space. If there is a function f : X → Y such that Cg ( f ) = 0/ and the set f (X) is countable then S there is a hereditary class A ⊂ P(X) such that A ∩ g = {0} / and X = n∈N Xn , where Xn ∈ A for n ∈ N. Theorem 9.29 ([11]). Let g be a GT on X and let (Y, d) be a metric space with at least one accumulation point. Let A ⊂ P(X) be a hereditary class such that S A ∩ g = {0} / and X = i∈N Xi , where Xi ∈ A. Let M ⊂ X. Then M = Cg ( f ) for T some f : X → Y if and only if M = n∈N Mn , where Mn ∈ g and Mn+1 ⊂ Mn for n ∈ N. If A is the family of sets with empty interiors, we obtain Theorem 9.30 ([11]). Let X be an almost resolvable topological space and let (Y, d) be a metric space with at least one accumulation point. Let g be a GT on X such that the interior of A is nonempty for each nonempty A ∈ g. Let M ⊂ X. T Then M = Cg ( f ) for some f : X → Y if and only if M = n∈N Mn , where Mn ∈ g and Mn+1 ⊂ Mn . Let (X, T ) be a topological space. A set A is said to be semi-open if A ⊂ cl (int(A)), pre-open if A ⊂ int (cl(A)), β -open if A ⊂ cl (int (cl(A))) and αopen if A ⊂ int (cl (int(A))). Denote the family of semi-open sets by SO(X), the family of pre-open sets by PO(X), the family of β -open sets by β (X) and the family of α-open sets by α(X). All SO(X), PO(X), β (X) and α(X) are GT’s (in fact, α(X) is a topology). A function f : X → Y (X and Y are topological spaces) is semi-continuous (pre-continuous, β -continuous, α-continuous) at x if for every open neighbourhood V of f (x) there is a set A from SO(X) (PO(X),

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β (X), α(X)) containing x such that f (A) ⊂ V , respectively. Denote by SO( f ), PO( f ), β ( f ), α( f ) the set of all semi-continuity, pre-continuity, β -continuity and α-continuity points of f , respectively. In fact SO( f ) = Q( f ) ([54]). Now, in Thereom 9.30, if g is the family of all open sets in X, we obtain the characterization of continuity points, if g is the family of all semi-open sets, we obtain the characterization of quasicontinuity points, if g is the family of all α-sets, we obtain the characterization of α-continuity points. If M is the family of nowhere dense sets, we obtain the characterization of pre-continuity and β -continuity points on spaces of first category ([11]). A function f : X × Y → Z (X,Y and Z are topological spaces) is said to be quasicontinuous at (x, y) with respect to first (second) coordinate if for all neighbourhoods U, V and W of x, y and f (x, y), respectively, there are nonempty open sets G and H such that x ∈ G ⊂ U, H ⊂ V (G ⊂ U, y ∈ H ⊂ V ) and f (G × H) ⊂ W . A function f is symmetrically quasicontinuous if it is quasicontinuous both with respect to the first and the second coordinate ([56]). Denote by Qsx ( f ), Qsy ( f ) and Qss ( f ) the set of all points at which f is quasicontinuous with respect to first coordinate, quasicontinuous with respect to second coordinate, symmetrically quasicontinuous, respectively. Then C( f ) ⊂ Qss ( f ) = Qsx ( f ) ∩ Qsy ( f ) ⊂ Qsx ( f ) ∪ Qsy ( f ) ⊂ Q( f ) ([20]). For A ⊂ X ×Y and x ∈ X (y ∈ Y ) let Ax = {v ∈ Y : (x, v) ∈ A},

Ay = {u ∈ X : (u, y) ∈ A}.

Denote SO1 (X,Y ) = {A ⊂ X ×Y : if (x, y) ∈ A then y ∈ cl ((int(A))x )} and   SO2 (X,Y ) = {A ⊂ X ×Y : if (x, y) ∈ A then x ∈ cl (int(A))y }}. Then (x, y) ∈ Qsx ( f ) ((x, y) ∈ Qsy ( f ), (x, y) ∈ Qss ( f )) if and only if for each neighbourhood W of f (x, y) there is a set A ∈ SO1 (X,Y ) (A ∈ SO2 (X,Y ), A ∈ SO1 (X,Y ) ∩ SO2 (X,Y )) containing (x, y) such that f (A) ⊂ W ([60]). It is easy to see that SO1 (X,Y ), SO2 (X,Y ) and SO1 (X,Y ) ∩ SO2 (X,Y ) are GT’s. So, according to Theorem 9.30 we obtain this characterization. Theorem 9.31. Let X and Y be topological spaces such that X ×Y is an almost resolvable topological space. Let (Z, d) be a metric space with at least one accumulation point. Let M ⊂ X ×Y . Then M = Qsx ( f ) (M = Qsy ( f ), M = Qss ( f )) for some f : X ×Y → Z if and only if M is the countable intersection of a de-

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creasing sequence of sets from SO1 (X,Y ) (SO2 (X,Y ), SO1 (X,Y )∩SO2 (X,Y )), respectively. Acknowledgements. The paper was supported by Grant VEGA 2/0177/12 and APVV-0269-11.

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JÁN B ORSÍK Mathematical Institute, Slovak Academy of Sciences Grešákova 6, 04001 Košice, Slovakia Katedra fyziky, matematiky a techniky FHPV, Prešovská univerzita v Prešove ul. 17. novembra 1, 08001 Prešov, Slovakia E-mail: [email protected], [email protected]

Chapter 10

On Extension Problem, Decomposing and Covering of Functions

ZBIGNIEW GRANDE, MARIOLA MARCINIAK

2010 Mathematics Subject Classification: 26A15, 26A21, 54C20, 54C30. Key words and phrases: extensions of maps, countable continuity, covering and decomposition of functions.

In this chapter, we review some problems related to extension, decomposition and covering of functions. We mainly do not give proofs of the results stated here.

10.1 An Extension Problem If (X, T ) is a topological normal space, then by well known Tietze extension theorem, for each nonempty closed set A ⊂ X and every continuous function g : A → [0, 1] there is a continuous function f : X → [0, 1] such that f  A = g. Let H ⊂ G be nonempty families of functions from X to Y and let A ⊂ X be a nonempty subset of X. A map W : G → H is said to be an extension operator from A onto X if the restrictions f  A and W ( f )  A are equal for each function f ∈ G. So, from Tietze theorem it follows that in the case of topological normal space X, for each nonempty closed set A ⊂ X, the family G of all real functions with restrictions to A being continuous and the family H of all continuous

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functions from X to R, there is an extension (from A onto X) operator W : G → H. We recall that a function h : A → R, where A ⊂ R is quasi-continuous at a point x0 ∈ A if for each positive real η > 0 and for each open set U containing x0 , there is an open set W ⊂ U such that W ∩ A 6= 0/ and h(W ∩ A) ⊂ (h(x0 ) − η, h(x0 ) + η). The function f is quasi-continuous if it is quasi-continuous at each point (see [38]). Denote by LA ( f , x) the set of limit numbers of a function f at x over the ˜ : ∃{x }⊂A\{x} xn → x ∧ f (xn ) → α}). Observe set A (i.e. LA ( f , x) = {α ∈ R n that if f : R → R is quasi-continuous and the set A is dense then for all x ∈ R, f (x) ∈ LA ( f , x). Consider the function f (x) = 1x for x 6= 0 and f (0) = 0. The function f is continuous on the set A = R \ {0} which is dense, but LA ( f , 0) ⊂ {−∞, +∞} so there is no quasi-continuous function g : R → R with g  A = f  A. Remark 10.1. Let A ⊂ R be a dense set and f : A → R be a quasi-continuous function. The function f can be extended on R to a quasi-continuous function g : R → R if and only if for every x ∈ R \ A, LA ( f , x) \ {−∞, +∞} 6= 0. / Proof. If there is a quasi-continuous function g : R → R such that g  A = f then for each x ∈ R \ A we have g(x) ∈ LA ( f , x) \ {−∞, +∞}. Now suppose that for every x ∈ R \ A, LA ( f , x) \ {−∞, +∞} 6= 0. / Let g : R → R be such that g(x) = f (x) for x ∈ A and g(x) ∈ LA ( f , x) \ {−∞, +∞} for x 6∈ A. Then the function g is quasi-continuous. t u Example 10.2. Let A ⊂ R be a dense set, G be the family of all functions f : R → R quasi-continuous on A and such that LA ( f , x) \ {−∞, +∞} 6= 0/ for each x ∈ R \ A, and H ⊂ G be the family of all quasi-continuous functions. By Remark 10.1 there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H. Remark 10.3. Let A = R \ {0}. We put: f1 (x) = 0 and f2 (x) = n for 1 1 1 1 x ∈ ( 2n , 2n−1 ], n = 1, 2, . . .; f1 (x) = n and f2 (x) = 0 for x ∈ ( 2n+1 , 2n ] for n = 1, 2, . . .; f1 (x) = f1 (−x), f2 (x) = f2 (−x) for x ∈ [−1, 0) and f1 (x) = f2 (x) = 0 for x ∈ (−∞, −1) ∪ (1, ∞). Then f1 , f2 : A → R are quasicontinuous; f1 and f2 can be extended onto R to quasi-continuous functions (we can take f1 (0) = 0 = f2 (0)), but LA ( f1 + f2 , 0) = {+∞} so f1 + f2 cannot be extended onto R to a quasi-continuous function. By Remark 10.3 the space G from Example 10.2 is not a linear space. Now we will consider families G of real functions which form linear spaces with

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the natural operations of addition of functions and multiplication by reals and discuss the problem of existence of linear extension operators. Let G be some linear space of bounded real functions defined on a topological space X, with the norm || f || = supx∈X | f (x)| for f ∈ G. Let H ⊂ G and A ⊂ X be a nonempty set. Suppose that for each function f ∈ G there is a function g : X → I f (where I f = [inf f , sup f ]) belonging to H such that f  A = g  A. Then there is an extension operator W : G → H from A onto X satisfying the condition ||W ( f )|| ≤ || f ||, f ∈ G, and such that W ( f ) = f for every f ∈ H, but this operator may be not linear. Observe that if there are functions f , g ∈ H ⊂ G with f + g ∈ G \ H then there is no a linear operator W : G → H such that W (h) = h for h ∈ H. Indeed, for such f , g we have W ( f + g) = W ( f ) +W (g) = f + g ∈ G \ H, contrary to W (G) ⊂ H. Remark 10.4 ([17]). Let A ⊂ R be a nonempty nowhere dense set. For each function f : R → R there is a quasi-continuous function g : R → R such that f  A = g  A and g is continuous at each point x ∈ R \ cl(A) . Moreover, if f is Lebesgue measurable (resp. of Baire α class, α ≥ 1), then g may have the same property. Example 10.5. Let A ⊂ R be a nonempty nowhere dense set. Assume that G is the family of all functions f : R → R and H is the family of all quasicontinuous functions, continuous on R \ cl(A). By Remark 10.4 there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H. Since the sum of two quasi-continuous functions may belong to G \ H, such operator cannot be linear. Example 10.6. Let A ⊂ R be a nonempty nowhere dense set. Assume that G is the family of all Lebesgue measurable functions f : R → R and H is the family of all Lebesgue measurable quasi-continuous functions, continuous on R \ cl(A). By Remark 10.4 there is an extension operator W : G → H such that W ( f ) = f for f ∈ H. Since the sum of two Lebesgue measurable quasicontinuous functions may belong to G \ H, such operator cannot be linear. Example 10.7. Let A ⊂ R be a nonempty nowhere dense set. Assume that G is the family of all functions f : R → R from Baire α class (α ≥ 1) and H is the family of all quasi-continuous functions of Baire α class, that are continuous on R \ cl(A). By Remark 10.4 there is an extension operator W : G → H such that W ( f ) = f for f ∈ H. Since the sum of two Baire α, quasi-continuous functions may belong to G \ H, such operator cannot be linear.

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Example 10.8. Let G be the family of all approximately continuous functions f : R → R and let H be the family of all approximately continuous and quasicontinuous functions g : R → R. Let A ⊂ R be a nonempty nowhere dense set of Lebesgue measure zero. It is known ([17]) that for every function f ∈ G there is a function g ∈ H such that f  A = g  A. So there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H. Since there is approximately continuous function f : R → R, that is not quasi-continuous and for each function f ∈ G there are functions f1 , f2 ∈ H with f = f1 + f2 , such operator W cannot be linear. Example 10.9. Let G be the family of all Baire 1 functions f : R → R and let H be the family of all Baire 1 and quasi-continuous functions. Let A ⊂ R be a nonempty nowhere dense set. It is known ([17]) that for every function f ∈ G there is a function g ∈ H such that f  A = g  A. So there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H. Since there is a Baire 1 function that is not quasi-continuous, and since for every function f ∈ G there are functions f1 , f2 ∈ G with f = f1 + f2 , such operator W cannot be linear. Remark 10.10. Let a nonempty Borel set A ⊂ R be such that its complement R \ A is c-dense in R. Using the similar construction as applied in the articles [14],[44] we can show that for each Lebesgue measurable (resp. having Baire property) function f : R → R there is a Lebesgue measurable (resp. with the Baire property) Darboux function g : R → R such that f  A = g  A. Similarly, for each function (resp. of Baire α class, where α ≥ 1 function), f : R → R there is a Darboux (resp. of Darboux and Baire α class) function g : R → R such that f  A = g  A. Example 10.11. Let a nonempty Borel set A ⊂ R be such that its complement R \ A is c-dense in R. Assume that G is the family of all Lebesgue measurable functions f : R → R and H is the family of all Lebesgue measurable Darboux functions g : R → R. Then G is a linear space, but for each f ∈ G there are two functions f1 , f2 ∈ H such that f = f1 + f2 . By Remark 10.10 there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H but it cannot be linear. Example 10.12. Let a nonempty Borel set A ⊂ R be such that its complement R \ A is c-dense in R. Assume that G is the family of all functions f : R → R from Baire α class, where α ≥ 1, and H is the family of all Darboux functions g : R → R of Baire α class. Then G is a linear space, but for each f ∈ G there are two functions f1 , f2 ∈ H such that f = f1 + f2 . By Remark 10.10 there is

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an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H, but it cannot be linear. Example 10.13. Let a nonempty Borel set A ⊂ R be such that its complement R \ A is c-dense in R. Assume that G is the family of all functions f : R → R with the Baire property and H is the family of all Darboux functions g : R → R with the Baire property. Then G is a linear space, but for each f ∈ G there are two functions f1 , f2 ∈ H such that f = f1 + f2 . By Remark 10.10 there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H but it cannot be linear. Example 10.14. Let a nonempty Borel set A ⊂ R be such that its complement R\A is c-dense in R. Assume that G is the family of all functions f : R → R and H is the family of all Darboux functions g : R → R. Then G is a linear space, but for each f ∈ G there are two functions f1 , f2 ∈ H such that f = f1 + f2 . By Remark 10.10 there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H but it cannot be linear. In the next example we will consider cliquish functions. Recall that a function f : R → R is cliquish if the set C( f ) of all its continuity points is dense. As above, the following remark is true. Remark 10.15. Let a nonempty Borel set A ⊂ R be such that its complement R \ A is c-dense in R. Using a similar construction as applied in the article [14] we can show that for each cliquish function f : R → R there is a Darboux cliquish function g : R → R such that g  A = f  A Example 10.16. Let a nonempty Borel set A ⊂ R be such that its complement R \ A is c-dense in R. Assume that G is the family of all cliquish functions f : R → R and H is the family of all Darboux cliquish functions g : R → R. Then G is a linear space, but for each f ∈ G there are two functions f1 , f2 ∈ H such that f = f1 + f2 ∈ G \ H. By Remark 10.15 there is an extension operator W : G → H from A onto X such that W (h) = h for h ∈ H but it cannot be linear. A general construction for linear spaces Now we consider the case in which H ⊂ G are families of real functions being both linear spaces with the natural operations of addition of functions and multiplication by reals. Proposition 10.17. Let H ⊂ G be linear spaces of real functions. Let A ⊂ X be a nonempty set and suppose that for every function g ∈ G there is a function f : X → R belonging to H such that f  A = g  A. Then there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H.

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Proof. Without loss of generality we can suppose that H 6= G and that H contains elements different from 0. Let B(H) be a basis of the linear space H and let B(G), such that B(G) ⊃ B(H) is a basis of the linear space G. For f ∈ B(H) we put W ( f ) = f and for g ∈ B(G) \ B(H) we take W (g) to be an element of H such that g  A = W (g)  A. Next, if h ∈ G is of the form h = ∑ki=1 ri hi , where hi ∈ B(G) and ri ∈ R for i = 1, 2, . . . , k, then we put W (h) = ∑ki=1 riW (hi ). Evidently, the operator W is well defined on G and its values belong to H. It is also linear. Since f = 1 · f , for f ∈ H, we have for such an f that W ( f ) = 1 ·W ( f ) = f . t u Example 10.18. Let X be a topological normal space and let A ⊂ X be a nonempty closed set. Denote by H the family of all continuous functions f : X → R and by G the family of all functions g : X → R, whose restrictions g  A are continuous. Then by the Tietze Theorem and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Recall that an F-space is a vector space V over the real (or complex) numbers together with a metric d : V ×V → R so that scalar multiplication in V is continuous with respect to d and the standard metric on R (or on C), addition in V is continuous with respect to d, the metric is translation-invariant and the metric space (V, d) is complete. Remark 10.19. Let H ⊂ G be families of real functions satisfying all requirements of Proposition 10.17. If G is an F-space and H has a complement in G, then there is a continuous linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Proof. Let W be a linear extension operator from A onto X such that W ( f ) = f for every f ∈ H. Because W 2 = W , the operator W obtained in the above construction is a projection. Because G is an F-space and H has a complement in G, we obtain that W is continuous. t u Let (X, ρ) be a metric space and A ⊂ X. We say that a function ω f : [0, ∞) → [0, ∞) is a modulus of continuity of the function f : A → R if | f (x1 ) − f (x2 )| < ω f (t) for all x1 , x2 ∈ A

with ρ(x1 , x2 ) < t

In [30] M. J. McShane proved the following theorem: Theorem 10.20. (McShane) If the function f defined on a subset A of metric space X has a concave modulus of continuity ω f such that lim ω f (t) = 0, then t→0

f can be extended to X preserving the modulus of continuity.

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Example 10.21. Let A be a nonempty set in a metric space (X, ρ). Denote by H the family of all functions f : X → R for which there exists some concave modulus of continuity ω f such that lim ω f (t) = 0. Let G be the family of all t→0

functions g : X → R, whose restrictions to A have a modulus of continuity ω f which is concave and such that lim ω f (t) = 0. Then by Theorem 10.20 and t→0

Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Let A be a nonempty set in a metric space (X, ρ). We say that a function f : A → R satisfies Hölder condition if | f (x1 ) − f (x2 )| ≤ M(ρ(x1 , x2 ))α

f or

x1 , x2 ∈ A

for some constants M > 0 and α > 0. Example 10.22. Let A be a nonempty set in a metric space (X, ρ) and 0 < α ≤ 1. Denote by H the family of all functions f : X → R satisfying Hölder condition with the exponent α and by G the family of all functions g : X → R whose restrictions g  A satisfy the Hölder condition with α on the set A. By McShane Corollary 1 from [30] and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Example 10.23. Let A be a nonempty set in a metric space (X, ρ). Denote by H the family of all bounded uniformly continuous functions and by G the family of all functions g : X → R whose restrictions g  A are bounded and uniformly continuous. By Corollary 2 from [30] and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. We recall that a vector L ∈ Rn is a derivative of a function f : F → R at a point a ∈ F ⊂ Rn if either a is an isolated point of F, or | f (x) − f (a) − (L, (x − a))| −→ 0 as x −→ a, x ∈ F, |x − a| where (x, y) denotes the scalar product of vectors x, y ∈ Rn . The vector L is a strict derivative of f at a point a ∈ F if either a is an isolated point of F, or | f (y) − f (x) − (L, (y − x))| −→ 0 as x, y −→ a, |y − x| (x, y ∈ F, x 6= y; x = a or y = a is allowed).

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In [3] the authors prove the following theorem: Theorem 10.24 (Aversa, Laczkovich, Preiss). Let F ⊂ Rn be a nonempty closed set, f : F → R and L : F → Rn be functions such that for each a ∈ F the vector L(a) is a derivative of f at a. Then f can be extended to an everywhere differentiable function g : Rn → R such that g0 = L on F if and only if the map L : F → Rn is Baire 1. In [28] the authors show the following theorem. Theorem 10.25 (Koc-Zaji´cek). Let F ⊂ Rn be a nonempty closed set, f : F → R and L : F → Rn be functions such that for each a ∈ F the vector L(a) is a derivative of f at a. Moreover, suppose that the mapping L : F → Rn is Baire 1. Then f can be extended to an everywhere differentiable function g : Rn → R such that g is C∞ on Rn \ F, the derivative g0 = L on F and g0 is continuous at all points a ∈ F at which L is continuous and L(a) is a strict derivative of f at a. These theorems are used in the next examples. Example 10.26. Let X = Rn and let A ⊂ Rn be a nonempty closed set. Denote by H the family of all differentiable functions g : Rn → R, and by G the family of all functions f : Rn → R, whose restrictions f  A are differentiable and its derivatives ( f  A)0 are Baire 1. By Theorem 10.24 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Example 10.27. Let X = Rn and let A ⊂ Rn be a nonempty closed set. Denote by H the family of all differentiable functions g : Rn → R, that are C∞ on Rn \ A, and by G the family of all functions f : Rn → R, whose restrictions f  A are differentiable and its derivatives ( f  A)0 are Baire 1. By Theorem 10.24 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Let A ⊂ Rn and x ∈ Rn . A vector v ∈ Rn is called tangent to A if there exist ∞ {xk }∞ k=1 ⊂ A and {αk }k=1 ⊂ [0, ∞) such that xk → x and αk (xk − x) → v. The set of all tangent vectors to A is called the contingent cone of A at x and will be denoted by Tan(A, x). In [3] the authors proved that for a function f : A → R differentiable at x, the derivative f 0 (x) is determined uniquely iff Tan(A, x) spans Rn . Remark 10.28 ([3]). There is a nonempty compact set A ⊂ R2 and a function f : A → R such that the contingent cone Tan(A, x) spans R2 for every x ∈ A and f has a derivative everywhere on A, but the derivative is not Baire 1 and thus f cannot be extended to R2 as an everywhere differentiable function.

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Theorem 10.29 ([28]). Let A ⊂ Rn be a nonempty closed set such that Tan(A, x) spans Rn for every x ∈ A. If a function f : A → R has a strict derivative everywhere on A, then f can be extended to Rn as an everywhere differentiable function. Remark 10.30 ([28]). There is a nonempty compact set A ⊂ R2 and a function f : A → R such that Tan(A, x) spans R2 for every x ∈ A and f has a strict derivative everywhere on A, but f cannot be extended to R2 as an everywhere continuously differentiable function. Example 10.31. Let A ⊂ Rn be a nonempty closed set such that Tan(A, x) spans Rn for each x ∈ A. Denote by G the set of all functions f : Rn → R, whose restrictions f  A have a strict derivative everywhere on A, and by H the family of all differentiable functions belonging to G. By Theorem 10.29 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. In the next examples we use the ordinary differentiation basis in Rn (see [11] and [45]). Recall that a point x ∈ Rn is a density point of a Lebesgue measurable set E ⊂ Rn if lim

r→0+

λn (E ∩ Q(x, r)) = 1, λn (Q(x, r))

where Q(x, r) denotes the cube with the center x and the length of edge equal r, and λn denotes the Lebesgue measure in Rn . Moreover, x is a density point of an arbitrary set H ⊂ Rn if there is a Lebesgue measurable set E ⊂ H such that x is a density point of E. The family Td of all sets M ⊂ Rn such that every point x ∈ M is a density point of M is a topology called the ordinary density topology. All sets belonging to Td are Lebesgue measurable. If Tnat denotes the natural topology in R then the ordinary approximate continuity of a function f : Rn → R denotes the continuity of f as a map from (Rn , Td ) into (R, Tnat ). A function f : Rn → R, locally Lebesgue integrable at a point x ∈ Rn , is said to be an ordinary derivative at x if R

lim+

r→0

Q(x,r)

f (t)dt

λn (Q(x, r))

= f (x).

It is well known that a Lebesgue measurable function f , locally bounded at x and approximately continuous at x is an ordinary derivative at x. All ordinary derivatives and all approximately continuous functions are Baire 1. Theorem 10.32 ([2]). If a set A ⊂ Rn has Lebesgue measure 0 and a function f : Rn → R is Baire 1 then there is an ordinary derivative g : Rn → R and an

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approximately continuous function h : Rn → R such that f  A = g  A = h  A. Moreover, if the image f (A) is contained in a closed interval K then g and h can be chosen so that their images are contained in K. Example 10.33. Let X = Rn and let A ⊂ Rn be a nonempty set of Lebesgue measure 0. Denote by G the family of all Baire 1 functions, and by H the family of all ordinary derivatives. By Theorem 10.32 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Example 10.34. Let X = Rn and let A ⊂ Rn be a nonempty set of Lebesgue measure 0. Denote by G the family of all Baire 1 functions, and by H the family of all approximately continuous functions. By Theorem 10.32 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Example 10.35. Let X = Rn and let A ⊂ Rn be a nonempty set of Lebesgue measure 0. Denote by G the family of all bounded Baire 1 functions, and by H the family of all bounded ordinary derivatives. By Theorem 10.32 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Example 10.36. Let X = Rn and let A ⊂ Rn be a nonempty set of Lebesgue measure 0. Denote by G the family of all bounded Baire 1 functions, and by H the family of all bounded approximately continuous functions. By Theorem 10.32 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Remark 10.37. Let a nonempty set A ⊂ Rn be such that its closure cl(A) is of Lebesgue measure zero. From the proof of Theorem 3 in [13] it follows that for each Baire 1 function f : Rn → R there is an approximately continuous function g : Rn → R, that is continuous at every point x ∈ Rn \ cl(A) and such that f (x) = g(x) for x ∈ A. Example 10.38. Let A ⊂ Rn be a nonempty set, whose closure cl(A) is of Lebesgue measure zero, H be the family of all approximately continuous functions, continuous at all points x ∈ Rn \ cl(A) and let G be the family of all Baire 1 functions. By Remark 10.37 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Remark 10.39. The above results concerning extensions of Baire 1 functions to approximately continuous functions and extensions of Baire 1 functions to

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almost everywhere continuous, approximately continuous functions were generalized for functions defined on some special metric spaces with measures (see [12]). In [24] the author considered the notion of B1 -retracts. A subset A of a topological space X is a B1 -retract of X if and only if for any topological space Y every continuous function f : A → Y can be extended to a Baire 1 function g : X → Y. Example 10.40. Let X be a topological space and let A ⊂ X be a B1 -retract of X. Denote by G the family of all functions f : X → R, whose restrictions to A are continuous, and by H the family of all Baire 1 functions g : X → R whose restrictions to A are continuous. Then the families G and H are linear spaces and by Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Remark 10.41. Let A ⊂ [a, b] be a nonempty set whose complement [a, b] \ A is countable and let f : [a, b] → R be a function with a bounded variation, continuous at each point x ∈ A. There is a right-continuous (resp. left-continuous) function f1 : [a, b] → R with a bounded variation and continuous at each point x ∈ A and such that f  A = f1  A. Proof. Since f is of bounded variation, there are two increasing functions g, h : [a, b] → R such that f = g − h and g, h are continuous at each point x ∈ A. If u ∈ [a, b) is a discontinuity point of g then we put g1 (u) = inf{g(t);t ∈ (u, b)}. For other points u ∈ [a, b] we put g1 (u) = g(u). Then the function g1 is rightcontinuous and continuous at each point x ∈ A, where g1 (x) = g(x). Similarly, if u ∈ [a, b) is a discontinuity point of h then we put h1 (u) = inf{h(t);t ∈ (u, b)}. For other points u ∈ [a, b] we put h1 (u) = h(u). Then the function h1 is right-continuous and continuous at each point x ∈ A, where h1 (x) = h(x). Moreover, the functions g1 and h1 are increasing. So the function f1 = g1 − h1 has bounded variation on [a, b], is right-continuous and f1  A = f  A. The case of left-continuity is similar. t u Example 10.42. Let A ⊂ [a, b] be a nonempty set such that its complement [a, b] \ A is countable. Denote by G the family of all functions f : [a, b] → R with a bounded variation on [a, b], that are continuous at each point x ∈ A. Moreover, let H be the family of all right-continuous (resp. left-continuous) functions g : [a, b] → R with bounded variation on [a, b] and continuous at each point x ∈ A. By Remark 10.41 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H.

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In [15] there was introduced the following classes of functions. A function f : R → R has property A3 at a point x if for each positive real r and each set U ∈ Td containing x there is an open interval I such that C( f ) ⊃ I ∩U 6= 0/ and | f (t) − f (x)| < r for t ∈ I ∩ U, where C( f ) denotes the set of all continuity points of f . A function f has property A3 if it has property A3 at each point x ∈ R. A function f : R → R has property A5 if for each nonempty set U ∈ Td there is an open interval I such that 0/ 6= I ∩U ⊂ C( f ). These classes were considered as some special notions of quasi- continuity and cliquishness using two topologies: natural topology Tnat and density topology Td . All functions of these classes are almost everywhere continuous and are very useful for considerations related to measurability of functions of two variables. For example there is a nonmeasurable (in the sense of Lebesgue) function f : R2 → R with approximately continuous vertical sections fx , for x ∈ R, and measurable horizontal sections f y , for y ∈ R, while the property A3 of vertical sections fx , for x ∈ R, and measurability of horizontal sections f y , for y ∈ R, imply the measurability of function f . Example 10.43. Denote by H the family of all functions f : R → R having property Ai (where i = 3 or i = 5) and by G the family of all almost everywhere continuous functions g : R → R. Let A ⊂ R be a nonempty set whose closure cl(A) is of Lebesgue measure 0. Then the families G and H are linear spaces and by Theorem 1 in [16] and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Remark 10.44. Let X be a metric space and let A ⊂ X be a nonempty Gδ -set. It is well known (see [29]) that for every Baire 1 function f : A → R there is a Baire 1 function g : X → R such that f  A = g  A. Example 10.45. Let X be a metric space and let A ⊂ X be a nonempty Gδ -set. Denote by G the family of all functions f : X → R whose restrictions to A are Baire 1 functions and by H the family of all Baire 1 functions g : X → R. Then families G and H are linear spaces and by Remark 10.44 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H. Remark 10.46 ([23]). Let X be a completely regular topological space and let A be a nonempty Lindelöf hereditarily Baire subset of X. If f : A → R is a Baire 1 function then there is a Baire 1 function g : X → R such that f  A = g  A. Example 10.47. Let X be a completely regular topological space and let A be a nonempty Lindelöf hereditarily Baire subset of X. Denote by G the family

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of all functions f : X → R whose restrictions to A are Baire 1 and by H the family of all Baire 1 functions g : X → R. Then families G and H are linear spaces and by Remark 10.46 and Proposition 10.17 there is a linear extension operator W : G → H from A onto X such that W ( f ) = f for every f ∈ H.

10.2 Decomposing and Covering of Functions Decomposing and covering of functions by continuous functions Let X,Y be topological spaces and k be a positive integer. We say that S f : X → Y is k-continuous if there exist sets {Xn }kn=1 such that X = kn=1 Xn and f  Xn is continuous for n = 1, . . . k. We say that f is finitely continuous if it is k- continuous for some k. We say that f is countably continuous if it is decomposable into countably many continuous functions, i.e. if there exists S∞ a sequence {Xn }∞ n=1 such that X = n=1 Xn and f  Xn is continuous for all n. We say that f is strongly k-continuous if its graph can be covered by k graphs of continuous functions, i.e. there exist continuous functions fi : X → Y , S i = 1, . . . k such that Gr( f ) ⊂ ki=1 Gr( fi ) where Gr( f ) denotes the graph of a function f . We say that f is strongly finitely continuous if it is strongly kcontinuous for some k. We say that f is strongly countably continuous if its graph can be covered by graphs of countably many continuous functions, i.e. if there exists the sequence { fi }∞ i=1 of continuous functions f i : X → Y such S∞ that Gr( f ) ⊂ i=1 Gr( fi ). In [39] R. J. O’Malley introduced the class of Baire-one-star function. We say that f ∈ B1∗ if for any nonempty closed set F ⊂ X there is an open set U ⊂ X such that U ∩ F 6= 0/ and the restriction f  F ∩U is continuous. If X is a complete metric space then functions from class B1∗ are of first Baire class. Moreover, f ∈ B1∗ iff it is piecewise continuous, i.e. there exists a S∞ sequence of nonempty closed sets {Xn }∞ n=1 such that X = n=1 Xn and f  Xn is continuous (see [22], [25]). Of course from Tietze Theorem we conclude that a piecewise continuous function is strongly countably continuous. It is easy to see that finitely continuous functions f : R → R have a nowhere dense graph, but of course even 2-continuous functions can be discontinuous everywhere (consider for example the Dirichlet function). In [40] R. J. Pawlak considered a nice subclass B1∗∗ of 2-continuous functions (we say that f belongs to B1∗∗ if is continuous or f  D( f ) is continuous, where D( f ) denotes the set of all discontinuity points of f ). He proved that if f ∈ B1∗∗ then the set of discontinuity points of f must be nowhere dense and B1∗∗ B1? .

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In [32] the author proves that the Darboux real function defined on a locally connected metric space is 2-continuous if and only if it belongs to the class B1∗∗ . In this paper we give an example of 3-continuous Darboux function that is not in the first class of Baire. However, in the case f : R → R (similar to that for functions from DB ∗1 ), if f is finitely continuous Darboux function then its set of discontinuity points must be nowhere dense and the set f (R) \ f (int(C( f ))) must be nowhere dense too [32], [33]. We say that f : Rn → Rk is a Hamel function if its graph is a Hamel basis for n+k R . Many authors considered Hamel functions with some nice properties. In [10] the authors show an example of Marczewski measurable Hamel function and an example of Hamel function which is both Lebesgue measurable and with the Baire property. In [36] T. Natkaniec gives an example of a quasicontinuous Hamel function. An example of finitely continuous Hamel function is given in the paper [43]. It is well known that for every measurable function f : I → R (where I is an S∞ interval) there exists a sequence of measurable sets {Xn }∞ n=1 such that n=1 Xn has full measure and f  Xn is continuous for all n. Lusin asked if any Borel function is necessarily countably continuous. The answer is negative and many authors give counterexamples (see [1], [9], [34], [46]). An interesting counterexample is the Lebesgue measure: in the paper [26] S. Jackson and R. D. Mauldin proved that the Lebesgue measure λ considered on the space of nonempty closed subsets of the unit interval with Hausdorff metric is not countably continuous. We say that f : R → R is symmetrically continuous if lim( f (x + t) − f (x − t)) = 0

t→0

f or

x ∈ R.

K. Ciesielski gives an example of symmetrically continuous function that is not countably continuous. The author uses the notion of Sierpi´nski function. The function f : A → R (where A ⊂ R) is of Sierpi´nski type if f  Y is discontinuous for every set Y ⊂ A of cardinality continuum. In the paper [7] the author observes that if there exists A ⊂ R of cardinality continuum, such that f  A is Sierpi´nski-Zygmund type, then f is not countably continuous. Next the author constructed a symmetrically continuous function f : R → R that "contains" a Sierpi´nski-Zygmund type function i.e. for some A ⊂ R of cardinality continuum f  A is a Sierpi´nski-Zygmund type. In the paper [47] S. Solecki proves the following dichotomy result: a Baire one function is countably continuous or "contains" (to be explained in detail in below) some complicated function. Let g : X1 → Y1 and f : X2 → Y2 . We write

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g v f if there exist embeddings (i.e. open continuous injections) φ : X1 → X2 and ψ : g[X1 ] → Y2 with ψ ◦ g = f ◦ φ . Recall that any ordinal number can be considered as a topological space by endowing it with the order topology. Let ω denote the first infinite ordinal. Define the Pawlikowski function P : (ω + 1)ω → ω ω by p(η) = γ, where  0 if η(n) = ω, γ(n) = η(n) + 1 if η(n) < ω. Theorem 10.48 ([47]). Let X be a Souslin space, Y be a separable metric space, and f : X → Y be a Baire one function. Then either f is countably continuous or P is "contained" in f, (i.e. P v f ). This result was generalized by Pawlikowski and Sabok. In the paper [42] the autors prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. In [47] the authors consider also decomposition of Baire one functions into continuous functions with closed domain, i.e. piecewise continuity and prove similar dichotomy result using Lebesgue functions L1 and L2 defined as follows. Let Q be the set of all points in 2ω that are eventually equal to 1. For each x ∈ Q fix a number ax > 0 so that: 1. if x, y ∈ Q, x 6= y, then ax 6= ay , 2. ax < 31n0 where n0 is the smallest natural number such that x(n) = 1 for n ≥ n0 . ω Let H : 2ω → [0, 1] be the embedding H(x) = 3x(n) n+1 , for x ∈ 2 . ω The functions L1 , L : 2 → R are defined by:  H(x) if x 6∈ Q, L(x) = H(x) + ax if x ∈ Q,  0 if x 6∈ Q, L1 (x) = ax if x ∈ Q.

Theorem 10.49 ([47]). Let X be a separable complete metric space, Y be a separable metric space, and f : X → Y be a Baire one function. Then either f is piecewise continuous or one of L, L1 is "contained" in f (i.e. L v f or L1 v f ).

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Strongly countably continuous functions were investigated in [20], [21], [37]. Let f be a monotone function that is discontinuous on a dense countable set (for example f (x) = ∑qn −2} and B∗ = R \ {n ∈ Z : n < 2} are closed. However, A∗ ∪ B∗ = R \ {−1, 0, 1} is not closed because {−1, 0, 1} 6∈ γℵ0 . Moreover, it is easy to see that intγℵ0 (R) = Q and clγℵ0 (0) / = R \ Q. From Property 1.3, Property 1.7 ([9]) and Lemma 1.1 ([10]) we obtain Property 11.1. If (X, γ) is GTS then (I.1) if A ⊂ B ⊂ X then intγ (A) ⊂ intγ (B), (I.2) intγ (A) ⊂ A for any A ⊂ X, (I.3) intγ (intγ (A)) = intγ (A) for any A ⊂ X, (C.1) if A ⊂ B ⊂ X then clγ (A) ⊂ clγ (B),

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(C.2) A ⊂ clγ (A) for any A ⊂ X, (C.3) clγ (clγ (A)) = clγ (A) for any A ⊂ X. Clearly, clγ (A) ∪ clγ (B) ⊂ clγ (A ∪ B) and intγ (A ∩ B) ⊂ intγ (A) ∩ intγ (B). However, there exists GTS such that the above two inclusions are proper. Indeed, if we consider the space (R, γℵ0 ) and sets A∗ , B∗ defined above, we obtain clγℵ0 (A∗ ) ∪ clγℵ0 (B∗ ) = A∗ ∪ B∗ = R \ {−1, 0, 1}. Simultaneously, clγℵ0 (A∗ ∪ B∗ ) = R. Moreover, intγℵ0 (R \ A∗ ) ∩ intγℵ0 (R \ B∗ ) = (R \ A∗ ) ∩ (R \ B∗ ) = / {−1, 0, 1} and intγℵ0 ((R \ A∗ ) ∩ (R \ B∗ )) = 0.

11.1.2 Nowhere densities and Baire spaces In case of topological space (X, τ), we can define a nowhere dense set in the following way: a set A ⊂ X is nowhere dense if intτ (clτ (A)) = 0. / This condition is equivalent to the following one: for any nonempty set U ∈ τ there exists a nonempty set V ∈ τ such that V ⊂ U and V ∩ A = 0. / However, in the case of generalized topological spaces these conditions lead to different concepts. Let (X, γ) be GTS and A ⊂ X. We shall say that A is a nowhere dense set if intγ (clγ (A)) = 0/ (e.g. [6], [31]). It is easy to see that a subset of a nowhere dense set is nowhere dense. However, a union of two nowhere dense sets does not have to be nowhere dense. Indeed, put X = [0, 2] and K = {0} / ∪ {[0, b) : b ∈ (0, 2]} ∪ {(a, 2] : a ∈ [0, 2)}. Obviously intγK (clγK ({0})) = 0/ and intγK (clγK ({2})) = 0, / so {0} and {2} are nowhere dense sets. Moreover, it is easy to see that intγK (clγK ({0, 2})) = [0, 2]. It implies that {0, 2} is not a nowhere dense set. What is more, for [0, 1) ∈ γK there is no set V ∈ γeK such that V ⊂ [0, 1) and V ∩ {0} = 6 0. / Therefore {0} is a nowhere dense set that does not satisfy the condition: for any U ∈ γeK there exists V ∈ γeK such that V ⊂ U and V ∩ {0} = 0. / For this reason, in the case of generalized topological space (X, γ) we introduce a new type of set. We shall say that A ⊂ X is a strongly nowhere dense set if for any U ∈ γe there exists V ∈ γe such that V ⊂ U and V ∩ A = 0. / The following statement shows a significant difference in the properties of nowhere dense sets and strongly nowhere dense sets in generalized topological spaces. Theorem 11.2. (a) There exists GTS (X, γ) and nowhere dense sets A, B ⊂ X such that A ∪ B is not a nowhere dense set.

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(b) For every two strongly nowhere dense sets A and B in an arbitrary GTS (X, γ) the union A ∪ B is a strongly nowhere dense set. Analogously to the topological case, we shall say that A ⊂ X is a meager (s-meager) set if there exists a sequence {An }n∈N of nowhere dense (strongly S nowhere dense) sets such that A = n∈N An . A set A is called second category (s-second category) set if it is not a meager (s-meager) set. A set A is said to be residual (s-residual) if X \ A is meager (s-meager). The above considerations lead to the different notions connected with the notion of Baire space in standard topological spaces. We shall say that GTS (X, γ) is • a weak Baire space (in short wBS) if each set U ∈ γe is an s-second category set, • a Baire space (in short BS) if each U ∈ γe is a second category set, • a strong Baire space (in short sBS) if V1 ∩ · · · ∩ Vn is a second category set for any V1 ,V2 , . . .Vn ∈ γ such that V1 ∩ · · · ∩Vn 6= 0. / Obviously, if we consider a topological space (X, τ) instead of a generalized topological space (X, γ) then the above notions are equivalent. In the case of generalized topological spaces they are not. It is easy to see that if GTS is a strong Baire space then it is a Baire space and a weak Baire space. Further, each Baire space is a weak Baire space. The converse implications are not true. The detailed considerations of these relationships (presented in Figure 11.1) are contained in [25] sBS
0.

1

Bowen-Dinaburg concept of entropy will be presented in section 11.3.

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Usually the base of logarithms is chosen either as 2 or as e. In fact, it does not matter which base we choose, as long as we use the same base greater then 1 all the time. To illustrate this definition, consider the function f : [0, 1] → [0, 1] from Figure 11.2 and γ f = {A ⊂ [0, 1] : A → A}. Obviously γ f is GT, A1 6∈ γ f f

1 A5 A4 A3 A2 A1 0 A1

A2

A3

A4

A5 1

Fig. 11.2 The graph of same function f : [0, 1] → [0, 1].

and Ai ∈ γ f for i ∈ {2, . . . , 5}. Moreover, for any i ∈ {1, . . . , 5} the set Ai is closed, because f ([0, 1] \ Ai ) ⊃ [0, 1] \ Ai . Putting K0 = {A1 , A2 , A3 , A4 , A5 } we have that A∗ = (A1 , A2 , A3 , A4 , A5 ) ∈ πK0 , A∗∗ = (A1 , A2 , A5 ) ∈ πK0 and A∗∗∗ = (A3 , A4 ) ∈ πK0 . Then   01100   0 1 1 1 0   0 1 0   11     MA∗ , f =  0 0 1 1 0  , MA∗∗ , f = 0 1 0 , MA∗∗∗ , f = . 11 1 1 1 1 1 001 00111 Thus, for example considering the natural logarithm and k = 1 we have 1 1 1 EK (A∗ ) = loge 4 ≈ 1.386294361 and EK (A∗∗ ) = EK (A∗∗∗ ) = loge 2 ≈ 0, f 0, f 0, f 10 10 (A ) ≈ 0.69314718. Moreover, if k = 10, then EK0 , f (A∗ ) ≈ 1.153051977, EK ∗∗ 0, f 10 0.069314718 and EK0 , f (A∗∗∗ ) ≈ 0.69314718. Furthermore, it is easy to see k k that EK (A∗∗ ) = 1k loge 2 and EK (A∗∗∗ ) = loge 2 for any k ∈ N. 0, f 0, f The (K, ξ )-entropy of the sequence A ∈ πK is the number k EK,ξ (A) = lim sup EK,ξ (A). k→∞

160

Anna Loranty, Ryszard J. Pawlak

Clearly, EK0 , f (A∗∗ ) = 0 and EK0 , f (A∗∗∗ ) > 0. Finally, the K-entropy of the map ξ is the number EK (ξ ) = sup EK,ξ (A). A∈πK

From above considerations one can conclude that EK0 ( f ) > 0 and so one can say that f is a chaotic function. In the case of new concepts in the theory of discrete dynamical systems, it is important to check the "invariance with respect to certain homeomorphisms". In the case of GTS (X, γ) we consider γ-homeomorphisms. We shall call a bijection f : X → X a γ-homeomorphism if both f and the inverse function f −1 are γ-continuous. As in the case of the topological space a function f : X → X is γ-continuous iff A ∈ γ implies f −1 (A) ∈ γ (see [10]). The first statement e presented below is related to a nonempty family K ⊂ P(X) invariant via γhomeomorphism, i.e ϕ(A) ∈ K for any A ∈ K and any γ-homeomorphism ϕ : X → X. e Theorem 11.5. Let (X, γ) be GTS and a nonempty family K ⊂ P(X) be invariant via γ-homeomorphism. If A = (A1 , A2 , . . . , An ) ∈ πK , then Aϕ = (ϕ(A1 ), ϕ(A2 ), . . . , ϕ(An )) ∈ πK for any γ-homeomorphism ϕ : X → X. In the theory of discrete dynamical systems, invariance of certain properties of conjugate functions plays a special role. For this reason the next two theorems are devoted to such kind of problems with respect to our previous considerations. But first, we will give the definition of conjugate functions. We will say that functions (multifunctions) f , g : X → X ( f , g : X ( X) are conjugate iff there exists a γ-homeomorphism ϕ : X → X such that ϕ ◦ f = g ◦ ϕ. In this case, we will also say that functions (multifunctions) f , g are conjugate via γ-homeomorphism ϕ. e Theorem 11.6. Let (X, γ) be GTS, a nonempty family K ⊂ P(X) be invariant via γ-homeomorphism and f , g : X → X be functions (or f , g : X ( X be multifunctions) conjugate via γ-homeomorphism ϕ : X → X. If A = (A1 , . . . , An ) ∈ πK , then MA, f = MAϕ ,g (where Aϕ = (ϕ(A1 ), . . . , ϕ(An ))). Now, let us assume that (X, γ) is GTS, f , g : X → X are conjugate via γe homeomorphism ϕ : X → X and a nonempty family K ⊂ P(X) is invariant via γ-homeomorphism. According to Theorem 11.5 and Theorem 11.6, we obtain

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tr(MkA, f ) = tr(MkAϕ ,g ) for any k ∈ N. Therefore EK ( f ) = supA∈πK EK, f (A) ≤ supA∈πK EK,g (A) = EK (g). By a similar argumentation we obtain that EK (g) ≤ EK ( f ). In this way an important theorem justifying consideration of generalized entropy was proved. Theorem 11.7. Let (X, γ) be GTS. If f , g : X → X are conjugate and a nonempty e family K ⊂ P(X) is invariant via γ-homeomorphism then EK ( f ) = EK (g). The above theorem is still true if we replace functions f , g : X → X by multifunctions f , g : X ( X. An important role in many considerations regarding practical use of mathematical theorems is played by finite spaces. Then the concept of entropy presented here makes possible some calculations. The following example is widely described in [30]. Here we will only indicate some elements related to these considerations. Let X = {a0 , a1 , . . . , a10 } and ν : X ( X be defined in the following way: ν(ai ) = {ai , ai+1 } for i ∈ {0, . . . , 4}, ν(a5 ) = {a5 , a0 }, ν(ai ) = {ai+1 } for i ∈ {6, . . . , 9} and ν(a10 ) = a6 . Of course, Fix(ν) = {a0 , . . . , a5 }. Putting γ = {B ⊂ X : B ⊂ ν(B)} we obtain that γ is GT in X and γ = {B ⊂ X : X \ B ⊂ Fix( f ) ∨ {a6 , a7 , a8 , a9 , a10 } ⊂ X \ B}. Set K = {{a0 }, {a1 }, {a2 }, {a3 }}. It folk (A) ≤ 1 log 4 for any A ∈ π and k ∈ N. This gives lows easily that EK,ν K k EK (ν) = 0. If we consider a permutation Π : X → X such that Π (Fix(ν)) = Fix(ν) and ζ = Π ◦ ν ◦ Π −1 then we check at once that ν and ζ are conjugate. Since K is invariant via γ-homeomorphism, we can deduce, according to Theorem 11.7, that EK (ζ ) = 0. The next theorem is connected with so called n-turbulent function (or nhorseshoe [2]) and it gives a convinient tool for lower estimation of a generalized entropy. e Theorem 11.8. Let (X, γ) be GTS, K ⊂ P(X) be a nonempty family, ξ :P(X) → P(X) and n ∈ N. If there exists a sequence A=(A1 , A2 , . . . , An )∈πK such that n [

i=1

then EK (ξ ) ≥ log n.

Ai ⊂

n \

i=1

ξ (Ai ),

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Anna Loranty, Ryszard J. Pawlak

11.3 Usual entropy and generalized entropy There is a natural question about the relationship between generalized entropy (introduced in the previous part of this chapter) and standard entropy for usual functions. The "covery" definition of entropy is connected with continuous functions, because its essence is the demand that inverse image of an open set is an open set ([1]). As mentioned in section 11.2 in compact metric spaces the above definition is equivalent to Bowen-Dinaburg’s definition of entropy ([7], [19]). So, we limit our considerations to compact metric spaces. Moreover, in recent times the results related to the entropy of discontinuous functions can be found in many papers. Therefore we will only give a definition of entropy formulated in the basic version in [7] and [19] (equivalent to the "covery" one) and transferred to a wider class of functions in [16]. Let (X, ρ) be a compact metric space, f : X → X, n ∈ N and ε > 0. We shall say that a set M ⊂ X is (n, ε)-separated if for any different points x, y ∈ M there exists i ∈ {0, 1, . . . , n − 1} such that ρ( f i (x), f i (y)) > ε. Moreover, a set E ⊂ X will be called (n, ε)-span if for every x ∈ X there is y ∈ E such that ρ( f i (x), f i (y)) ≤ ε for any i ∈ {0, 1, . . . , n}. Set sn (ε) = max{card(M) : M ⊂ X is (n, ε) − separated set}, rn (ε) = min{card(E) : E ⊂ X is (n, ε) − span set}, s(ε) = lim supn→∞ 1n log sn (ε) and r(ε) = lim supn→∞ n1 log rn (ε). The topological entropy of f is the number 1 h( f ) = lim+ lim sup log sn (ε). ε→0 n→∞ n It is possibile that h( f ) = +∞. Moreover, let us remark ([16], Lemma 3.2) that if ε1 < ε2 , then s(ε1 ) ≥ s(ε2 ). Furthermore, in [16] one can find the following fact Remark 11.9. For any compact metric space (X, ρ) and any f : X → X we have that lim+ s(ε) = lim+ r(ε) = h( f ). ε→0

ε→0

Obviously, if (X, ρ) is a compact metric space and f (x) = x for x ∈ X, then e h( f ) = EK ( f ) = 0 for any nonempty family K ⊂ P(X). The following theorem shows that the generalized entropy is the lower estimation of standard one. We will present only a sketch of the proof of this theorem (the complete proof of this theorem one can find in [30]). Theorem 11.10. If (X, ρ) is a compact metric space and f : X → X then e EK ( f ) ≤ h( f ), for any nonempty family K ⊂ P(X).

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Proof. If EK ( f ) = 0 then the above inequality is obvious, so let us assume that EK ( f ) = p > 0 and suppose contrary to our claim that h( f ) ∈ [0, p). There exists a sequence A0 = (A1 , A2 , . . . , Ak ) ∈ πK such that EK, f (A0 ) >

h( f ) + p > 0. 2

Moreover, there exists a strictly increasing sequence {mn }n∈N of positive integers such that 1 h( f ) + p n mn > . log(tr(Mm A0 , f )) 2 (m )

n n Fix n ∈ N. With the notation Mm A0 , f = [xi, j ]i, j≤k we have that if

(m )

(m )

xt,t n > 0, then there exist xt,t n various sequences (At , As1 , . . . , Asmn −1 , At ) such that s1 , . . . , smn −1 ∈ {1, 2, . . . , k}, At → As1 , Asi → Asi+1 for i ∈ {1, . . . , mn − 2} f

f

(mn )

and Asmn −1 → At . Let us denote the set of these sequences by Yt f

(mn )

Yt

(mn )

= {Y1t , . . . ,Y t(mn ) }). There exists an injective function ξt : Yt xt,t

(mn )

such that if Yit = (At , Ais1 , . . . , Aismn −1 , At ) ∈ Yt

(set

→ At

, then

ξt (Yit ) ∈ At , f (ξt (Yit )) ∈ Ais1 , . . . , f mn −1 (ξt (Yit )) ∈ Aismn −1 , f mn (ξt (Yit )) ∈ At . (m )

Putting Z = {t ∈ {1, . . . , k} : xt,t n > 0} and Qmn = that n card(Qmn ) = tr(Mm A0 , f ).

(mn ) ) t∈Z ξt (Yt

S

we obtain

Moreover, there exist ε ∗ > 0 such that Qmn is (mn , ε)-separated set for any ∗ n ε ∈ (0, ε ∗ ). Therefore smn (ε) ≥ tr(Mm A0 , f ) for any ε ∈ (0, ε ). Thus h( f ) ≥ lim sup( n→∞

1 1 n log smn (ε)) ≥ lim sup( log tr(Mm A0 , f )) > h( f ), mn mn n→∞

which is impossible.

t u

11.4 Generalized Vietoris topology and generalized entropy As it was already indicated if we have fixed function then it is possible to consider some multifunction, map, etc. Opportunities in this area are much wider (e.g. [8]). For this reason, recently there appeared many papers in which

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authors compared the properties of dynamic of functions and suitable multifunctions and maps (e.g. [46], [21], [32]). In this section we refer to these ideas. Let (X, γ) be GTS. We will denote by CL(X) the family of all nonempty, closed subsets of X. Consider the family Vγ ⊂ P(CL(X)) consisting of sets α ∈ P(CL(X)) such that α = 0/ or for any A ∈ α there exist sets U1 , . . . ,Un ∈ γ such S that A ∩ Ui 6= 0, / for any i ∈ {1, . . . , n} and {B ∈ CL(X) : B ⊂ ni=1 Ui and B ∩ Ui 6= 0/ for any i ∈ {1, . . . , n}} ⊂ α. We check at once that the family Vγ is a generalized topology in the space CL(X). This generalized topology can be called generalized Vietoris topology, because the above definition agrees with the classical definition of Vietoris topology in usual topological spaces. Let A ∈ P(X). If there exists a set B ∈ CL(X) such that B ⊂ A, then put c d(A) = {B ∈ CL(X) : B ⊂ A}. Otherwise put d(A) = 0. / Moreover, set CL(X) = {d(A) : A ∈ CL(X)}. Let A X be a γ-closed set. Obviously, CL(X) \ d(A) = {P ∈ CL(X) : P ∩ (X \ A) 6= 0} / = 6 0/ and X \ A ∈ γ. Moreover, we have that W ∩ X 6= 0/ and W ∩ (X \ A) 6= 0/ for any W ∈ CL(X) \ d(A). We check at once that {C ∈ CL(X) : C ⊂ X ∪ (X \ A) ∧C ∩ X 6= 0/ ∧C ∩ (X \ A) 6= 0} / = CL(X) \ d(A). Therefore, we obtain that CL(X) \ d(A) ∈ Vγ , so d(A) is Vγ -closed. Result of the above considerations can be saved in the form of theorem Theorem 11.11. Let (X, γ) be sGTS. If a set A is γ-closed, then d(A) is Vγ closed. In the papers [46], [21], [47] a special kind of multifunction from CL(X) into itself connected with a function f : X → X was considered. The authors considered topological spaces or compact metric spaces X and the multifunction ψ df : CL(X) ( CL(X) defined by the formula ψ df (A) = d( f (A)) for A ∈ CL(X) in connection with various problems of chaos. We can also investigate this multifunction for T1 -sGTS (X, γ) and f : X → X or f : X ( X. For example we have Theorem 11.12. Let (X, γ) be T1 -sGTS. For any multifunction f : X ( X (function f : X → X) we have ECL(X) ( f ) = ECL(X) (ψ df ). c

(11.1)

The proof of this theorem was divided into three parts. In the first one, it was shown that a sequence A = (A1 , A2 , . . . An ) belongs to πCL(X) if and only if Ad = (d(A1 ), d(A2 ), . . . d(An )) belongs to πCL(X) . Next, it was proved that c

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MA, f = MAd ,ψ df . Finally, using the properties shown in the first and second parts, it was shown that ECL(X) ( f ) ≤ ECL(X) (ψ df ) and ECL(X) (ψ df ) ≤ ECL(X) ( f ). c c For more details we refer the reader to [30]. Fixed points and periodic points of functions play an important role in the theory of combinatorial dynamics. It is particularly important to find a relationship between entropy of a function and the number of its fixed or periodic points. Now, we present the relationship between the fixed points of certain multifunctions and the generalized entropy of these multifunctions. We start with the definition. e Let (X, γ) be GTS and K ⊂ P(X) be a nonempty family. We say that multifunction ψ : X ( X has the property IK if for every sequence (A1 , A2 , . . . , An , A1 ) ∈ πK such that Ai → Ai+1 for i = 1, 2, . . . , n − 1 and An → A1 there ψ

ψ

exists a sequence (x1 , x2 , . . . , xn ) such that xi ∈ Ai and xi+1 ∈ ψ(xi ) for i = 1, 2, . . . , n − 1 and x1 ∈ ψ(xn ). It is worth noting that if we consider a function f : R → R instead of multifunction ψ and a family K of all Tnat -closed intervals then the above definition agrees with the one given in ([3], [51]). In the context of these considerations the following theorem seems to be interesting. Theorem 11.13. Let (X, γ) be GTS and ψ : X ( X be a multifunction having the property ICL(X) . Then   1 n ECL(X) (ψ) ≤ lim sup max 0, log (card (Fix(ψ ))) . n n→∞ As mentioned at the beginning of section 11.2, to each function f : X → X, the map ξ f generated by f may be assigned. Some properties of a function ξ f  CL(X) for a compact metric space X (so called "functions induced by f ") one can find, for example, in papers [46], [21], [32], [47]. Moreover, if we consider CL(X) equipped with Vietoris topology then ξ f  CL(X) is a continuous function whenever f is a continuous function. From Theorem 11.10 and results contained in papers [42], [26], [32] it may be concluded Theorem 11.14. If (X, ρ) is a compact metric space and f : X → X is a cone tinuous function, then for an arbitrary nonempty family K ⊂ P(X) we have EK ( f ) ≤ h( f ) ≤ h(ξ f  CL(X)).

(11.2)

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Taking into account the examples presented in the papers [42], [32], [30], it is easily seen that the inequality in (11.2) may be strict. On the other hand we have the following theorem Theorem 11.15. Let X = [0, 1]m and Ar(X) be the set of all arcs in X. The set { f ∈ C(X) : EAr(X) ( f ) = h( f ) = h(ξ f  CL(X))} is dense in the space C(X) with the metric of uniform convergence. Therefore, for any continuous function f : [0, 1]m → [0, 1]m there exists a continuous function which is "arbitrarily close" to f and for which all entropies considered in Theorem 11.14 have the same value. Notice that the above theorem is still true if we replace Ar(X) with a family of all nonempty closed or connected or Borel subsets of X. Moreover, we can consider non-singleton, convex and compact subset of Rm in place of X or some manifold instead of Rm . Of course, in last situation we must replace the set X with a set being a homeomorphic image of a suitable convex set. These considerations are a continuation of research initiated in the papers [44] and [39].

11.5 Transitive multifunctions We start this section with some definitions. Let (X, γ) be GTS and Φ : X ( X. Similarly to the case of usual functions, we shall say that a multifunction Φ is transitive if for any pair of nonempty open sets U,V ⊂ X there exists k ∈ N such that V ∩ Φ k (U) 6= 0. / Moreover, we shall say that a set (sequence) ΘΦ (x0 ) = {x0 , x1 , x2 , . . . } is an orbit of x0 under Φ if xi ∈ Φ(xi−1 ) for any i ∈ N. It is worth noting that, unlike in the case of a function, there may exist a lot of different orbits of x0 under multifunction Φ. From now on, ΘΦa (x0 ) stands for the family of all orbits ΘΦ (x0 ) of x0 under multifunction Φ. Clearly, we have the following property. Property 11.16. Let (X, γ) be GTS. If ΘΦ (x0 ) = {x0 , x1 , x2 , . . . } is an orbit of x0 under Φ : X ( X then xi ∈ Φ i (x0 ) for i ∈ N. In the paper [31] one can find the example showing that the converse statement is not true. However one can prove the following fact: Theorem 11.17. Let (X, γ) be GTS, Φ : X ( X and x0 ∈ X. If α ∈ Φ m (x0 ) for some positive integer m, then there exists an orbit ΘΦ (x0 ) = {x0 , x1 , x2 , . . . } of x0 under Φ such that xm = α.

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Many mathematicians have investigated relationship between transitivity and existence of a dense orbit of a function (e.g. [43], [17], [38], [33]). For instance, in [43] one can find the example of a transitive function f which does not have a dense orbit. On the other hand there exists a function f with dense orbit which is not a transitive function ([17]). Therefore these two notions are independent in general. However in some cases there are equivalent ([17], [38]). It is clear that there is no connection between transitivity and existence of a dense orbit for multifunctions in general, too. To see this it suffices to consider multifunctions Φ(x) = { f (x)}, where f is one of the functions mentioned above. However, we have the following theorem Theorem 11.18. Let (X, γ) be GTS and Φ : X ( X. A multifunction Φ is transitive if and only if for any U,V ∈ γe there exist x0 ∈ U and the orbit ΘΦ (x0 ) such that ΘΦ (x0 ) ∩V 6= 0. / The next theorem presents some condition equivalent to transitivity for a lower semicontinuous multifunction Φ : X → X i.e multifunction Φ : X → X such that for any x ∈ X and any U ∈ γ such that Φ(x) ∩ U 6= 0/ there exists V ∈ γ(x) such that Φ(t) ∩U 6= 0/ for any t ∈ V . Theorem 11.19. Let GTS (X, γ) be a Baire space with a countable base. A lower semicontinuous multifunction Φ : X ( X is transitive if and only if S the set {x ∈ X : clγ ( ΘΦa (x)) = X} is residual. One can ask if we can consider, in the above theorem, a set of all points x ∈ X such that there exists a dense orbit of x under Φ instead of the set S {x ∈ X : clγ ( ΘΦa (x)) = X}. The following example shows (see [31]) that answer to this question is negative. If we consider (R, Tnat ) and an arbitrary sequence {qi }i∈N of all rational numbers, then it is easy to see that the multifunction Φ : R ( R such that Φ(x) = Q if x ∈ R \ Q and Φ(x) = {q1 , q2 , . . . , qi } if x = qi is transitive and lower semicontinuous. On the other hand there is no dense orbit of x under Φ for any x ∈ R.

11.6 Strongly transitive multifunctions Let (X, γ) be a Baire GTS and Φ : X ( X (Φ : X → X). A multifunction (function) Φ is strongly transitive if for any U,V ∈ γe we have that {x ∈ U : ΘΦ (x) ∩V 6= 0/ for some ΘΦ (x) ∈ ΘΦa (x)} is a second category set.

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Obviously, each strongly transitive multifunction (function) is transitive. On the other hand, there exists a transitive multifunction (function) which is not strongly transitive. Indeed, if we consider (R, Tnat ) then the multifunction ( {π} for x ∈ R \ Q, Ω (x) = Q for x ∈ Q is transitive because V ∩ Ω (U) ⊃ V ∩ Q 6= 0/ for any U,V ∈ Tenat . Moreover, the set of all x ∈ (0, 1) such that ΘΩ (x) ∩ (1, 2) 6= 0/ for some orbit ΘΩ (x) ∈ ΘΩa (x) is countable, so we have that Ω is not strongly transitive. The same is true if we consider the function f : [0, 2] → [0, 2] presented in Example in the paper [43] instead of the multifunction Ω . Now we will focus on a particular type of multifunction, which has been also studied, for example by Crannell, Frantz and LeMasurier ([8]). Let us start with definition of the Cartesian product of generalized topological spaces introduced by Császár in [15]. Let T be a nonempty set and Xt 6= 0/ for any t ∈ T . Furthermore, let (Xt , γt ) be GTS for any t ∈ T and X = ∏t∈T Xt be the Cartesian product of the sets Xt . Moreover, let B be a set of all sets of the form S ∏t∈T Mt , where Mt ∈ γt for each t ∈ T and Mt 6= γt only for a finite number of t from T . We call γB the product of generalized topologies γt . Obviously, γB is a generalized topology in X. For any GTS (X, γ) and any function f : X → X we define a multifunction f¯ : X ( X in the following way f¯(x) = {y ∈ X : (x, y) ∈ clγ×γ (Γ ( f ))}, where Γ ( f ) is a graph of f and γ × γ is the product of generalized topologies. Our definition agrees with the one given in [8], in the case of topological spaces. To illustrate this concept, consider the space (R, Tnat ) and two functions f1 , f2 : R → R defined in the following way: f1 (x) = f2 (x) = sin 1x for x 6= 0, f1 (0) = 0 and f2 (0) = 1. Then f¯1 (x) = f¯2 (x) = sin 1x for x 6= 0 and f¯1 (0) = f¯2 (0) = [−1, 1]. The following two statements describe some connection between orbits and properties of a function f and orbits of a suitable function f¯. Proposition 11.20. Let (X, γ) be GTS, f : X → X and x0 ∈ X. The set Θ f (x0 ) = {x0 , f (x0 ), f 2 (x0 ), . . . } is an orbit of x0 under multifunction f¯. Proposition 11.21. Let (X, γ) be T2 -GTS and f : X → X. (a) If x0 ∈ C( f ) then f¯(x0 ) = { f (x0 )}.

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(b) If Θ f (x0 ) ⊂ C( f ) for some x0 ∈ X then Θ f (x0 ) is the unique orbit of x0 under multifunction f¯. Using the above statements, we can show the following theorem related to cm-function f : X → X i.e function f : X → X such that D( f ) is a countable set and f −m (x) = {z ∈ X : f m (z) = x} is a meager set for any x ∈ D( f ) and m ∈ N ∪ {0}. Theorem 11.22. Let GTS (X, γ) be a strong Baire space with a countable base such that for any U ∈ γe and any finite set A ⊂ U there exists a set V ∈ γe such that V ⊂ U \ A. Let f : X → X be a cm-function. The following conditions are equivalent: (A) f is strongly transitive, (B) there exists x0 ∈ X such that Θ f (x0 ) is a dense set and Θ f (x0 ) ⊂ C( f ), (C) f¯ is strongly transitive, (D) there exists x0 ∈ X such that there exists an orbit Θ f¯(x0 ) which is a dense set and Θ f¯(x0 ) ⊂ C( f ). In the proof of this theorem an important role also plays the following property: if (X, γ) is GTS and f : X → X is a cm-function then the set {x ∈ X : Θ f (x) ∩ D( f ) 6= 0} / is a meager set.

References [1] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological Entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. [2] L. Alsedá, J. Llibre, M. Misiurewicz Combinatorial Dynamics and Entropy in Dimension One, World Sci., 1993. ˇ ˘ Theorem for connecti[3] J. Andres, P. Snyrychowá, P. Szuca, Szarkovskii’s vity Gδ -relations, Int. J. Bifurkation Chaos 16 (2006), 2377–2393. [4] L. S. Block, W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992. [5] J. Borsik, Generalized oscillations for generalized continuities, Tatra Mt. Math. Publ. 49 (2011), 119–125. [6] J. Borsik, Points of generalized continuities, Tatra Mt. Math. Publ. 52 (2012), 153–160. [7] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971) 401–414. [8] A. Crannell, M. Frantz, M. LeMasurier, Closed relations and equivalence classes of quasicontinuous functions, Real Anal. Exchange 31(2) (2005/06), 409–423. [9] Á. Császár, Generalized open sets, Acta Math. Hungar. 75(1-2) (1997), 65–87.

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[10] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar. 96(4) (2002), 351–357. [11] Á. Császár, γ-connected sets, Acta Math. Hungar. 101(4) (2003), 273–279. [12] Á. Császár, Separation axioms for generalized topologies, Acta Math. Hungar. 104(1-2) (2004), 63–69. [13] Á. Császár, Modyfication of generalized topologies via hereditary classes, Acta Math. Hungar. 115(1-2) (2007), 29–36. [14] Á. Császár, Normal generalized topologies, Acta Math. Hungar. 115(4) (2007), 309–313. [15] Á. Császár, Product of generalized topologies, Acta Math. Hungar. 123(1-2) (2009), 127–132. ˇ [16] M. Ciklová, Dynamical systems generated by functions with Gδ graphs, Real Anal. Exchange 30 (2004/2005), 617–638. [17] N. Deˇgirmenci, S˛. Koc˛ak, Existence of a dense orbit and topological transitivity: when are they equivalent?, Acta Math. Hungar. 99(3)(2003), 185–187. [18] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Lecture Notes in Math. 1513, Springer, Berlin, 1992. [19] E. I. Dinaburg, Connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR 35 (1971), 324–366 (in Russian). [20] E. Ekici, Generalized hyperconnectedness, Acta Math. Hungar. 133(1-2) (2011), 140–147. [21] R. Gu, Kato’s chaos in set valued discrete systems, Chaos, Solitions & Fractals 31 (2007), 765–771. [22] J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys. 70 (1979), 133–160. [23] X. Huang, X. Wen, F. Zeng, Pre-image entropy of nonautonomous dynamical systems, Jrl. Syst. Sci. & Complexity 21 (2008), 441–445. [24] T. Jyothis, J. J. Sunil, µ-Compactness in Generalized Topological Spaces, Research Article, Journal of Advanced Studies in Topology 3(3) (2012), 18–22. [25] E. Korczak-Kubiak, A. Loranty, R. J. Pawlak, Baire generalized topological spaces, generalized metric spaces and infinite games, Acta Math. Hungar. 140(3) (2013), 203–231. [26] D. Kwietniak, P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces, Chaos, Solitons & Fractals 33 (2007), 76–86. [27] M. Lampart, P. Raith, Topological entropy for set valued maps, Nonlinear Analysis 73 (2010), 1533–1537. [28] J. Li, Generalized topologies generated by subbases, Acta Math. Hungar. 114(1-2) (2007), 1–12. [29] J. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992. [30] A. Loranty, R. J. Pawlak, The generalized entropy in the generalized topological spaces, Topology Appl. 159 (2012), 1734–1742. [31] A. Loranty, R. J. Pawlak, On the transitivity of multifunctions and density of orbits in generalized topological spaces, Acta Math. Hungar. 135(1-2) (2012), 56–66. [32] X. Ma, B. Hou, G. Liao, Chaos in hyperspace system, Chaos, Solitions & Fractals 40 (2009), 653–660. [33] J. -H. Mai, W. -H. Sun, Transitivities of maps of general topological spaces, Topology Appl. 157 (2010), 946–953.

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[34] M. Marteli, Discrete Dynamical Systems and Chaos, Longman Scientific and Technical, London, 1992. [35] W. K. Min, Generalized continuous functions defined by generalized open sets on generalized topological spaces, Acta Math. Hungar. 128(4) (2010), 299–306. [36] J. M. Mustafa, µ-semi compactness and µ-semi Lindelöfness in generalized topological spaces, IJPAM 78(4) (2012), 535–541. [37] J. C. Oxtoby, Measure and Category, Springer - Verlag, New York, 1980. [38] H. Pawlak, R. J. Pawlak, Transityvity, dense orbits and some topologies finer than the natural topology of the unit interval, Tatra Mt. Math. Publ. 35 (2007), 1–12. [39] R. J. Pawlak, A. Loranty, A. Bakowska ˛ On the topological entropy of continuous and almost continuous functions, Topology Appl. 158 (2011), 2022–2033. [40] Z. Pawlak, Rough sets, International Journal of Computer and Information Science 11(5) (1982), 341-356. [41] Z. Pawlak, Rough classification, International Journal of Man-Machine Studies 20 (1984), 469-483. [42] J. S. C. Peña, G. S. López, Topological entropy for induced hyperspace maps, Chaos, Solitions & Fractals 28 (2006), 979–982. [43] A. Peris, Transitivity, dense orbit and discontinuous functions, Bull. Belg. Math. Soc. 6 (1999), 391–394. [44] J. A. Pomykała, Approximation operations in approximation space, Bull. of the Polish Academy of Sci. Mathematics, Theoretical Computer Science 15(9-10) (1987), 653–662. [45] P. L. Power, K. L. Rajak, Some New Concepts of Continuity in Generalized Topological Space, International Journal of Computer Applications 38(5) (2012), 12–17. [46] H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos, Solitons & Fractals 17 (2003), 99–104. [47] H. Román-Flores, Y. Chalco-Cano, G. Nunes-Silva, Chaos induced by turbulent and erratic functions, IFSA-EUSFLAT (2009), 231–233. [48] M. S. Sarsak, Weak separation axioms in generalized topological spaces, Acta Math. Hungar. 131(1-2) (2011) 110–121. [49] M. S. Sarsak, Weakly µ-compact spaces, Demonstratio Mathematica XLV (4) (2012), 929–938. ˘ theorem holds for some discontinuous functions, Fund. Math. [50] P. Szuca, Szarkovskii’s 179 (2003), 27–41. [51] S. Wiggins, Chaotic Transport in Dynamical Systems, Springer-Verlag, Interdisc. Applied Math. vol. 2, New York 1991.

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A NNA L ORANTY Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

RYSZARD J. PAWLAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 12

On rings of Darboux-like functions. From questions about the existence to discrete dynamical systems

EWA KORCZAK-KUBIAK, HELENA PAWLAK, RYSZARD J. PAWLAK

2010 Mathematics Subject Classification: 26A15, 26A18, 54C40, 54C70, 54H25. Key words and phrases: Darboux function, ring of functions, ideal of a ring of functions, ´ atkowski ´ atkowski Swi ˛ property, strong Swi ˛ property, Sharkovsky property, entropy point, generalized Fleissner condition, almost continuous function, first return continuity.

Introduction A combination of considerations regarding algebraic structures of functions and topological properties of examined transformations is a common topic of many scientific papers. A lot of mathematical research and important theories are based on it. On the other hand, limiting considerations connected with topological aspects or measure theory to some algebraic structures gives completely new possibilities (e.g. in the context of dynamical systems it is visible in [2], [13], [14]). The facts mentioned above lead us in obvious way to the necessity of analyzing algebraic properties of classes of functions widely examined in the real functions theory. In this theory, Darboux-like functions play a particular role (e.g. basic properties of Darboux functions are presented at the beginning of the classical monograph connected with real functions theory [4]). Discovery that each real function of a real variable is a sum of two Darboux functions ([24]) became a starting point of looking for the answers to many questions connected with algebraic operations (addition, multiplication,

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lattice operations) performed on Darboux-like functions (e.g. [5], [6], [12], [20], [27], [29]). In this chapter we will concentrate on the considerations connected with the rings of Darboux-like functions. It is a very wide issue so we have to limit it to basic topics. Note in fact, that with problems regarding rings of functions and its ideals (in algebraic sense) one can strictly relate the issues connected with ideals of sets and theory of density points and approximately continuous functions ([34]). However these considerations go beyond the scope of this chapter. We will mainly focus on pointing out assumptions which guarantee the existence of rings of functions contained in fixed families of Darboux-like functions, examining its basic properties and, taking into account the directions signalled at the beginning, applying them in research connected with the discrete dynamical systems. To avoid analysis of very detailed issues we will sometimes only indicate the literature containing regarded facts. Throughout this chapter we will use the classical symbols and notions. However, in order to avoid misunderstandings, we will present basic denotation, symbols and definitions which will be used in the next parts of the chapter. Let f be a function. If A is a subset of the domain of f then the symbol f  A will stand for the restriction of f to A. The set of all continuity (discontinuity) points of f will be denoted by C( f ) (D( f )). Moreover we will use the notation C ∗ ( f ) = X \ D( f ). If f is a real valued function then let us denote by Z( f ) the zero set of f ; i.e., Z( f ) = f −1 (0). If F is a family of functions f : X → R then S put D[F] = f ∈F D( f ) and Z[F] = {Z( f ) : f ∈ F}. For a function f : R → R and x0 ∈ R we will use the following notations: − R ( f , x0 ) = {α ∈ R : f −1 (α)∩(x0 −δ , x0 ) 6= 0/ for any δ > 0} and R+ ( f , x0 ) = {α ∈ R : f −1 (α) ∩ (x0 , x0 + δ ) 6= 0/ for any δ > 0}. If f : X → R then f (x) = f (x) and f (x) = f (x)· f (x) for n > 1. If f : X → X then put f 0 (x) = x and f n (x) = f ( f n−1 (x)) for n > 1. A point x such that f M (x) = x, but f n (x) 6= x, for n ∈ {1, 2, . . . , M −1} is called a periodic point of f of prime period M. The set of all periodic points of f of prime period M will be denoted by PerM ( f ). The symbol constX,Y α will stand for the constant function from X to Y assuming value α. If A is a subset of the domain of f : X → Y and B ⊂ Y , then we shall say that a set A f -covers a set B (denoted by A → B) if B ⊂ f (A). f

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The distance between a set A ⊂ R and a point x ∈ R (in the natural metric) will be denoted by dist(A, x). In this paper we will consider several classes of functions, apart from the family of continuous functions, we will deal with Darboux functions or almost continuous functions. It should be noted that in our case, we limit most of these definitions (except continuous function and Darboux function) to the case of real functions of a real variable. However, these definitions can be naturally extended to the more general case. We start with definition of Darboux function in general case. Let (X, TX ) and (Y, TY ) be topological spaces. We shall say that f : X → Y is a Darboux function if an image of any connected set A ⊂ X is a connected set. In the case of a real function f of a real variable the above definition is equivalent to the following intermediate value property: if x and y belong to the domain of f and α is any number between f (x) and f (y) then there exists a number z between x and y such that f (z) = α. Let f : X → R. We say that f is a Baire one function (or f is of the first class of Baire) if for any a ∈ R the sets {x ∈ X : f (x) < a} and {x ∈ X : f (x) > a} are Fσ type. We say that a function f belongs to the class B1∗∗ if D( f ) = 0/ or f  D( f ) is continuous ([40]). It is worth noting that the family B1∗∗ has been introduced in a connection with research regarding rings of functions and it is wider than the class of all continuous functions and is included in the class B1∗ ([31]). Now, let X,Y be the unit intervals or R (with natural topology) and f : X → Y be a function. A function f is approximately continuous if for any x ∈ X there exists a Lebesgue measurable set Ax ⊂ X such that lim

h→0

λ (Ax ∩ [x − h, x + h]) = 1 and f (x) = lim f (t). t→x, 2h t∈Ax

Obviously if X = [0, 1] and x = 0 or x = 1 we consider limh→0 λ (Ax ∩[x,x+h]) =1 h λ (Ax ∩[x−h,x]) = 1, respectively. This kind of functions was considered or limh→0 h for the first time by A. Denjoy in 1915 ([8]). Clearly, the family of all continuous functions from X to Y is a proper subset of the family of all approximately continuous functions from X to Y . The next kind of functions we will consider are derivatives. It is known that the class of all approximately continuous functions is not contained in the class of derivatives but every bounded approximately continuous function is

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a derivative ([4]). In 1959 J. Stallings in paper [47] introduced the notion of almost continuity. We call a function f almost continuous if for any open set U ⊂ X ×Y containing the graph of f , U contains the graph of some continuous function g : X → Y . It is worth noting that every derivative (approximately continuous function) is an almost continuous function. ´ atkowski ´ atkowski We shall say that f has the Swi ˛ property (or is a Swi ˛ function) if for any x, y ∈ X such that x < y and f (x) 6= f (y) there exists z ∈ C( f ) such that z ∈ (x, y) and f (z) belongs to the open interval of the ends f (x) and ´ atkowski f (y). This kind of functions was introduced by T. Swi ˛ and T. Ma´nk in 1982 ([28]). In the paper [26] one can find the following definition. A function ´ atkowski f has the strong Swi ˛ property if for any x, y ∈ X such that x < y and f (x) 6= f (y) and any α between f (x) and f (y) there exists z ∈ C( f ) such that ´ atkowski z ∈ (x, y) and f (z) = α. This function is also called a strong Swi ˛ func´ atkowski ´ atkowski tion. If f is a strong Swi ˛ function then it has the Swi ˛ property and moreover it is a Darboux and quasi-continuous function ([27]). We will use the following symbols for families of considered functions: Const(X) - the family of all constant functions defined on X, C(X,Y ) - the family of all continuous functions f : X → Y , D(X,Y ) - the family of all Darboux functions f : X → Y , ´ atkowski S(X,Y ) - the family of all functions f : X → Y having the Swi ˛ property, ´ atkowski sS(X,Y ) - the family of all functions f : X → Y having the strong Swi ˛ property, B1 (X,Y ) - the family of all functions f : X → Y of first Baire class, B1∗∗ (X,Y ) - the family of all functions f : X → Y from the class B1∗∗ , 40 (X,Y ) - the family of all derivatives from X to Y which are not approximately continuous functions, 0 (X,Y ) - the family of all approximately continuous functions f : X → Y Cap which are not continuous functions, A(X,Y ) - the family of all almost continuous functions f : X → Y . In all the above notations if X = Y we will write only one X, e.g. D(X) instead of D(X, X), S(X) instead of S(X, X) etc. If additionally X = Y = R then we will write shortly D, S etc. For brevity, if we wish to consider the intersection of two or three classes of functions, we shall write them next to each other (e.g. DS(X,Y ) or DB1 (X)). The ring R of real functions defined on [0, 1] is called a complete ring if it contains the class of all continuous functions and the following condition is fulfilled:

12. On rings of Darboux-like functions

if f , g ∈ R, then max( f , g) ∈ R and min( f , g) ∈ R.

177

(12.1)

If F is a fixed class of functions and f ∈ F then the symbol ℜF ( f ) will stand for the family of all rings of functions from F containing the function f . If we additionally assume that considered rings are extensions of some ring W then we will write ℜW F ( f ). Moreover, if ℜ is a family of rings, then we will b write ℜ to denote that all the rings belonging to ℜ satisfy condition (12.1). For brevity of notation in the next parts of the chapter we will use the following rule. If f : X → Y then writing ℜW F ( f ) we will assume that all the functions from the ring W and the family F are defined on X and their values belong to Y . For example, if f : [0, 1] → R is a Darboux function then we will C([0,1],R) write ℜCD ( f ) instead of ℜD([0,1],R) ( f ). Let R be a ring. We will denote by I(R) the set of all ideals of R. If f ∈ R then the symbol ( f )R will stand for the ideal generated by f . An ideal J ∈ I(R) will be called an extension (restriction) of an ideal J1 ∈ I(R) if J1 ⊂ J (J ⊂ J1 ). An ideal J will be called a z-ideal if f ∈ R and Z( f ) ∈ Z[J ] T implies f ∈ J . Moreover, if J2 ∈ I(R) is a z-ideal such that Z[J2 ] is a nonempty closed set belonging to Z[J2 ], then we will called it z’-ideal. The set of all z’-ideals of R will be denoted by Iz0 (R). An ideal J is prime if f g ∈ J implies f ∈ J or g ∈ J . A nonzero ideal J0 ∈ I(R) is called essential if it intersects every nonzero ideal nontrivially. For A ⊂ R we write Ann(A) to denote the set {ξ ∈ R : ξ · A = {const0 }}, where const0 stands for the constant function assuming value 0.

12.1 Rings of the real Darboux-like functions defined on topological spaces The results presented in this part are based on the paper [43]. It is known that the family of all continuous functions defined on a topological space is a ring. Since each continuous function is a Darboux function, then for any topological space X one can create a ring of Darboux functions defined on X. In the context of our considerations this case is less interesting. That is why the question arises whether for each topological space there exists a ring of real Darboux functions defined on X containing at least one discontinuous function (we call such rings essential Darboux rings). We can extend the question: is there for any topological space a discontinuous Darboux function

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defined on it? The following theorem shows that even in the case of spaces with "very nice properties" such Darboux functions may not exist. Theorem 12.1. There exists a connected, uncountable, Hausdorff topological space X such that every Darboux function f : X → R is constant. From the above theorem it is easy to conclude the following Corollary 12.2. There exists a connected, uncountable, Hausdorff topological space X for which there are no essential Darboux rings of real functions defined on X. In the context of the above results and the questions posed at the beginning of the section, the following problem seems to be fundamental: what kind of assumptions should we impose on the space X to obtain the existence of essential Darboux ring of real functions defined on X? The answer to this question is Theorem 12.3. If X is a connected and locally connected topological space such that there exists a nonconstant continuous function f : X → R, then there exists an essential Darboux ring of functions from X to R. Of course obtaining the answer to one of the questions generates new problems, for example connected with the existence of essential Darboux rings consisting of such functions f that D( f ) ⊂ Z( f ) (essential rings with this property will be called ∗-rings). Theorem 12.4. Let X be a non-singleton, connected and locally connected, perfectly normal topological space. Then for every point x0 ∈ X there exists a Darboux ∗-ring R of real functions defined on X such that D[R] = {x0 }. Of course the properties of such rings and properties of families of such rings may be examined. For example in [43] some properties of rings connected with cardinal functions were examined. However, the detailed considerations regarding these problems are beyond the scope of this chapter.

12.2 Rings of the real Darboux-like functions defined on the unit interval. From now on till the end of the chapter we will refer the Darboux property exclusively to the natural topology. So if a topology T is given and we will write that each T -continuous function (i.e. continuous when we consider topology T in [0, 1]) has the Darboux property then we will mean that each T -continuous function has the intermediate value property.

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´ atkowski 12.2.1 Rings of Darboux and Swi ˛ functions. The main results of this section are based on the statements contained in the papers [37] and [32]. ´ atkowski At first one can notice that for Darboux functions, the Swi ˛ property is equivalent to other properties frequently examined in real analysis. ´ atkowski Theorem 12.5. A Darboux function f : [0, 1] → R has the Swi ˛ property if and only if for any x ∈ [0, 1] there exists a sequence {xn }n∈N ⊂ C( f ) such that limn→∞ xn = x and limn→∞ f (xn ) = f (x). In [32] the statement being used in the proofs of theorems connected with rings ´ atkowski of Swi ˛ functions was proved. Before formulating this theorem we will briefly recall two notions. We call a function f : [0, 1] → [0, 1] quasi-continuous if for any x ∈ [0, 1] and any neighbourhood U of x and any neighbourhood V of f (x) there exists a nonemty open set W ⊂ U such that f (W ) ⊂ V . We say that f : [0, 1] → [0, 1] has a strong Blumberg set B iff B is dense in [0, 1], f  B is continuous and for any nonempty open set U ⊂ [0, 1] the set f (U ∩ B) is dense in f (U). Theorem 12.6. For Darboux function f : [0, 1] → [0, 1] the following conditions are equivalent: ´ atkowski (i) f has the Swi ˛ property, (ii) f is quasi-continuous, (iii) f has a strong Blumberg set. ´ atkowski It is worth noting that in [32] the Swi ˛ property was defined also for 2 functions defined on R and the theorem analogous to Theorem 12.6 was proved. From Theorem 12.3 it follows immediately that there exists discontinuous Darboux function f such that ℜD ( f ) 6= 0. / In this case we can ask another question: what kind of assumptions should we impose on a Darboux function f to have ℜCD ( f ) 6= 0? / In particular one can ask whether the fact that f is a ´ Darboux and Swiatkowski ˛ function is a sufficient condition for the existence of a ring belonging to ℜCD ( f ). The following theorem shows that the answer to this question is negative. Theorem 12.7. There exists a Darboux function f : [0, 1] → R having the ´ atkowski Swi ˛ property such that ℜCD ( f ) = 0. / Indeed, let C denote the classical Cantor set and C∗ denote the set of all endpoints of the intervals "removed" from [0, 1] in construction of C in even

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steps. For any component (a, b) of the set [0, 1] \ C "removed" from [0, 1] in the (2n + 1)-th (n = 0, 1, 2, . . . ) step we will use the symbol hba to denote a continuous function defined on (a, b) such that for any c ∈ (a, b) we a ]. For any component (a, b) of the set have hba ((a, c)) = hba ((c, b)) = [0, a − n+1 [0, 1] \ C "removed" from [0, 1] in the 2n-th (n = 1, 2, . . . ) step we will use the symbol hba to denote a continuous function defined on (a, b) such that for any c ∈ (a, b) we have hba ((a, c)) = hba ((c, b)) = [b + bn , 2]. Define f : [0, 1] → R as follows:  for x ∈ C∗ ,  2 f (x) = 0 for x ∈ (C \ C∗ ),   b ha (x) for x ∈ (a, b), where (a, b) is a component of [0, 1] \ C. ´ atkowski The function f is a Darboux function and it has the Swi ˛ property, but C ℜD ( f ) = 0. / The details of this example are presented in [37]. It is not difficult to check that the function constructed above is not of first Baire class. The question is whether the assumption that a considered function is of first class of Baire may improve the situation. The answer is positive: Proposition 12.8. If f : [0, 1] → R is Darboux and first class of Baire, then ´ atkowski ℜCD ( f ) 6= 0. / Moreover if f also has the Swi ˛ property, then ℜCS ( f ) 6= 0. / Indeed, let K be the set of all functions h of the form h = h0 f + h1 f + · · · + hm−1 f + hm , where h0 , h1 , . . . , hm ∈ C([0, 1], R) and m ∈ N. It is easy to see that f ∈ K, C([0, 1], R) ⊂ K and K is a ring of functions. Applying the Young condition ([53]) we can show that K is a Darboux ring. Similarly, Theorem 12.5 implies that K is a ring of functions having the ´ atkowski Swi ˛ property, whenever f has this property. It should be mentioned here that in the case of Darboux Baire one functions it is possible to construct rings of functions from DB1 ([0, 1]) in another way presented in [19] and [36]. ´ atkowski) However, let us notice that there exist Darboux (Swi ˛ functions not belonging to B1 ([0, 1], R) and for which construction of such a ring is possible. Theorem 12.9. Let f : [0, 1] → R be a Darboux function such that D( f ) is a nowhere dense set and the following condition is satisfied

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181

for any x0 ∈ [0, 1] and for any ε > 0 there exists δ > 0 such that for any component J of the set C( f ) if dist(J, x0 ) < δ then dist( f (J), f (x0 )) < ε. ´ atkowski Then f is a Swi ˛ function and there exists a topology T in [0, 1] such C b b C ( f ). that C(([0, 1], T ), R) ∈ ℜD ( f ) ∩ ℜ S ´ atkowski All the above considerations regarding rings of Swi ˛ functions were connected directly with the similar considerations regarding Darboux functions. This situation is not accidental, which is shown by the next theorem. ´ atkowski Theorem 12.10. Let f : [0, 1] → R be a Swi ˛ function such that D( f ) = C b {x0 }. Then ℜ ( f ) 6= 0/ if and only if f is a Darboux function. S

In [17], [21], [52] the following issue was examined: at what assumptions regarding topology T finer than the natural topology of the real line do we have the equality C = C(([0, 1], T ), R) (this problem was also investigated in the case of more general spaces e.g. in [30])? The paper [22] presents a synthesis of the results on this issue. Natural complement to the considerations presented above is examining the possibility of creating a topology T finer than the natural topology of the real line such that the classes of real continuous and T -continuous functions are different but the families of Baire one functions in both topologies coincide. Due to the considerations of this section, it seems to be natural to demand from the family of T -continuous real functions to consist ´ atkowski only of Darboux and Swi ˛ functions. Then of course this family will be an essential and complete ring being an extension of the ring C([0, 1], R). Now let us formulate an adequate theorem. Theorem 12.11. There exists a topology T ∗ finer than a natural topology of [0, 1] fulfilling the following conditions: 1. C([0, 1], R) C(([0, 1], T ∗ ), R), 2. the families of all functions of first Baire class with respect to natural topology and to topology T ∗ coincide, ´ atkowski 3. the ring C(([0, 1], T ∗ ), R) consists of Darboux and Swi ˛ functions.

´ atkowski 12.2.2 Rings of strong Swi ˛ functions. ´ atkowski As it was pointed out in the introduction, functions with the strong Swi ˛ property are Darboux functions, so considering rings consisting of strong ´ atkowski Swi ˛ functions in this chapter is entirely justified.

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The considerations of this section are based on [23] and [44]. ´ atkowski A sum of function with the strong Swi ˛ property and linear func´ atkowski tion may not have the strong Swi ˛ property. One can ask the following questions: for which functions f is there a complete ring of functions hav´ atkowski ing the strong Swi ˛ property and containing f and what is the form of functions belonging to such ring? Searching for answers to these questions has led to formulation of "generalized Fleissner condition". We say that a function f : [0, 1] → R fulfills the generalized Fleissner condition, if f is a continuous function or f  D( f ) = const and for each x ∈ D+ ( f ) there exists a sequence {xn } ⊂ C ∗ ( f ) such that xn & x and f (xn ) = f (x), n = 1, 2, . . . and for each x ∈ D− ( f ) there exists a sequence {yn } ⊂ C ∗ ( f ) such that yn % x and f (yn ) = f (x), n = 1, 2, . . . . It is easy to show that the family of all functions fulfilling the generalized Fleissner condition is a proper subset of the class of all functions with the ´ atkowski strong Swi ˛ property. Theorem 12.12. If f fulfills the generalized Fleissner condition then b sSB∗∗ ( f ) 6= 0. ℜ / 1 The proof of the above theorem is based on the observation that if f fulfills the generalized Fleissner condition and D( f ) 6= 0/ then there exists α ∈ R such D( f ),R

that f  D( f ) = constα

.

12.2.3 Ideals of rings of almost continuous functions. According to the earlier considerations (e.g. Proposition 12.8) and due to the results presented in [19] and [36] it is easy to conclude the existence of rings of almost continuous functions containing discontinuous functions (so called: essential almost continuous rings). In reference to the previous section we can also formulate the following theorem. Theorem 12.13. If a function f fulfills the generalized Fleissner condition then b A ( f ) 6= 0. ℜ / Further considerations regarding existence of rings of almost continuous functions may be found in [44], while the next part of this section will be based on the paper [39].

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183

In the study of algebraic properties of rings, ideals play a special role (e.g. [15]). In the remaining part of the section we examine this issue in relation to the rings of almost continuous functions. We shall consider the properties of ideals of some rings of almost continuous functions, being extensions of rings of continuous functions. Due to other observations in this section, these results can also be applied easily to the other classes of functions. Let f ∈ A([0, 1], R) be a function such that D( f ) = D( f ) ⊂ Z( f ). We will e C ( f ) = {R ∈ ℜC ( f ) : D(g) ⊂ D( f ) for g ∈ R}. use the following notation ℜ A A e C ( f ) then we always assume that In the further considerations, if we write ℜ A f is a fixed function belonging to A([0, 1], R), such that 0/ 6= D( f ) = D( f ) ⊂ Z( f ). Our considerations start with the observation that the results included in the papers [3], [29], [33], [37], [38], [50] show, that for a function f satisfying the e C ( f ) 6= 0/ and, moreover, ℜ e C ( f ) contains more than one above assumptions, ℜ A A ring. The following theorem also shows some relationship between the ideals of the appropriate rings. Theorem 12.14. For each countable and closed set P ⊂ [0, 1], there exists a function f : [0, 1] → R such that f ∈ A([0, 1], R) and D( f ) = P, for which e C ( f ) such there exist two families of rings {Rη : η < c}, {Hη : η < c} ⊂ ℜ A that Rη1 6= Rη2 , Hη1 6= Hη2 , ( f )Hη1 6= ( f )Hη2 (η1 , η2 < c and η1 6= η2 ) and ( f )Rη1 = ( f )Rη2 (η1 , η2 < c). In many papers and monographs (e.g. [1], [15], [16]) the authors investigated the ideals of rings of continuous functions (often defined on more abstract space than R). So, to begin with, let us note the relations between ideals e C ( f ). of the rings of continuous functions and ideals of the rings belonging to ℜ A First let us make some preliminary observations. e C ( f ), there exists an ideal J0 of Remark 12.15. For an arbitrary ring R ∈ ℜ A the ring C([0, 1], R) such that J0 6∈ I(R). In fact. Let [a, b] ⊂ (0, 1) be a nondegenerate interval such that [a, b] ∩ D( f ) = 0/ and x0 ∈ (a, b). Putting J0 = {h ∈ C([0, 1], R) : h(x0 ) = 0} we obtain that J0 ∈ I(C([0, 1], R)). Now, we consider a function k : [0, 1] → R defined by  for x = x0 ,  0 k(x) = 1 for x ∈ [0, 1] \ (a, b),   linear in the segments [a, x0 ] and [x0 , b]. Note that k ∈ J0 but f · k 6∈ J0 .

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Since in this section rings of functions containing the family of all continuous functions are examined, so the following result seems to be interesting. e C ( f ). For an arbitrary z’-ideal J ∈ I(C([0, 1], R)) Theorem 12.16. Let R ∈ ℜ A T for which Z[J ] is not a singleton, there exist: (A) an extension J ∗ of J which is a z’-ideal of C([0, 1], R), such that J ∗ 6∈ I(R), (B) a restriction J∗ of J which is a z’-ideal of C([0, 1], R), such that J∗ ∈ I(R). The above theorem suggests considering in some sense opposite situation e C ( f ). Does there i.e. the following problem. We have a fixed ideal J of R ∈ ℜ A exist a restriction J∗ of J such that J∗ is an ideal of C([0, 1], R) and R? The following theorem gives the answer to this question. Theorem 12.17. Let f be a function for which D( f ) is a countable set. For e C ( f ), there exists a restriction J∗ ∈ I(C([0, 1], R)) ∩ each ideal J of R ∈ ℜ A I(R). Moreover, if J is an essential ideal of R, then we may assume that J∗ is also an essential ideal of R. β

Let us introduce some more notations. For a function ξ : R → R let ξα (α < β ) denote a function defined as follows (e.g. [4], p. 36):  if ξ (x) ≥ β ,  β β ξα (x) = ξ (x) if ξ (x) ∈ [α, β ],   α if ξ (x) ≤ α.

Let F be a fixed family of functions. The symbol Fb will stand for the set β {ξα : ξ ∈ F ∧ α < 0 < β }. Moreover, if (X, ρ) is a metric space, M ⊂ X and , where for fix R > 0, γ(x, R, M) x ∈ X, then p(M, x) = 2 · lim supR→0+ γ(x,R,M) R is a supremum of the set of all positive r such that there exists z ∈ X such that B(z, r) ⊂ B(x, R) \ M (here B(y, δ ) denotes an open ball i.e. B(y, δ ) = {w ∈ X : ρ(y, w) < δ } for y ∈ X and δ > 0). We shall say that M is uniformly porous if there exists m > 0 such that p(M, x) ≥ m for any x ∈ X. e C ( f ). Then the set Theorem 12.18. Let J be a nontrivial ideal of a ring R ∈ ℜ A A = Ann(J ) has the following property: Ab is uniformly porous (in Rb which is endowed with the metric of uniform convergence). e C ( f ). If J ∈ Iz0 (R0 ), then J is an intersection of Theorem 12.19. Let R0 ∈ ℜ A prime ideals.

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The proof is similar to that of Theorem 2.8 from [15] for rings of continuous functions. e C ( f ), then we can consider the set Iz0 (R) with the Note that, if R ∈ ℜ A T T metric ρ0 such that ρ0 (J1 , J2 ) = ρH ( Z[J1 ], [J2 ]) for J1 , J2 ∈ Iz0 (R), where ρH (A, B) = max(supa∈A (dist(a, B), supb∈B (dist(b, A))) for any closed sets A, B ⊂ [0, 1]. It is not hard to give an example of a z’-ideal which is not prime. So, the question arises whether this phenomenon is rare or frequent. The successive theorem is the answer to this question. e C ( f ). Theorem 12.20. Let P be the set of all prime ideals of a ring R0 ∈ ℜ A Then P ∩ Iz0 (R0 ) is a uniformly porous set in the space (Iz0 (R0 ), ρ0 ).

12.3 Rings of Darboux-like functions and problems connected with discrete dynamical systems. 12.3.1 The Sharkovsky property. In [41] the following statement of M. Misiurewicz was quoted: Combinatorial Dynamics has its roots in Sharkovsky’s Theorem. The basic version of this theorem concerns exclusively continuous functions. In [48] and [49] the theorem was generalized to the case of functions with connected and Gδ graphs (obviously such functions have the Darboux property). This part of the chapter will be based on the papers [35] and [41]. Initial considerations are intended to highlight the main ideas connected with the issues presented in this section. It is very useful to introduce the following notions. Let (I1 , I2 , . . . , IM ) be a finite sequence of continuums (Ii ⊂ R for i = 1, 2, . . . , M) and let f1 , f2 , . . . , fM : R → R. We say that (I1 , I2 , . . . , IM ) is ( f1 , f2 , . . . , fM )-cycle if I1 → I2 → I3 → f1

f2

f3

. . . → IM → I1 . If f1 = f2 = · · · = fM = f , we say that ( f1 , f2 , . . . , fM )-cycle fM−1

fM

(I1 , I2 , . . . , IM ) is ( f )-cycle. If x0 ∈ I1 is a point such that ( fi ◦ · · · ◦ f1 )(x0 ) ∈ Ii+1 for i ∈ {1, 2, . . . , M}, we shall say that x0 is connected with an ( f1 , f2 , . . . , fM )cycle (I1 , I2 , . . . , IM ). We shall say that ( f )-cycle (J1 , J2 , . . . , JM ) predominates ( f1 , f2 , . . . , fM )cycle (I1 , I2 , . . . , IM ) if for each i ∈ {1, 2, . . . , M}, there exists a homeomorphic

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embedding ξi : Ji → Ii such that ( fi ◦ · · · ◦ f1 )(ξ1 (x)) = ξi+1 ( f i (x)) for each point x connected with ( f )-cycle (J1 , J2 , . . . , JM ). A family of functions F is substituted by a class of functions F1 if for any M ∈ N and any arbitrary ( f1 , f2 , . . . , fM )-cycle (I1 , I2 , . . . , IM ), where f1 , f2 , . . . , fM ∈ F, there exists an ( f )-cycle (J1 , J2 , . . . , JM ) which predominates ( f1 , f2 , . . . , fM )-cycle (I1 , I2 , . . . , IM ) such that f ∈ F1 . We shall say that a family of functions F has the property J1 if for any ( f )cycle (I1 , I2 , . . . , IM ) ( f ∈ F1 ) there exists a point x0 connected with this cycle and such that f M (x0 ) = x0 . If F is a family of real functions of a real variable then we shall denote c F = { fn ◦ fn−1 ◦ · · · ◦ f1 : f1 , f2 , . . . , fn ∈ F, n ≥ 1}. First, we are going to establish two classes of functions PC and PD , which will form a model for our considerations. Let us note that some functions belonging to PC (PD ) were considered in many papers and monographs (e.g. [4], [5]). Let P be an arbitrary Cantor-like set in [0, 1] (additionally we assume that 0, 1 ∈ P) and let P0 ⊂ P. Then we can distinguish some properties of functions fP0 ,P : R → R which are connected with the sets P0 and P. (P1 ) fP0 ,P (x) = 0 if x ∈ P \ P0 and if P0 6= 0, / then fP0 ,P (x) = 1, if x ∈ P0 . (P2 ) fP0 ,P  [a, b] is a continuous function and fP0 ,P ([a, b]) = [0, 1] for any connected component (a, b) of [0, 1] \ P. 0 (P2 ) fP0 ,P  [a, b] is a continuous function, fP0 ,P  [a, b] is a Darboux function and fP0 ,P ([a, b]) = [0, 1] for any connected component (a, b) of [0, 1] \ P. (P3 ) fP0 ,P (x) = fP0 ,P (0), for x < 0 and fP0 ,P (x) = fP0 ,P (1), for x > 1. Let us denote by PC (PD ) a family of all functions fP0 ,P fulfilling conditions (P1 ), (P2 ), (P3 ) ((P1 ), (P20 ), (P3 )) for all possible pairs of sets (P, P0 ). It is easy to see that PC ⊂ D (PD ⊂ D) and, moreover, both classes contain nonmeasurable (in the Lebesgue sense) functions, if the measure of P is positive and P0 is a nonmeasurable set. Moreover, one can remark that the family PC is substituted by the family C and the family PD is substituted by the family DB1 . It should be mentioned here that we can consider various modifications of our models. For example, we can replace the condition (P2 ) (and (P20 )) with (P200 ) fP0 ,P  [a, b] ∈ DB1 and f ([a, b]) = [0, 1], for any component of [0, 1] \ P. Then such a family is also substituted by the family DB1 . Moreover, the assumption 3 suggests that one can consider the functions mapping [0, 1] into itself.

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Now, we can define the notion of Sharkovsky function. First we should consider the following Sharkovsky ordering of the set of all positive integers: 3 ≺ 5 ≺ 7 ≺ · · · ≺ 2·3 ≺ 2·5 ≺ · · · ≺ 22 ·3 ≺ 22 ·5 ≺ · · · ≺ 23 ≺ 22 ≺ 2 ≺ 20 = 1. We shall say that f is a Sharkovsky function provided that if PerM ( f ) 6= 0/ and M ≺ N, then PerN ( f ) 6= 0. / Theorem 12.21. Let us suppose that F is a family of functions substituted by a family F1 and the family F1 has the property J1 . Then each function f ∈ F c is a Sharkovsky function. Since C and DB1 have the property J1 , then both families PC and PD consist of Sharkovsky functions. At the end of twentieth century, a team of American mathematicians considered issues related to the theory, which can generally be called: "first return" ([9], [10], [11]). It is worth noting that the first return continuous functions have the Darboux property. In the next part we will use this theory to build our own solutions leading to defining wide class of Darboux functions (see Theorem 12.22). A set H ⊂ R is called an od-set if H is an open and dense subset of R. Let H be an od-set and f : R → R be a function. We shall say that a set H f -replaces R (denoted by H → R) if for any nondegenerated interval [α, β ] ⊂ R there f −r

exists (a, b) ⊂ [α, β ] ∩ H such that [a, b] → f ([α, β ]). f

The idea of the notions below derives from [7], [9], [10] and [11]. Let H be an od-set in R and {dn }n∈N ⊂ H be a fixed H-trajectory (i.e. {dn }n∈N is a sequence of distinct points such that {dn : n ∈ N} is dense set in H). For x ∈ R the left first return path to x based on {dn }n∈N , Pxl = {tk : k ∈ N} is defined as follows: t1 is the first element of the sequence {dn }n∈N in the set (−∞, x), for t ∈ {2, 3, . . . } the element tk+1 is the first element of the sequence {dn }n∈N in the set (tk , x). The right first return path to x based on {dn }n∈N , Pxr = {sk : k ∈ N} is defined analogously. A function f : R → R is first return continuous from the left (right) at x with respect to the H-trajectory {dn }n∈N if   lim f (t) = f (x)  lim f (t) = f (x) .

t→x t∈Pxl

t→x t∈Pxr

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A function f : R → R is an (H, {dn }n∈N )-first return continuous function ( f ∈ FRC(H, {dn }n∈N )) if it is first return continuous with respect to the Htrajectory {dn }n∈N from the left and right at each point x ∈ H and for any component (a, b) of the set H, f is first return continuous with respect to the H-trajectory {dn }n∈N from the right (left) at a (b). We shall call f an S(H, {dn }n∈N )-function ( f ∈ S(H, {dn }n∈N )) if H → R f −r

and f ∈ FRC(H, {dn }n∈N ). We say that f : R → R is an S-function ( f ∈ S) provided that there exists an od-set H and an H-trajectory {dn }n∈N such that f ∈ S(H, {dn }n∈N ). The following theorem justifies considering the class S in the context of the Darboux-like functions. Theorem 12.22. If f : R → R is an S-function, then f is a Darboux function. With reference to our considerations and the above statement it seems to be interesting to ask the following question: what kind of assumption should we impose on f in order to have guaranteed the existence of a ring belonging to b Const ( f )? ℜ S b Const ( f ). Theorem 12.23. Let f ∈ S. Then there exists a ring R ∈ ℜ S

12.3.2 Rings of Darboux-like functions and entropy points. In the introduction to this chapter it was noted that in the case of dynamical systems, some algebraic structures of functions are often considered (e.g. [46], [51]). This section will deal with rings of Darboux-like functions in the context of local interpretation of entropy. The results presented here are based on the papers [42] and [45]. We will start with introducing the concept of almost fixed point. Let f : [0, 1] → [0, 1] be a Darboux function. We will say that a point x0 is an almost fixed point of f if x0 ∈ int(R− ( f , x0 )) ∪ int(R+ ( f , x0 )). If x0 = 0 or x0 = 1, then we only consider R+ ( f , x0 ) or R− ( f , x0 ), respectively. From now on, aFix( f ) stands for the set af all almost fixed points of f and Fix( f ) denotes the set af all fixed points of f .

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It should be mentioned here that the notion of almost fixed point was created on the basis of conception of Darboux point presented by J. Lipi´nski in [25]. In the theory of discrete dynamical systems, the question regarding symmetry of properties of conjugate functions is essential. The following statement refers to this question with respect to the possession of almost fixed points by such functions. Theorem 12.24. If f , g : [0, 1] → [0, 1] are topologically conjugate via a homeomorphism φ (i.e. φ ◦ f = g ◦ φ ), and x0 ∈ aFix( f ), then φ (x0 ) ∈ aFix(g). It is easy to see that the function f : [0, 1] → [0, 1] defined by the formula: f (0) = 12 and f (x) = | sin 1x | for x ∈ (0, 1] belongs to the class DB1 and 0 ∈ aFix( f )\ Fix( f ). However, the next theorem shows, that in the case of function f ∈ DB1 ([0, 1]), in every neighborhood of any almost fixed point of f one can find a fixed point of f . Theorem 12.25. Let f ∈ DB1 ([0, 1]) and let x0 ∈ aFix( f ). Then (x0 − ε, x0 + ε) ∩ Fix( f ) 6= 0/ for each ε > 0. Our considerations are limited to the real functions defined on the interval [0, 1]. However, it should be noted that all the following definitions, Theorem 12.27 and Remark 12.28 may be formulated for more general spaces ([44]). Let f : [0, 1] → [0, 1]. An f -bundle B f is a pair (F, J) consisting of a family F of pairwise disjoint (nonsingletons) continuums in [0, 1] and a connected set J ⊂ [0, 1] (fibre of bundle) such that A → J for any A ∈ F. Let ε > 0, f

n ∈ N and B f = (F, J) be an f -bundle. A set M ⊂ F is (B f , n, ε)-separated if for each x, y ∈ M, x 6= y there is 0 ≤ i < n such that f i (x), f i (y) ∈ J and ρ( f i (x), f i (y)) > ε. Let S

maxsep[B f , n, ε] = max{card(M) : M ⊂ [0, 1] is (B f , n, ε)-separated set}. The entropy of the f -bundle B f is the number   1 h(B f ) = lim lim sup log (maxsep[B f , n, ε]) . ε→0 n→∞ n We shall say that a sequence of f -bundles Bkf = (Fk , Jk ) converges to a point x0 (written Bkf −→ x0 ), if for any ε > 0 there exists k0 ∈ N such that k→∞

S

Fk ⊂ B(x0 , ε) and B( f (x0 ), ε) ∩ Jk 6= 0/ for any k ≥ k0 . Putting E f (x) = {lim sup h(Bnf ) : Bnf −→ x} n→∞

n→∞

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we obtain a multifunction E f : X ( R ∪ {+∞}. We shall say that a point x0 ∈ [0, 1] is an entropy point of f if h( f ) ∈ E f (x0 ) (where h( f ) denotes an entropy1 of a function f ). If in addition we require that x0 ∈ Fix( f ), then such a point will be called a strong entropy point of f . The family of all functions f : [0, 1] → [0, 1] having an entropy point (a strong entropy point) will be denoted by E([0, 1]) (Es ([0, 1])). Theorem 12.26. Let f be a Darboux function. If x0 ∈ aFix( f ) ∩ Fix( f ) then x0 is a strong entropy point of f . The following theorem shows that the notion of an almost fixed point is "dynamically invariant". Theorem 12.27. Let functions f : [0, 1] → [0, 1] and g : [0, 1] → [0, 1] be topologically conjugate. Then f ∈ E([0, 1]) if and only if g ∈ E([0, 1]). The above theorem is still true if we replace E([0, 1]) with Es ([0, 1]). Let F be some class of functions from the unit interval into itself. We shall say that a function f : [0, 1] → [0, 1] is TΓ -approximated by functions belonging to F if for each open set U f containing the graph of f , there exists g ∈ F such that the graph of g is a subset of U f . We shall say that a function f : [0, 1] → [0, 1] is Tu -approximated by functions belonging to F if there exists a sequence { fn }n∈N ⊂ F uniformly convergent to f . If we consider the family A([0, 1]) or C([0, 1]) as the family F in above definitions, then we have Remark 12.28. (a) If f ∈ A([0, 1]), then the function f can be TΓ -approximated by continuous functions from Es ([0, 1]). (b) If f ∈ A([0, 1]), then the function f can be TΓ -approximated by discontinuous but almost continuous functions from Es ([0, 1]). (c) If f ∈ C([0, 1]), then the function f can be Tu -approximated by continuous functions from Es ([0, 1]). Let Per∞ ( f ) denote the set of all points x ∈ Fix( f ) such that for any open neighborhood V of x and each n ∈ N there exists yx ∈ Pern ( f ) for which O f (yx ) = { f n (yx ) : n = 0, 1, 2, . . . } ⊂ V . f If Per∞ ( f ) 6= 0, / then we will say that f has the local periodic property. The family of all functions having local periodic propery will be denoted by Per∞ . Theorem 12.29. If f ∈ DB1 ([0, 1]) then there exists a ring K ∈ RDB1 ([0,1],R) ( f ) such that 1

A definition of entropy of a function can be found in section 11.3.

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(a) the function f can be TΓ -approximated by functions belonging to KEs 40 Per∞ ([0, 1]). (b) the function f can be TΓ -approximated by functions belonging to 0 Per∞ ([0, 1]). KEs Cap Now, following [19], we will introduce another class of functions whose definition is based on the notions of an od-set, H-trajectory and (H, {dn }n∈N )first return continuity presented in Section 12.3.1. Let H ⊂ [0, 1] be an od-set in [0, 1], {dn }n∈N be an H-trajectory and f : [0, 1] → R. We say that function f is HC -connected with respect to H-trajectory {dn }n∈N if f ∈ FRC(H, {dn }n∈N ), {dn }n∈N ⊂ C( f ) and for any x ∈ [0, 1] \ H and any ε > 0 there exists δ ∈ (0, ε) such that for any component I of the set H the following condition is fulfilled: (I ∩ (x − δ , x + δ ) 6= 0) / ⇒ ( f ({dn : n = 1, 2, ...} ∩ I ∩ (x − δ , x + δ )) ∩ ( f (x) − ε, f (x) + ε) 6= 0). / The symbol ConnC will denote the family of all functions f : [0, 1] → R such that there exist an od-set H( f ) and an H( f )-trajectory {dn }n∈N such that f is H( f )C -connected with respect to {dn }n∈N . Theorem 12.30. If f ∈ ConnC ([0, 1]) then there exists a ring K ∈ RConnC ([0,1],R) ( f ) such that (a) the function f can be TΓ -approximated by functions belonging to KEs ([0, 1]). (b) the function f can be Tu -approximated by functions belonging to KEs ([0, 1]).

References [1] F. Azarpanah, Essential ideals in C (X), Period. Math. Hungar. 31(2) (1995), 105– 112. [2] A. Bi´s, P. Walczak Entropies of hyperbolic groups and some foliated spaces, Foliations: Geometry and Dynamics, World Sci. Pub., 2002, 197–211. [3] J. Brown, Almost continuous Darboux functions and Reed’s pointwise convergence criteria, Fund. Math. 86 (1974), 1–7. [4] A. M. Bruckner, Differentiation of Real Functions, CRM Monogr. Ser., vol. 5, AMS, Providence, RI, 1994. [5] A. M. Bruckner, J. G. Ceder, Darboux continuity, Jahresber. Deutsch. Math.-Verein. 67 (1965), 93–117. [6] A. M. Bruckner, J. G. Ceder, On the sum of Darboux functions, Proc. Amer. Math. Soc. 51 (1975), 97–102.

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´ ´ atek, [7] I. Cwiek, R. J. Pawlak, B. Swi ˛ On some subclasses of Baire 1 functions, Real Anal. Exchange 27(2) (2001/2002), 415–422. [8] A. Denjoy, Mémoire sur les dérivés des fonctions continues, Journ. Math. Pures et Appl. 1 (1915), 105–240. [9] U. B. Darji, M. J. Evans, R. J. O’Malley, A first return characterization of Baire one functions, Real Anal. Exchange 19 (1993/1994), 510–515. [10] U. B. Darji, M. J. Evans, R. J. O’Malley, Universally first return continuous function, Proc. Amer. Math. Soc. 123(9) (1995), 2677–2685. [11] U. B. Darji, M. J. Evans, P. D. Humke, First return approachability, J. Math. Anal. Appl. 199 (1996), 545–557. [12] R. Fleissner, A note on Baire 1 Darboux function, Real Anal. Exchange 3 (1977-78), 104–106. [13] S. Friedland, Entropy of graphs, semigroups and groups, in: Ergodic theory of Z d Actions, M. Policott and K. Schmidt (eds.), London Math. Soc. Lecture Notes Ser. 228, Cambridge Univ. Press, 1996, 319–343. [14] E. Ghys, R. Langevin, P. Walczak Entropie geometrique des feuilletages, Acta Math. 160 (1988), 105–142. [15] L. Gillman, M. Jerison, Rings of continuous functions, Springer-Verlag, 1976. [16] O. A. S. Karamzadeh, M. Rostami, On the intrinsic topology and some related ideals of C (X), Proc. Amer. Math. Soc. 93(1) (1985), 179–184. [17] E. Kocela, Properties of some generalizations of the notion of continuity of a function, Fund. Math. 78 (1973), 133–139. [18] E. Korczak-Kubiak, Pier´scienie funkcji H-spójnych, Doctoral Thesis, Łód´z University, 2009 (in Polish). [19] E. Korczak-Kubiak, R. J. Pawlak, Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions, Czech. Math. Journ., to appear. [20] J. Kosman, A. Maliszewski, Quotiens of Darboux-like function, Real Anal. Exchange 35(1) (2010), 243–251. [21] B. Koszela, On the equality of classes of continuous functions with different topologies in the set of real numbers, Demonstratio Math. 10(4) (1977), 617–627. ´ atkowski, [22] B. Koszela, T. Swi ˛ W. Wilczy´nski, Classes of continuous real functions, Real Anal. Exchange 4 (1978-79), 139–157. ´ atkowskiego, [23] J. Kucner, Funkcje posiadajace ˛ silna˛ własno´sc´ Swi ˛ Doctoral Thesis, Łód´z University, 2002 (in Polish) [24] A. Lindenbaum, Sur quelques propriétés des fonctions de variable réelle, Ann. Soc. Math. Polon. 6 (1927), 129–130. [25] J. Lipi´nski, On Darboux points, Bull. Acad. Pol. Sci. Sér. Math. Astronom. Phys. 26(11) (1978), 869–873. ´ atkowski [26] A. Maliszewski, On the limits of strong Swi ˛ functions, ZNPŁ, Matematyka, 27(719) (1995), 87–93. [27] A. Maliszewski, Darboux property and quasi-continuity. A uniform approach, Wyz˙ sza Szkoła Pedagogiczna w Słupsku, 1996. ´ atkowski, [28] T. Ma´nk, T. Swi ˛ On some class of functions with Darboux’s characteristic, ZNPŁ 301, Matematyka z.11 (1977), 5–10. [29] T. Natkaniec, Almost continuity, habilitation thesis, Bydgoszcz, 1992. [30] H. Nonas, Stronger topologies preserving the class of continuous functions, Fund. Math. CI (1978), 121–127.

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[31] R. J. O’Malley, B∗1 Darboux functions, Proc. Amer. Math. Soc. 60 (1976), 187–192. ´ atkowski [32] H. Pawlak, R. Pawlak, On some conditions equivalent to the condition of Swi ˛ for Darboux functions of one and two variables, ZNPŁ 413 (1983), 33–40. [33] H. Pawlak, R. J. Pawlak, Fundamental rings for classes of Darboux functions, Real Anal. Exchange 14 (1988/1989), 189–202. [34] H. Pawlak, R. J. Pawlak, On m-rings of functions and some generalizations of the notion of density points, Real Anal. Exchange 17, 1991-92, 550–570. [35] H. Pawlak, R. J. Pawlak, First-return limiting notions and rings of Sharkovsky’s functions, Real Anal. Exchange 34(2) (2008/2009), 549–564. [36] H. Pawlak, R. J. Pawlak, On TΓ approximation of functions by means of derivatives and approximately continuous functions having local periodic property, Real Functions, Density Topology and Related Topics, 2011, Łód´z University Press, 101–110. [37] R. J. Pawlak, Przekształcenia Darboux, Habilitation Thesis, Łód´z, 1985 (in Polish). [38] R. J. Pawlak, On rings of Darboux functions, Colloq. Math. 53 (1987), 289–300. [39] R. J. Pawlak, On ideals of extensions of rings of continuous functions, Real Anal. Exchange 24(2) (1998/1999), 621–634. [40] R. J. Pawlak, On some class of functions intermediate between the class of B∗1 and the family of continuous functions, Tatra Mt. Math. Publ. 19 (2000), 135–144. [41] R. J. Pawlak, On the Sharkovsky’s property of Darboux functions, Tatra Mt. Math. Publ. 42 (2009), 95–105. [42] R. J. Pawlak, On the entropy of Darboux functions, Colloq. Math. 116(2) (2009), 227–241. [43] R. J. Pawlak, E. Korczak, On some properties of essential Darboux Rings of real functions defined on topological spaces, Real Anal. Exchange 30(2) (2004/2005), 495–506. [44] R. J. Pawlak, J. Kucner, On some problems connected with rings of functions, Atti. Sem. Mat. Fis. Univ. Modena e Reggio Emilia LII (2004), 317–329. [45] R. J. Pawlak, A. Loranty, A. Bakowska ˛ On the topological entropy of continuous and almost continuous functions, Topology Appl. 158 (2011), 2022–2033. [46] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. [47] J. Stallings, Fixed point theorem for connectivity maps, Fund. Math. 47 (1959), 249–263. [48] P. Szuca, Punkty stałe odwzorowa´n typu Darboux, Doctoral Thesis, Gda´nsk, 2003 (in Polish). [49] P. Szuca, Sharkovskii’s theorem holds for discontinuous functions, Fund. Math. 179 (2003), 27–41. [50] A. Tomaszewska, On the set of functions possessing the property (top) in the space ´ atkowski of Darboux and Swi ˛ functions, Real Anal. Exchange 19(2) (1993/1994), 465–470. [51] P. Walczak, Dynamics of foliations, groups and pseudogroups, Mon. Mat. PAN, vol. 64, Birkhäuser Verlag, 2004. [52] W. Wilczy´nski, Topologies and classes of continuous real functions of a real variable, Rend. Circ. Mat. Palermo 26.1 (1977), 113–116. [53] J. Young, A theorem in the theory of functions of a real variable, Rend. Circ. Mat. Palermo 24 (1907), 187–192.

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E WA KORCZAK -K UBIAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

H ELENA PAWLAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

RYSZARD J. PAWLAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 13

I -approximate differentiation of real functions

EWA ŁAZAROW

2010 Mathematics Subject Classification: 26A24, 26A21, 54C30. Key words and phrases: Baire property, I-density topology, I-approximately Dini derivatives, I-approximately differentiable.

The notion of I-approximate differentiation [9] is based upon the notion of an I-density point which was introduced in [10]. For any a ∈ R and A ⊂ R, we denote aA = {ax : x ∈ A} and A − a = {x − a : x ∈ A}. Definition 13.1 ([10]). Let A ⊂ R be a set having the Baire property and x ∈ R. We say that x is an I-density point of a set A (I-d(A, x) = 1) if for each increasing sequence {nm }m∈N of positive integers, there exists a subsequence {nm p } p∈N such that n o t ∈ [−1, 1] : χnm p ·(A−x)∩[−1,1] (t) 9 1 is a set of the first category. A point x is called an I-dispersion point of a set A (I-d(A, x) = 0) if x is an I-density point of the set R \ A. If in the above definition, the interval [−1, 1] is replaced either by [0, 1] or by [−1, 0], we obtain the definitions of right-hand I-density and I-dispersion points (I-d + (A, x) = 1 and I-d + (A, x) = 0) or left-hand I-density and I-dispersion points (I-d − (A, x) = 1 and I-d − (A, x) = 0), respectively.

196

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We shall need the following characterization of the I-dispersion point of an open set. Lemma 13.2 ([5]). Let G ⊂ R be an open set. Then 0 is an I-dispersion point of G if and only if for each n ∈ N, there exist k ∈ N and a real number δ > 0 such that, for each h ∈ (0, δ ) and for each i ∈ {1, ..., n}, there exist two numbers j ∈ {1, ..., k} and j0 ∈ {1, ...k} such that      i−1 j−1 i−1 j + h, + h = 0/ G∩ n nk n nk and

      j0 i − 1 j0 − 1 i−1 + h, − + h = 0. / G∩ − n nk n nk

For each A ⊂ R having the Baire property, let ΦI (A) = {x ∈ R : x is an I-density point of A} . Recall that, for any sets A ⊂ R and B ⊂ R having the Baire property, we have the following: 1. 2. 3. 4.

ΦI (A)4A is a set of the first category, if A4B is a set of the first category, then ΦI (A) = ΦI (B), ΦI (0) / = 0/ and ΦI (R) = R, ΦI (A ∩ B) = ΦI (A) ∩ ΦI (B),

where A4B denotes the symmetric difference of the set A and B (see [10]). Further, the family TI = {A ⊂ R : A has the Baire property and A ⊂ ΦI (A)} is a topology on the real line, called I-density topology (see [10]). The topology TI is stronger then the natural topology. It is Hausdorff topology but not regular. The family of all functions continuous with respect to I-density topology we call I-approximately continuous. We recall that a set B ⊂ R is said to be the Baire cover of a set A ⊂ R if the set B has the Baire property, A ⊂ B and, for each set C ⊂ B \ A having the Baire property, the set C is of the first category. Definition 13.3. A point x ∈ R is called an exterior I-density point of a set A ⊂ R (I-de (A, x) = 1) if there exists the Baire cover B of the set A such that I-d(B, x) = 1.

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A point x ∈ R is called an exterior I-dispersion point of a set A ⊂ R (I-de (A, x) = 0) if there exists the Baire cover B of the set A such that I-d(B, x) = 0. A point x ∈ R is called an exterior right-hand I-density point of a set A ⊂ R (I-de+ (A, x) = 1) if there exists the Baire cover B of the set A such that I-d + (B, x) = 1. A point x ∈ R is called an exterior right-hand I-dispersion point of a set A ⊂ R (I-de+ (A, x) = 0) if there exists the Baire cover B of the set A such that I-d + (B, x) = 0. In a similar way we define and we denote exterior left-hand I-density points and exterior left-hand I-dispersion points of a set A ⊂ R. Definition 13.4. Let f : R → R and x ∈ R. The upper right-hand I-approximate Dini derivative of a function f at a point x (I-D+ f (x)) is defined as the greatest lower bound of the set      f (t) − f (x) > α ,x = 0 . α ∈ R : I-de+ t >x: t −x The lower right-hand I-approximate Dini derivative of a function f at a point x (I-D+ f (x)) is defined as the least upper bound of the set      f (t) − f (x) < α ,x = 0 . α ∈ R : I-de+ t >x: t −x The left-hand I-approximate Dini derivatives are defined similarly and denoted by I-D− f (x) and I-D− f (x). The ordinary Dini derivatives of a function f : R → R at a point x ∈ R, we denote by D+ f (x), D+ f (x), D− f (x) and D− f (x), respectively. To study the properties of the I-approximate Dini derivatives we shall consider only the upper right-hand I-approximate Dini derivative, because we can obtain analogous properties for other I-approximate Dini derivatives by the following 1. if for each x ∈ R, g(x) = − f (x) then for each x ∈ R,  I-D+ f (x) = − I-D+ g(x) , 2. if for each x ∈ R, g(x) = f (−x) then for each x ∈ R,  I-D− f (x) = − I-D+ g(−x) ,

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3. if for each x ∈ R, g(x) = − f (−x) then for each x ∈ R, I-D− f (x) = I-D+ g(−x). It is easy to see that Theorem 13.5. Let f : R → R. Then for each point x ∈ R, I-D+ f (x) ≤ D+ f (x). Moreover we have the following theorem Theorem 13.6 ([4]). If f : R → R is continuous, then the set  x ∈ R : I-D+ f (x) 6= D+ f (x) is of the first category. The above theorem is not true for an arbitrary function having the Baire property. If we consider the characteristic function of the set rational numbers then for each irrational numbers x, we have D+ f (x) = +∞ and I-D+ f (x) = 0. Theorem 13.7 ([9]). If f : R → R is a monotone function, then I-D+ f (x) = D+ f (x), for each point x ∈ R. Theorem 13.8 ([4]). If f : R → R and m ∈ R. If a set  A ⊂ x ∈ R : I-D+ f (x) < m is of the second category and the function f|A is continuous, then there exists an interval [a, b] ⊂ R such that the set [a, b] ∩ A is of the second category and the function h(x) = f (x) − mx is nonincreasing on [a, b] ∩ A. By taking into consideration the characteristic function of the Bernstein set it is easy to see that the I-approximate Dini derivatives of the function which does not have the Baire property may not have this property, either. But the following theorem is true. Theorem 13.9 ([3]). If a function f : R → R has the Baire property, then its I-approximate Dini derivatives have the Baire property, too. Moreover, if f is continuous, then they are of the Baire class 3.

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Additionally, by the following theorem Theorem 13.10 ([3]). Let f : R → R and  A ⊂ x ∈ R : I-D+ f (x) < +∞ . If A is a set of the second category then there exists a set W ⊂ A such that the set A \W is of the first category and the function f|W is continuous. we obtain Theorem 13.11 ([3]). Let f : R → R. If  R \ x ∈ R : I-D+ f (x) < +∞ is a set of the first category, then the function f has the Baire property. The relations between the ordinary Dini derivatives of an arbitrary real function of real variable were described in the Denjoy-Young-Saks Theorem: Theorem 13.12 ([11]). Let f : R → R and E1 = {x ∈ R : f is differentiable at x} ,  E2 = x ∈ R : D+ f (x) = D− f (x) = +∞, D+ f (x) = D− f (x) = −∞ ,  E3 = x ∈ R : D+ f (x) = D− f (x) are finite, D+ f (x) = −∞, D− f (x) = +∞ ,  E4 = x ∈ R : D+ f (x) = D− f (x) are finite, D+ f (x) = +∞, D− f (x) = −∞ . Then the set R \ (E1 ∪ E2 ∪ E3 ∪ E4 ) has Lebesgue measure zero. Theorem 13.13 ([12]). Let f : R → R be a Lebesgue measurable function and E1 = {x ∈ R : f is differentiable at x} ,  E2 = x ∈ R : D+ f (x) = D− f (x) = +∞, D+ f (x) = D− f (x) = −∞ . Then the set R \ (E1 ∪ E2 ) has Lebesgue measure zero. It is worth mentioning that the theorems remain true if we replace there the ordinary Dini derivatives by the approximate Dini derivatives (see [1]). The following example shows that the relations given in Danjoy-YoungSaks Theorem are not satisfied for I-approximate Dini derivatives even if we assume the measurability in the sense of Baire and Lebesgue.

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Example 13.14 ([4]). Let A be a set of the first category such that R \ A has Lebesgue measure zero and f be the characteristic function of the set A. Then for each x ∈ A, I-D+ f (x) = I-D+ f (x) = −∞ and I-D− f (x) = I-D− f (x) = +∞. By Theorem 13.6, it immediately follows that if the upper and lower Dini derivatives of a continuous function f : R → R are equal to +∞ and −∞, resp., then the upper and lower I-approximate Dini derivatives are equal to +∞ and −∞, respectively, on a residual set. Moreover, we have Theorem 13.15 ([4]). If a continuous function f : R → R has no finite derivative at any point, then there exists a residual set E ⊂ R such that, for each x∈E I-D+ f (x) = I-D− f (x) = −∞ and I-D− f (x) = I-D− f (x) = ∞. Theorem 13.16 ([4]). If f : R → R has the Baire property, then the sets  x ∈ R : I-D+ f (x) 6= I-D− f (x) , {x ∈ R : I-D+ f (x) 6= I-D− f (x)} are of the first category. The above theorem is not true for an arbitrary real function of a real variable, for example if we consider the characteristic function of the Bernstein set, then for each x ∈ R \ B, I-D− f (x) = I-D+ f (x) = 0, I-D+ f (x) = +∞ and I-D− f (x) = −∞. Theorem 13.16 is a category version of the Denjoy-Young-Saks Theorem, for functions having the Baire property, establishing a relation between the Iapproximate Dini derivatives. In the next theorem, it is shown that this result cannot be improved even if we assume continuity of the function f . Theorem 13.17 ([4]). For any a and b such that −∞ ≤ a < b ≤ +∞, there exists a continuous function f : [0, 1] → R and a residual set E on an interval [0, 1] such that for each point x ∈ E, I-D+ f (x) = I-D− f (x) = a and I-D+ f (x) = I-D− f (x) = b. Definition 13.18. Let f : R → R and x ∈ R.

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We say that f has a right-hand I-approximate derivative at a point  a function 0 + x I- f+ (x) , if I-D f (x) = I-D+ f (x). Then I- f+0 (x) is the common value of I-D+ f (x) and I-D+ f (x). We say that  a function f has a left-hand I-approximate derivative at a point x I- f−0 (x) , if I-D− f (x) = I-D− f (x). Then I- f−0 (x) is the common value of I-D− f (x) and I-D− f (x). We say that a function f has an I-approximate derivative at a point x (I- f 0 (x)), if I- f+0 (x) = I- f−0 (x). Then I- f 0 (x) is the common value of I- f+0 (x) and I- f−0 (x). We say that a function f is I-approximately differentiable at a point x if |I- f 0 (x)| < +∞. We say that a function f is I-approximately differentiable if f is Iapproximately differentiable everywhere. The ordinary derivative of a function f : R → R at a point x ∈ R, we denote by f 0 (x). Lemma 13.19. Let f : R → R and  A ⊂ x ∈ R : I-D+ (x) < +∞ . If the set A is dense subset of R and the function f|A is differentiable, then 0 (x), for each x ∈ A. I-D+ f (x) ≤ f|A 0 = s. We suppose that there exists β > 0 such that Proof. Let x ∈ A and f|A

I-D+ f (x) > s + β . Then x is not an exterior right-hand I-dispersion point of the set   f (t) − f (x) W = t >x: > s+β . t −x Therefore, for each δ > 0, W ∩ (x, x + δ ) is a set of the second category. Let 0 < α < β . By our assumption the function f|A is differentiable at x and therefore there exists a real number δ > 0 such that   f (t) − f (x) < s+α . A ∩ (x, x + δ ) ⊂ t > x : t −x Let V be the Baire cover of the set W . Then there exists an open interval (a, b) ⊂ (x, x + δ ) such that (a, b) \V is a set of the first category. Let y ∈ A ∩ (a, b). Then I-de+ (W, y) = 1 and for each t ∈ (y, b) ∩W ,

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f (t) − f (x) > (s + β )(t − x) and f (x) − f (y) > (−s − α)(y − x). Therefore

  f (t) − f (y) t −x y−x > s+ β −α , t −y t −y t −x

for each t ∈ (y, b) ∩W . Hence lim +

t→y , t∈W

f (t) − f (y) = +∞ t −y

0 (x). and I-D+ f (y) = +∞, a contradiction. Thus I-D+ f (x) ≤ f|A

t u

Lemma 13.20. Let f : R → R and  A ⊂ x ∈ R : I-D+ (x) < +∞ . If there exists the Baire cover B of the set A such that R \ B is a set of the first category and the function f|A is differentiable, then 0 I-D+ f (x) ≥ f|A (x),

for each x ∈ A. 0 = s. We suppose that there exists η > 0 such that Proof. Let x ∈ A and f|A

I-D+ f (x) < s − η. Then x is an exterior right-hand I-dispersion point of the set   f (t) − f (x) > s−η . S= t >x: t −x Therefore there exists the Baire cover P of a set S such that x is a right-hand I-density point of the set W = R \ P and   f (t) − f (x) W ⊂ t >x: ≤ s−η . t −x Let n ∈ N. The set W is a subset of the second category of the  interval x, x + n1 , hence there exists an open interval (an , bn ) ⊂ x, x + 1n such that (an , bn ) \ W is a set of the first category. Since A ∩ (an , bn ) is a subset of the second category of the interval (an , bn ), we have

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  1 W ∩ A ∩ x, x + 6= 0. / n  We choose a point tn ∈ W ∩ A ∩ x, x + 1n . In this way we define a sequence of real numbers {tn }n∈N such that limn→∞ tn = x, for each n ∈ N, tn ∈ A ∩ (x, +∞) and f (tn ) − f (x) ≤ s − η. tn − x By the assumption the function f|A is differentiable at x, therefore 0 0 f|A (x) ≤ f|A (x) − η, 0 (x). a contradiction. Hence I-D+ f (x) ≥ f|A

t u

By Lemmas 13.19 and 13.20 we have the following Theorem 13.21. Let f : R → R and A ⊂ R. If there exists the Baire cover B of the set A such that R \ B is a set of the first category, the function f|A is differentiable and, for each x ∈ A,  −∞ < min {I-D+ (x), I-D− (x)} ≤ max I-D+ (x), I-D− (x) < +∞, then the function f is I-approximately differentiable at each point x ∈ A. We observe that in the definition of I-approximate derivative of a function f at a point x in [9], [2], [5], [6] and [8] it was assumed that f has the Baire property in some neighborhood of x. We have defined I-approximate derivative without this assumption. But by Theorem 13.11 we know that every I-approximately differentiable function has the Baire property. Therefore if a function f is I-approximately differentiable then it is I-approximately continuous function. Moreover we have the following theorem. Theorem 13.22 ([8]). For every I-approximately continuous function f : R → R and ε > 0 there exists an I-approximately differentiable function h : R → R such that | f (x) − h(x)| < ε, for each x ∈ R. Corollary 13.23 ([8]). The uniform closure of the family of all I-approximately differentiable functions coincides with the family of all I-approximately continuous functions. Now we give the several properties of a function f : R → R which is Iapproximately differentiable.

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Theorem 13.24 ([9]). Let f : R → R. If a function f is an I-approximately differentiable then it is Baire ∗ 1, which means that there exists a sequence of closed sets {An }n∈N such that for each n ∈ N, f|An is a continuous function and S R = n∈N An . By the above we know that if a function f has a finite I-approximate derivative at each point x ∈ R then the function f is of the first class of Baire. This result is not true if a function possesses infinite I-approximate derivatives. Then we have the following theorems: Theorem 13.25 ([2]). Let f : R → R be a function having the Baire property. If a function f has an I-approximate derivative at each point x ∈ R then it is of the second class of Baire. Theorem 13.26 ([2]). There exists a function f : R → R having the Baire property such that f has an I-approximate derivative at each point x ∈ R and f is not of the first class of Baire. Theorem 13.27 ([9]). Let f : R → R. If a function f is an I-approximately differentiable then it has the Darboux property. Theorem 13.28 ([9]). Let f : R → R. If a function f is an I-approximately differentiable and I- f 0 (x) ≥ 0 at each x ∈ R, then f is nondecreasing. Theorem 13.29 ([9]). Let f : R → R be Baire 1 and Darboux. Suppose that 1. 2.

I- f 0 (x) exists except on a denumerable set, I- f 0 (x) ≥ 0 almost everywhere (with respect to the Lebesgue measure).

Then f is a nondecreasing and continuous function. Now we give the several properties of a function I- f 0 . Theorem 13.30 ([9]). Let f : R → R. If a function f is an I-approximately differentiable then the function I- f 0 has the Darboux property. Theorem 13.31 ([5]). Let f : R → R. If a function f is an I-approximately differentiable then the function I- f 0 is of Baire class one. Theorem 13.32 ([7]). Let f : R → R. If a function f is an I-approximately differentiable then there exists a sequence of perfect sets {Hn }n∈N and a sequence of differentiable functions {hn }n∈N defined on R such that 1. 2.

hn = f over Hn , h0n = I- f 0 over Hn ,

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3.

S

n∈N Hn

205

= R.

By Theorem 13.28 and Theorem 13.7 we have the following Theorem 13.33. Let f : R → R. If a function f is an I-approximately differentiable and I- f 0 is bounded above or below then for each x ∈ R, I- f 0 (x) = f 0 (x). We assume that a function f is I-approximately differentiable. Since the I-approximate derivative of f possesses the Darboux property, the above theorem forces I- f 0 to attain every value indeed infinitely often on any interval where I- f 0 is not f 0 . Thus I- f 0 must oscillate between positive and negative values whose absolute value may be as large as desired. On the other hand, since I-approximate derivative of f is a function of Baire class one, we know that there exists an open dense set G on which f 0 = I- f 0 . So the question arises whether the oscillation mentioned in the above occurs on the component intervals of the set G. In what follows, an affirmative answer is furnished to this question. Theorem 13.34 ([6]). Let f : R → R. Suppose f has a finite I-approximate derivative I- f 0 (x) at each point x ∈ (a, b) and let M ≥ 0. If I- f 0 (x) attains both M and −M on (a, b), then there exists a subinterval (c, d) ⊂ (a, b) on which I- f 0 = f 0 and f 0 attains both M and −M on (c, d). Now we give applications of the above theorem. Theorem 13.35 ([6]). Let f : R → R. Suppose f has a finite I-approximate derivative I- f 0 (x) at each point x ∈ (a, b) and let α be a real number. If  x ∈ (a, b) : I- f 0 (x) = α 6= 0/ then there exists x0 ∈ int ({x ∈ (a, b) : f is differentiable function at x}) such that f 0 (x0 ) = α. Corollary 13.36 ([6]). Let f : R → R. Suppose f has a finite I-approximate derivative I- f 0 (x) at each point x ∈ (a, b). If a set {x ∈ (a, b) : f (x) = 0} is dense in (a, b), then f is identically zero on (a, b). Corollary 13.37 ([6]). Let f : R → R and g : R → R. Suppose f and g have a finite I-approximate derivative I- f 0 (x) and I-g0 (x) at each point x ∈ (a, b). If a set {x ∈ (a, b) : f (x) = g(x)} is dense in (a, b), then f = g on (a, b).

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Corollary 13.38 ([6]). Let f : R → R and g : R → R. We assume that a function f has a finite I-approximate derivative I- f 0 (x) and a function g has a finite derivative g0 , at each point x ∈ (a, b). If f 0 = g0 on int ({x ∈ (a, b) : f is differentiable function at x}) , then f 0 = g0 on (a, b). Theorem 13.39 ([6]). Let W be a property of functions saying that any function which is differentiable and possesses W on an interval (c, d) is monotone on (c, d). Let f : R → R. If a function f has a finite I-approximate derivative I- f 0 (x) at each x ∈ (a, b) and if f has the property W on (c, d), then the function f is monotone on (a, b). Now we shall prove the relationships between I-approximate derivative and ordinary derivative. Lemma 13.40 ([3]). Let h : R → R. If D ⊂ R is a residual set such that the function h|D is continuous, then, for each open interval J ⊂ [0, +∞), the set A = {x ∈ D : (x + J) ∩ {t > x : h(t) > h(x)} is a set of the second category } is open relative to D. Lemma 13.41. Let f : R → R be a function having the Baire property, (c, d) be an open interval and b ∈ R. Let  E = x ∈ (c, d) : f is I-approximately differentiable at x and I- f 0 (x) < b . If (c, d)\E is a set of the first category, then there exist an open interval (a, b) ⊂ (c, d) and a set D ⊂ (a, b) such that (a, b) \ D is a set of the first category and for any x ∈ D and y ∈ D, if x 6= y then f (x) − f (y) < b. x−y Proof. Put g(x) = f (x) − bx for each x ∈ R. For each x ∈ R, let P(x) = {t > x : g(t) < g(x)} and L(x) = {t < x : g(t) > g(x)}. For n ∈ N, p ∈ N and h > 0 we define a set Anph in the following way: x ∈ Anph if and only if there exists j ∈ {1, ..., n} such that

13. I -approximate differentiation of real functions



207

 j−1 j · h + x, · h + x \ P(x) n n

is a set of the first category and, for each j ∈ {1..., n},   −j −j+1 · h + x, · h + x ∩ L(x) n n is a set of the second category. By Lemma 13.40 we know that for each n ∈ N, p ∈ N and h > 0, a set Anph has the Baire property. Let x ∈ E. Then x is a right-hand I-density point of the set P(x) and x is a left-hand I-density point of the set L(x). Therefore by Lemma 13.2, we have that [ [ \ Anph . E⊂ n∈N p∈N 0 g(y). Hence, for any x ∈ D and y ∈ D, if x < y then g(x) > g(y). Therefore, for any x ∈ D and y ∈ D, if x 6= y then f (y)− f (x) < b. t u y−x Lemma 13.42. Let f : R → R be a continuous function. If R \ {x ∈ R : f is I-approximately differentiable at x} is a set of the first category, then the set of all points of continuity of upper and lower derivatives of f is everywhere dense on R.

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Proof. Let (a, b) be an arbitrary open interval. Put E = {x ∈ (a, b) : f is I-approximately differentiable at x} and for each n ∈ N, En = {x ∈ E : |I- f 0 (x)| < n}. By Theorem 13.9 we know that, for each n ∈ N, the set En has the Baire S property and by our assumption E = n∈N En . Therefore there exist a positive integer n and an open interval (a1 , b1 ) ⊂ (a, b) such that (a1 , b1 ) \ En is a set of the first category. Thus, by Lemma 13.41 and by continuity of the function f , there exists a closed interval [c1 , d1 ] ⊂ (a1 , b1 ) such that for any x ∈ [c1 , d1 ] f (y) | ≤ n. Hence and y ∈ [c1 , d1 ], if x 6= y then | f (x)− x−y   −n ≤ inf f 0 (x) : x ∈ [c1 , d1 ] ≤ sup f 0 (x) : x ∈ [c1 , d1 ] ≤ n, where f 0 and f 0 denote the lower and the upper derivative of the function f , respectively. Let   1 0 A = x ∈ (c1 , d1 ) ∩ E : − n < I- f (x) < n 2 and

  1 B = x ∈ (c1 , d1 ) ∩ E : −n < I- f 0 (x) < n . 2

Since for each x ∈ E, I- f 0 (x) = I-D+ f (x), then by Theorem 13.9, the sets A and B have the Baire property and one of these is a set of the second category. We assume it is the former. Then there exists an open interval (a2 , b2 ) ⊂ [c1 , d1 ] such that   1 (a2 , b2 ) \ x ∈ (c1 , d1 ) ∩ E : − n < I- f 0 (x) < n 2 is a set of the first category. In the similar way as the above, by Lemma 13.41, we can show that there exists a closed interval [c2 , d2 ] ⊂ (a2 , b2 ) such that   1 − n ≤ inf f 0 (x) : x ∈ [c2 , d2 ] ≤ sup f 0 (x) : x ∈ [c2 , d2 ] ≤ n. 2 If the second set is a set of the second category, then we have a closed interval [c2 , d2 ] ⊂ (a2 , b2 ) such that   1 −n ≤ inf f 0 (x) : x ∈ [c2 , d2 ] ≤ sup f 0 (x) : x ∈ [c2 , d2 ] ≤ n. 2

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209

Thus   3 sup f 0 (x) : x ∈ [c2 , d2 ] − inf f 0 (x) : x ∈ [c2 , d2 ] ≤ · 2n. 4 By induction, we may define a sequence of closed intervals {[ck , dk ]}k∈N such that for each k ∈ N, [ck+1 , dk+1 ] ⊂ [ck , dk ] ⊂ (a, b) and  k−1  0 3 sup f (x) : x ∈ [c2 , d2 ] − inf f (x) : x ∈ [c2 , d2 ] ≤ 2n . 4 

Let x ∈ at x.

T

0

k∈N [ck , dk ].

Then x ∈ (a, b) and functions f 0 and f 0 are continuous t u

Theorem 13.43. Let f : R → R be a continuous function. If R \ {x ∈ R : f is I-approximately differentiable at x} is a set of the first category, then R \ {x ∈ R : f is differentiable at x} is a set of the first category, too. 0

Proof. By Lemma 13.42, a set A of points of continuity of the function f is dense and, of course, a Gδ set. Therefore A is a residual subset of R. We know 0 that the function f is differentiable at each point of continuity of f . Thus f is differentiable at each point belonging to A. t u Theorem 13.44. Let f : R → R be a function having the Baire property and E = {x ∈ R : f is I-approximately differentiable at x} . If R \ E is a set of the first category, then there exists a set M such that R \ M is a set of the first category, the function f|M is differentiable and at each point 0 (x) = I- f 0 (x). x ∈ M, f|M Proof. We consider a sequence of sequences of open intervals {{(ank , bnk }k∈N : n ∈ N} such that 1. for each n ∈ N, R = k∈N (ank , bnk ), 2. for any n ∈ N and k ∈ N, bnk − ank < n1 . S

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Let n ∈ N. We define Kn to be the family of all open intervals J n ⊂ R such that there exist k(J n ) ∈ N and a set E(J n ) ⊂ J n for which a. b.

J n \ E(J n ) is a set of the first category, for any x ∈ E(J n ) and y ∈ E(J n ), if x 6= y then ank(J n ) ≤

f (y) − f (x) ≤ bnk(J n ) . y−x

By Lemma 13.41, there exists a sequence {Jpn } p∈N ⊂ Kn such that S R \ p∈N Jpn is a set of the first category and for any p ∈ N and p0 ∈ N, if S / We put Mn = E ∩ p∈N E(Jpn ). Then R \ Mn is a set of p 6= p0 then Jpn ∩ Jpn0 = 0. the first category. Let x ∈ Mn . There exists p ∈ N such that x ∈ E(Jpn ). Therefore for each y ∈ E(Jpn ), if x 6= y then ank(Jpn ) ≤

f (y) − f (x) ≤ bnk(Jpn ) . y−x

We suppose that I- f 0 (x) < ank(J n ) . Then there exists λ > 0 such that (x − λ , x + p λ ) ⊂ Jpn and   f (y) − f (x) n < ak(Jpn ) (x − λ , x + λ ) ∩ y ∈ R : x 6= y and y−x is a set of the second category. It is impossible since R \ Mn is a set of the first category. Therefore I- f 0 (x) ≥ ank(J n ) . In a similar way we can show that p I- f 0 (x) ≤ bnk(J n ) . p Hence for each y ∈ E(Jpn ), if y 6= x then f (y) − f (x) 1 0 − I- f (x) < bnk(Jpn ) − ank(Jpn ) < . y−x n Let M = n∈N Mn . Then R \ M is a set of the first category. Let x ∈ M and n ∈ N. There exists p ∈ N such that x ∈ E(Jpn ). Then for each y ∈ Jpn ∩ M ⊂ Jpn ∩ Mn = E(Jpn ) such that x 6= y, we have f (y) − f (x) 1 0 − I- f (x) < bnk(Jpn ) − ank(Jpn ) < . y−x n T

0 (x) = I- f 0 (x). Therefore f|M

t u

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211

Now we shall consider functions I- f+0 and I- f−0 . By Theorem 13.9 we know that if a function f : R → R has the Baire property then I- f+0 and I- f−0 have the Baire property, too. Theorem 13.45. Let f : R → R be a continuous function. If the function f has a right-hand (left-hand) I-approximate derivative (finite or infinite) at each x ∈ R, then the function I- f+0 (I- f−0 ) is of the first class of Baire. Proof. Consider to fix the ideas, the derivative I- f+0 . Now we suppose that f is not of the first class of Baire. Then there exist a perfect set P and a real numbers b and d such that d < b and  D = x ∈ P : I- f+0 (x) < d is a set of the second category in P and  B = x ∈ P : I- f+0 (x) > b is dense in P. We denote, for any x ∈ R, n ∈ N and r ∈ N, D(x) = {t ∈ [x, +∞) : f (t) − f (x) ≤ d(t − x)} , B(x) = {t ∈ [x, +∞) : f (t) − f (x) ≥ b(t − x)} , and Dnr =

\

[

0 ω1 and there is a function f such that f |Z is not Borel for any uncountable set Z ⊆ R (for details see [36]). By classical theorems of Luzin and Nikodym, a function from Theorem 14.8 is nonmeasurable and does not have Baire property. Moreover, although it is possible to construct it to be injective, it is nowhere monotone in the sense that its restriction to any set of cardinality c is not monotone. Let us introduce the following notion. Definition 14.9. We say that a function f : K → K is Sierpi´nski-Zygmund function if for every set A ⊆ K of cardinality c, the restriction f |A is not a Borel map.

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The set of all Sierpi´nski-Zygmund functions f : R → R (denote it by SZ(R)) was firstly considered, in the context of algebrability, by J. L. GámezMerino, G. A. Munoz-Fernández, V. M. Sánchez and J. B. Seoane-Sepúlveda in [21]. The authors proved that this set is c+ -lineable and, also, c-algebrable. This was the motivation to ask a question if one can prove 2c -algebrability of the set SZ(R) in ZFC. Here we recall a result due to A. Bartoszewicz, S. Gła¸b, D. Pellegrino and J. B. Seoane-Sepúlveda (cf. [16]). Before stating it let us recall the following notion. Definition 14.10. Let κ be an infinite cardinal number. Let A and B be any subsets of R with cardinality κ. We say that A and B are almost disjoint provided that card(A ∩ B) < κ. The existence of 2c pairwise almost disjoint subsets, of size c, of R follows, for example, from (CH) or from (MA). It is also known that in ZFC in every set of cardinality κ there is an almost disjoint family (a family consisting of pairwise almost disjoint sets) of cardinality κ + . The main result in this context from [16] is the following. Theorem 14.11. The set SZ(R) is strongly κ-algebrable, provided there exists an almost disjoint family in R of cardinality κ. Moreover, card(SZ(R)) = 2c . Before proving the Theorem 14.11 let us state a useful lemma. Lemma 14.12 ([16]). Let P be a family of nonzero polynomials without constant term and X be a subset of R, both of cardinality less than c. Then there exists a set Y = {yξ : ξ < c}, of cardinality c, such that P(yξ1 , ..., yξn ) ∈ / X, for any polynomial P ∈ P in n variables and any distinct ξ1 < ... < ξn < c. Now we are able to prove Theorem 14.11. Proof. Let {gα : α < c} be a numeration of all Borel functions, {xα : α < c} a numeration of R and {Pα : α < c} a numeration of all nonzero polynomials without constant term. Let us inductively define a family {Yα : α < c} of subsets of R with cardinality c, by putting at the step α < c as Yα the set which existence implies Lemma 14.12 used for X = {gλ (xα ) : λ ≤ α} and P = {Pλ : λ ≤ α}. For each α < c consider a numeration Yα = {yαξ : ξ < c}. Note that Y = ∏α γ and α > β , P( fζ1 , ..., fζn ) is different from gγ at the point xα . Therefore P( fζ1 , ..., fζn ) is a Sierpi´nski-Zygmund function. t u Notice here also a simple observation. Remark 14.13. Any additive group A ⊆ SZ(R) ∪ {0} of cardinality κ generates an almost disjoint family in the plane R × R (by considering graphs of f ∈ A as members of this family). Hence the result described above and obtained by A. Bartoszewicz, S. Gła¸b, D. Pellegrino and J. B. Seoane-Sepúlveda with strong algebrability on the level of cardinality of maximal almost disjoint family in c is the best possible in that case. J. L. Gámez-Merino, J. B. Seoane-Sepúlveda noted in [23] that "κ = 2c " is independent with ZFC. This implies that the sentence "SZ(R) is strongly card(SZ(R))-algebrable (lineable)" is independent with ZFC. Moreover, it was the first time, when strong algebrability was proved in ZFC on the level higher than c (since in ZFC there is an almost disjoint family of cardinality c+ ). So the considerations took the authors to the question if there exists a free algebra of 2c generators in RR or CC . The answer to this was given by A. Bartoszewicz, S. Gła¸b and A. Paszkiewicz in their work [15]. We will come back to this in the Section Method of large free algebras.

14.4 Some general methods in algebrability and strong algebrability We will recall some general methods of constructing algebras and free algebras of real and complex functions.

14.4.1 Independent Bernstein sets The following general construction was described by A. Bartoszewicz, M. Bienias and S. Gła¸b in [12] but firstly was used by A. Bartoszewicz, S. Gła¸b,

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D. Pellegrino and J. B. Seoane-Sepúlveda in [16] in the proof of 2c -algebrability of the set PES(C). Let us recall the idea of the method. For a nonempty set X and A ⊆ X let us denote A0 = X\A and A1 = A. We name a family A to be B-independent (where B be is a family of subsets of X) if Aε11 ∩ ... ∩ Aεnn ∈ B for any distinct Ai ∈ A, any εi ∈ {0, 1} for i ∈ {1, ..., n} and n ∈ N. We say that A is independent if it is P(X) \ {0}-independent. / Well known theorem of Fichtenholz and Kantorovich (generalized to the case of any complete Boolean algebra, by B. Balcar and F. Franˇek in [9]) says that for any infinite set of cardinality κ there is an independent family of 2κ subsets of this set. Let us recall the well known definition of a Bernstein set. Definition 14.14. A subset B of a Polish space is called a Bernstein set if B ∩ P 6= 0/ 6= B0 ∩ P for every perfect subset P. Denote by B the family of all Bernstein sets. We say that a family A is an independent family of Bernstein sets provided that A ⊆ B and A is B-independent. Our aim is to construct an independent family of Bernstein sets of cardinality 2c . Repeating the idea from [12] consider the decomposition of R into c pairwise disjoint Bernstein sets {Bα : α < c}. It is easy to check that for any s ⊆ c with s 6= 0/ and c\s 6= 0, / the S set α∈s Bα is Bernstein. Let {Nξ : ξ < 2c } be an independent family in c such that for every ξ1 < ... < ξn < 2c and for any εi ∈ {0, 1} (i ∈ {1, ..., n}), the set Nξε11 ∩ ... ∩ Nξεnn has cardinality c. To construct the desired family of Bernstein sets let us put Bξ = α∈Nξ Bα , for ξ < 2c . Then every set Bξ is Bernstein. Note that for every ξ1 < ... < ξn < 2c and any εi ∈ {0, 1} for i ∈ {1, ..., n} the set S

(Bξ1 )ε1 ∩ ... ∩ (Bξn )εn =

[



ε α∈Nξ 1 ∩...∩Nξεnn 1

is Bernstein, too. That means {Bξ : ξ < 2c } is an independent family of Bernstein sets. Having the independent family of Bernstein sets, we can define 2c linearly independent functions: for α < c, let gα : Bα → C (or R) be a nonzero function defined on a Bernstein set Bα (where {Bα : α < c} is the decomposition of R into pairwise disjoint Bernstein sets). Then for every ξ < 2c let us put ( gα (x) , if x ∈ Bα and α ∈ Nξ ; fξ (x) = 0 , otherwise. Then the family { fξ : ξ < 2c } is linearly independent. Finally (cf. [12]) by spanning the algebra by the functions { fξ : ξ < 2c } we obtain an algebra

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of 2c generators. It is worth mentioning here that the independent Bernstein sets method cannot be used to prove strong 2c -algebrability. Indeed, one can consider a nonzero polynomial P(x1 , ..., xn ) = ∏k6=l (xk − xl ) ∏k xk , for which, for any collection of functions fξ1 , ..., fξn , of the above type, we have P( fξ1 , ..., fξn ) = 0. Hence, an algebra spanned by { fξ : ξ < 2c } is not a free algebra. Using the method described above it is possible to get the 2c -algebrability of several families of functions in RR or CC , namely PES(C), SES(C) \ PES(C), EDD(R). To present this method let us establish the following Theorem 14.15. The set EDF(R) is 2c -algebrable. Proof. Consider an independent family of Bernstein sets {Bξ : ξ < 2c }. For ξ < 2c let fξ be the characteristic function of the set Bξ . We will show that the family { fξ : ξ < 2c } ⊆ EDF(R) generates an algebra in EDF(R). Let P be any nonzero polynomial in n variables without a constant term and let ξ1 < ξ2 < ... < ξn < 2c . Suppose P( fξ1 , ..., fξn ) is nonzero. Notice that each fξi is constant on every set of the form (Bξ1 )ε1 ∩ ... ∩ (Bξn )εn so is P( fξ1 , ..., fξn ). Since P( fξ1 , ..., fξn ) is nonzero, there are εi ∈ {0, 1} for i ∈ {1, ..., n} such that P( fξ1 , ..., fξn )|(Bξ1 )ε1 ∩...∩(Bξn )εn 6= 0. Clearly P( fξ1 , ..., fξn )|(Bξ1 )0 ∩...∩(Bξn )0 = 0. Therefore, P( fξ1 , ..., fξn ) is everywhere discontinuous (the set of the type (Bξ1 )ε1 ∩ ... ∩ (Bξn )εn is Bernstein so it is dense). Hence EDF(R) is 2c -algebrable. t u By similar argument as in Example 14.3, it is easy to see that the set EDF(R) is not even strongly 1-algebrable. Since finite sets are compact, we have that EDF(R) ⊆ EDC(R), therefore the following holds. Corollary 14.16. The set EDC(R) is 2c -algebrable. One can ask if this result can be strengthen. The answer is: it cannot. T. Banakh, A. Bartoszewicz, M. Bienias and S. Gła¸b proved in [10] that any compact-preserving nowhere continuous function f cannot take infinitely many values on every interval. So for any function f ∈ EDC(R) there is an interval I, such that f (I) is finite. And again the same argument as in Example 14.3 shows that this set cannot be even strongly 1-algebrable. Hence EDC(R) is not strongly κ-algebrable for any cardinal κ > 0. Therefore, the result from Corollary 14.16 is the best possible in the sense of algebrability.

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14.4.2 The method of large free algebras Let us come back to the question (from the Section Sierpi´nski-Zygmund functions): does there exist a free algebra of 2c generators in RR or CC ? The positive answer to this question was given by A. Bartoszewicz, S. Gła¸b and A. Paszkiewicz in their work [15]. Moreover, they described a new general method of proving the strong 2c -algebrability, that was very useful in improving some results to the highest possible level. Here we recall some of their results. Let us start with the answer. Theorem 14.17. Let X be an infinite set of cardinality κ with κ ω = κ. Let I be a subset of K with a nonempty interior. Then there exists a free linear subalgebra of KX with 2κ generators { fξ : ξ < 2κ }, such that P( fξ1 , ..., fξk ) maps X onto P(I k ) for every nonzero polynomial P in k variables without constant term, any ξ1 , ..., ξk < 2κ and any k ∈ N. For the reader’s convenience we give a sketch of their proof. Proof. Let Y = ([0, 1] × κ)N and let {Aξ : ξ < 2κ } be an independent family of subsets of κ. For each ξ < 2κ let us define a function f¯ξ : Y → [0, 1] by a formula ∞ χ (yn ) A f¯ξ (t1 , y1 ,t2 , y2 , ...) = ∏ tn ξ , n=1

where tn ∈ [0, 1], yn ∈ κ and χA stands for the characteristic function of a set A (assume here that 00 = 1). The condition κ ω = κ implies that κ ≥ c so card(Y ) = card(X). I has nonempty interior, therefore card(I) = c and there are bijections φ : X → Y and ψ : [0, 1] → I. Let fξ = ψ ◦ f¯ξ ◦ φ : X → K for ξ < 2κ . Then { fξ : ξ < 2κ } is a set of free generators in KX . Indeed, take any ξ1 < ... < ξk < 2κ . Let n 1 Y0 = (t1 , y1 ,t2 , y2 , ...) ∈ Y : t1 , ...,tk ∈ [0, 1],ti = for i > k, 2 o [ \ yi ∈ Aξi \ Aξ j for i ≤ k and yi ∈ A0ξ j for i > k . j6=i

j

Consider a nonzero polynomial P in k variables without a constant term. Let x = φ (t1 , y1 ,t2 , y2 , ...) ∈ X0 = φ −1 (Y0 ). Then P( fξ1 , ..., fξk )(x) = P(ψ ◦ f¯ξ1 , ..., ψ ◦ f¯ξk )(φ (x)) = P(ψ(t1 ), ..., ψ(tk )).

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Notice here that φ |X0 is onto I. Since I has a nonempty interior, P is nonzero on I k . Therefore, P( fξ1 , ..., fξk ) is nonzero on X0 , so it is on X. To finish the proof observe that each function fξ is onto I. t u Let us introduce a notion that generalizes the notion of strongly everywhere surjective functions (cf. [15]). Definition 14.18. [15] Let F ⊆ P(X) and I ⊆ K. We say that a function f ∈ KX is I-strongly everywhere surjective with respect to F (in short f ∈ SES(I, F)), if for every F ∈ F there are k ∈ N and polynomial P in k variables without constant term such that f (F) = P(I k ) and card({x ∈ F : f (x) = y}) = card(X) for every y ∈ f (F). In particular we have that: • for a family F consisting of all nonempty open subsets of C, the set SES(C, F) is the set of all strongly everywhere surjective complex functions; • for a family F consisting of all nonempty perfect subsets of C, the set SES(C, F) is the set of all perfectly everywhere surjective complex functions. A. Bartoszewicz, S. Gła¸b and A. Paszkiewicz proved the strong algebrability of SES(I, F) for some families F of subsets of X and I ⊆ K. They were using a classical result due to Kuratowski and Sierpi´nski (see [32]). Proposition 14.19. (Disjoint Refinement Lemma) Let κ ≥ ω. For any family {Pα : α < κ} of sets of cardinality κ there is a family {Qα : α < κ}, such that for every distinct α, β < κ • Qα ⊆ Pα ; • Qα ∩ Qβ = 0. / The family {Qα : α < κ} is called a disjoint refinement of the family {Pα : α < κ}. Thanks to Disjoint Refinement Lemma and existence of a large free subalgebra in RX , CX , the authors obtained the following general result. Theorem 14.20 ([15]). Let X be an infinite set of cardinality κ with κ ω = κ. Let I be a subset of K with a nonempty interior and F ⊆ P(X). Assume that card(F) ≤ κ and card(F) = κ for every F ∈ F. Then the set SES(I, F) is strongly 2κ -algebrable.

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Proof. Consider a numeration {Fα : α < κ} of F, such that each set F occurs κ times. Let {Qα : α < κ} be its disjoint refinement. Without loss of generality, S we may assume that α 0 there exists an η > 0 such that for any pairwise nonoverlapping intervals [a1 , b1 ], . . . , [an , bn ], with both endpoints in E, n

n

∑ (bi − ai ) < η

i=1



∑ |F(bi ) − F(ai )| < ε.

i=1

∞ We call sequence of sets {Ei }∞ i=1 with E = i=1 Ei an E-form. If, moreover, all Ei are closed (measurable) we say the E-form is closed (measurable). A function F : E → R is said to be an ACG-function if there exists an E-form {Ei }i such that for each i, F  Ei is an AC-function. If the E-form can be chosen closed, F is said to be [ACG]. In an analogous manner one can define VB, VBG, and [VBG] properties. As it was mentioned by Lusin, the class of indefinite D-integrals coincides with the class of all continuous ACG-functions F (on a given [a, b]). As not all such functions are almost everywhere differentiable, the ordinary derivative is no longer a tool for recovery of integrand. It remains possible however, by Lebesgue theorem, to differentiate F at almost every x along Ei ⊂ [a, b], provided F  Ei is AC and x ∈ Ei is an accumulation point of Ei . As long as F is, moreover, continuous, all sets Ei from the definition of ACG can be chosen closed, so measurable, thus for almost all x the derivation along Ei can be seen (Lebesgue Density Theorem) as derivation along a set with density point at x. This leads to the notion of approximate derivative: we say F : [a, b] → R is approximately differentiable at x if there exists a set E ⊂ [a, b] with density 1 at x, such that the limit

S

lim

t→x,t∈E

F(t) − F(x) 0 = Fap (x), t −x

called the approximate derivative of F at x, exists. One can show that this definition is unique, i.e., it does not depend on E. (The notion of approximate (asymptotique) derivative has been brought first, as above, by Khintchine [27] in connection with the constructive approach to D-integral.) Theorem 15.1 (Lusin). A function f : [a, b] → R is D-integrable if and only if the following condition is satisfied:

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there exists a continuous ACG-function F : [a, b] → R 0 such that Fap (x) = f (x) at almost every x ∈ [a, b].

The further development of integration related to approximate derivatives was usually motivated in connection with the above result.

15.2 Derivation bases It will be convenient to consider definitions of some integrals using the concept of derivation basis. We introduce two bases: the approximate basis and the approximate symmetric basis. To do this we need an auxiliary concept of density gauge. A density gauge ∆ on a set E ⊂ R is defined to be an indexed collection ∆ = {∆x : x ∈ E} of subsets ∆x ⊂ [0, ∞) with the property that for each x ∈ E there is a measurable Dx ⊂ ∆x with the right density one at x. Clearly, ∆ can be identified with the plane set ∆ = {(x, h) : x ∈ E, h ∈ ∆x } ⊂ R2 . While defining the approximate symmetric basis, the latter is moreover assumed to be measurable. Having a fixed density gauge ∆ on E we define the set β∆ of all possible pairs (hx, x ± hi, x) with (x, h) ∈ ∆ . For all possible ∆ on E = R these sets β∆ will appear as elements of the approximate basis Bap . Similarly, the approximate symmetric basis Bap.s is defined as the collection of elements β∆s for all possible measurable (as plane sets) density gauges ∆ on R, where β∆s is the set of all pairs ([x − h, x + h], x) with (x, h) ∈ ∆ for a fixed ∆ . If B denotes any of the bases Bap and Bap.s , then given β ∈ B we say that (I, x) ∈ β is a β -fine tagged interval. A finite collection of β -fine tagged intervals {(Ii , xi )}ni=1 with Ii ∩ int I j = ∅ for i 6= j, is called a β -fine division. It is said to be tagged in a set E ⊂ R if xi ∈ E for each i = 1, . . . , n. It is said to be S a β -fine partition of an [a, b] if ni=1 Ii = [a, b]. For β ∈ B and a set E ⊂ R we define β [E] = {(I, x) ∈ β : x ∈ E}. Let F : [a, b] → R and the interval function ∆ F be defined by ∆ F(I) = F(d) − F(c) for all intervals I = [c, d] ⊂ [a, b] such that (I, x) ∈ β for some x and some β . We define the upper and lower derivative of F at a point x with respect to the basis B by setting 0

∆ F(I) , β ∈B (I,x)∈β [{x}] λ (I)

FB (x) = inf

sup

∆ F(I) . β ∈B (I,x)∈β [{x}] λ (I)

F 0B (x) = sup

inf

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If FB (x) = F 0B (x), then the common value is called the derivative of F at x with respect to the basis B and is denoted by FB0 (x). Derivative with respect to the basis Bap coincides with the approximate 0 (x) defined in previous section. Respectively, the derivative with derivative Fap respect to the basis Bap.s is called the approximate symmetric derivative and is 0 (x). A similar notation will be used for respective upper and denoted as Fap.s lower derivatives. We shall also use the notions of approximate continuity and approximate symmetric continuity which can be introduced in an obvious way.

15.3 Burkill’s AP-integral Most of approaches to defining integrals that solve the problem of recovering a function from its ordinary derivative, including constructive and descriptive definitions coming from the original papers by Denjoy, Lusin and Khintchine, Perron approach, the Riemann-type Kurzweil–Henstock definition and some others, were eventually proved to define the same class of integrable functions (see [50], [55]). For integration of approximate derivative the situation turned out to be more complicated. Most of researchers’ effort in this field was exerted into finding relations between approximate Perron-type integrals and the Denjoy– Khintchine integral and its approximately continuous generalizations. The approximately continuous Perron integral (AP-integral) was introduced by John Burkill in [7]. Let f : [a, b] → R. An approximately continuous function M : [a, b] → R is an ap-major function of f on [a, b] if M(a) = 0 and M 0ap (x) ≥ f (x) for all x ∈ [a, b]. An approximately continuous function m : [a, b] → R is an ap-minor function of f on [a, b] if m(a) = 0 and m0ap (x) ≤ f (x) for all x ∈ [a, b]. Definition 15.2. A function f : [a, b] → R is (Burkill’s) AP-integrable on [a, b] if f has at least one ap-major function and at least one ap-minor function on [a, b] and infM M(b) = supm m(b), where inf and sup range over all ap-major functions and ap-minor functions of f , respectively. This common value is the R AP-integral of f on [a, b] and is denoted by (AP) ab f . It is straightforward that AP-integral recovers any approximately differentiable function from its (approximate, of course) derivative F (M = m = F can be taken). A natural question is whether D-integral can serve for the same purpose. As a function having everywhere a finite approximate derivative need

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not be continuous, the answer is "no". But assuming the antiderivative is continuous, the recovery with D-integral is possible. It is a result of G. P. Tolstov [64] who proved that every approximately differentiable function has [ACG] property. As an opposite problem one can ask the question as to whether Burkill’s AP-integral, solving the recovery problem, is powerful enough to encompass D-integral, or, in other words, whether every continuous ACG-function is an indefinite AP-integral. Again Tolstov [65] provided an example that refutes such a hypothesis. Let C ⊂ [0, 1] be the Cantor ternary set. Put  at x ∈ C,  0 (m) 1 F(x) = m at x = mid Ii ,     (m) (m)  (m) (m)  linear on both min Ii , mid Ii and mid Ii , max Ii , (m)

where Ii , i = 1, . . . , 2m−1 , is one of the closed intervals of mth rank contiguous to C in [0, 1]. So defined F on [0, 1] is clearly continuous and ACG. In order to prove F cannot be an integral in Burkill’s sense, Tolstov proved that any indefinite AP-integral should fulfil the following property: for each perfect E ⊂ [0, 1] and each ε > 0 there is a portion E 0 of E such that, if (an , bn ), n ∈ N, are intervals contiguous to E 0 , for each n there is a measurable Enε ⊂ (an , bn ) such that λ (Enε ) ≥ (1 − ε)(bn − an ) and  ∑ |F(xn ) − F(an )| + |F(yn ) − F(bn )| < ∞ n∈N

for any choice of xn , yn ∈ Enε , n ∈ N. The above function F clearly does not satisfy it. P. Bullen [5], inspired by the property, gave a descriptive definition of an integral claiming that it is equivalent to the AP-integral. But this claim has been refuted: K. Liao in [35] constructed a fairly simple example of continuous nearly everywhere differentiable function without the above property. So the Bullen integral turned out to be more restrictive than the AP one. The comparison of D- and AP-integrals results in neutral: they are incomparable. This initiated the seek for a common counterpart for both these integrals, which would unify, in particular, integration of all approximate derivatives with D-integration. The first (and simplest) quest in this direction was made as early as in 1934 by J. Ridder. He called a function f : [a, b] → R, β -integrable [45] if there exists an approximately continuous [ACG]-function F : [a, b] → R with R 0 (x) at almost every x ∈ [a, b], b f := F(b) − F(a). As the class f (x) = Fap a of approximately continuous [ACG]-functions has monotonicity property with

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respect to approximate derivation, this definition is proper (i.e., the integral is defined uniquely). For a continuous F, if F  E is AC, then F  cl E is so; therefore β -integral is a straightforward extension of D-integral (via Lusin’s result, Theorem 15.1). Ridder generalized also Burkill’s AP-integral (D4 -integral of [44]) and proposed a proof that β -integral and D4 -integral are equivalent. The same definitions (respectively AD-integral in [29], AP∗ -integral in [30]) one can find in later papers by Y. Kubota. Ridder and Kubota claimed to have proved that these two generalizations are equivalent. However, they used a similar fallacious argument in justification of this claim. Then there were a few incorrect attempts to repair Ridder and Kubota’s proof (for details see [24], [48]). Eventually C.-M. Lee in [33] and D. N. Sarkhel in [48] proved that each indefinite AP-integral F on [a, b] is a Baire one star function, i.e., there is an [a, b]-form {Ei }∞ i=1 such that for each i, F  Ei is continuous. Consequently, F is an [ACG]-function (a generalization of Tolstov’s result [64]) and so AP-integral is included in Ridder’s β -integral. We say a linear space L of functions on [a, b] has the monotonicity property 0 (x) ≥ 0 for a.e. x ∈ [a, b] with respect to approximate derivation if for F ∈ L, Fap implies F is nondecreasing. Any such L induces a descriptive definition of R 0 (x) almost everywhere integral: (L) ab f = F(b) − F(a) as long as f (x) = Fap on [a, b] and F ∈ L. As examples of L, classes of continuous ACG-functions and approximately continuous [ACG]-functions (L1 ), inducing respectively Dintegral and β -integral, can be mentioned. Before the β -integral has been finally established as a generalization of AP-integral [33], [48], some other, weaker than [ACG], Lusin-type conditions defining L were considered in order to provide such generalization: L2 consisting of approximately continuous [VBG]-functions satisfying condition (N), L3 of approximately continuous ACG-functions, and L4 of approximately continuous VBG-functions satisfying (N). The integrals defined by classes L2 to L4 are known as Tap -integral [49], AD-integral [28], and AK(N) -integral [24], respectively. These integrals, at the cost of more sophisticated proof of monotonicity property, were easier to show to cover the AP-integral. The classes Li , i = 1, 2, 3, 4, do not coincide, so do the corresponding classes of integrands (see [49], [58]). Chart 1, borrowed from [58], depicts interrelations between them. For other descriptively defined approximately continuous integrals see e.g. [11], [21], [59].

15. On approximately continuous integrals

239

L2 ∩ L3

6⊃

L1

6⊂

L2 L2 + L3

L4

L3

Chart 1 Relations between Li -integrals.

15.4 Approximate Kurzweil–Henstock integral We discuss in this section an approximately continuous variant of the renowned concept of Riemann-type integral due to J. Kurzweil and R. Henstock. To define a Riemann-type integral with respect to the basis Bap it is important to notice that this basis has the partitioning property: for each β ∈ Bap and each [a, b] there exists a β -fine partition of [a, b] (see [22], Lemma 3). Definition 15.3. We say a function f : [a, b] → R is AH-integrable if there is a R number I = (AH) ab f , the value of AH-integral, such that for each ε > 0 we can find β ∈ Bap with the property that n (15.1) ∑ f (xi )λ (Ii ) − I < ε i=1 holds provided {(Ii , xi )}ni=1 is a β -fine partition of [a, b]. The value (AH) ab f is unique because Bap is filtering, i.e., β3 ⊂ β1 ∩ β2 for all β1 , β2 ∈ Bap and some β3 ∈ Bap . The above definition can be used in fact for any basis with the partitioning property giving a corresponding B-integral. The basis Bap.s does not possess the partitioning property. So a Riemann-type integral with respect to this basis requires more sophisticated definition, which will be considered in Section 15.6. For a wide class of bases, Riemann-type integral is equivalent to the appropriately defined Perron-type integral (see [41], [55]). This is also true for the basis Bap . But the Perron-type integral constructed according to that general scheme is defined by major and minor functions which are not supposed to be approximately continuous (see [55]), unlike our ap-major and ap-minor functions (page 236). The question as to whether this integral, and so also AH-integral, is equivalent to AP-integral, is still open. We can say only that Burkill’s AP-integral is covered by AH-integral. R

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We consider now some kind of descriptive characterizations of the AHintegral. For this we need a few additional definitions. The variation of a function F : R → R on a set E ⊂ R with respect to an element β ∈ B is defined as n

V (F, E, β ) = sup ∑ |∆ F(I j )|, j=1

where sup ranges over all divisions {(I j , x j )}nj=1 ⊂ β [E]. The variational measure of a set E ⊂ R, generated by F, with respect to a basis B is defined to be VB (15.2) F (E) = inf V (F, E, β ). β ∈B

If B stands for the basis Bap , we shall call (15.2) the approximate variational ap measure of E, and denote it as VF (E). Similarly, in the case of Bap.s , (15.2) is called the approximate symmetric variational measure of E and is denoted by ap.s VF (E). ∞ We say VB F is σ -finite on a set E ⊂ R if there is an E-form {En }n=1 such B B B that VF (En ) < ∞ for each n. We say VF is absolutely continuous if VF (N) = 0 for each nullset N. The class of functions generating absolutely continuous variational measure is widely used in the Kurzweil–Henstock theory of integration. In the case of approximate basis some authors (see for example [37]) use the term ASL functions (after approximate strong Lusin condition) to name functions of this class. Another class, which also plays an important role in this theory, is the class of functions of generalized absolute continuity with respect to a basis. A function F is said to be ACB on a set E if for any ε > 0 there exist η > 0 and an element β ∈ B such that ∑ni=1 |F(bi ) − F(ai )| < ε holds for any division {([ai , bi ], xi )}ni=1 ⊂ β [E] with ∑ni=1 (bi − ai ) < η. F is said to be ACGB on E if there is an E-form {Ek }∞ k=1 such that F is ACB on Ek for each k. For AH-integral many properties, known also for more general classes of bases, hold. In particular (see [3], [4], [11], [22], [23], [66]), the AH-indefinite integral F of f : [a, b] → R is approximately continuous at each point of [a, b] 0 (x) = f (x) a.e. on [a, b]. Two and it is approximately differentiable a.e. with Fap important general properties of indefinite B-integrals run as follows: P1) A function F : [a, b] → R, F(a) = 0, is the indefinite B-integral of a function f on [a, b] if and only if F generates absolutely continuous VB F and F is B-differentiable a.e. with FB0 (x) = f (x) a.e. on [a, b].

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241

P2) A function F : [a, b] → R, F(a) = 0, is the indefinite B-integral of a function f on [a, b] if and only if F is an ACGB -function B-differentiable a.e. with FB0 (x) = f (x) a.e. on [a, b]. In order to make the definitions of VB F and ACGB compatible for a function F : [a, b] → R, we accept that F(x) = F(b) for x > b and F(a) = 0 for x < a. Property P1) was proved in a more general local system setting in [66]. Property P2) is known to be equivalent to P1) for a wide class of bases B, in particular for Bap (see [14], Theorem 5.1). The properties P1) and P2) are examples of so-called partial descriptive characterizations of the indefinite integral (see [42]). A certain drawback of those characterizations is that B-differentiability a.e. of the functions in the class of primitives is included into characterization as an additional assumption. That is why they are called “partial" to distinguish them from a deeper result called “full descriptive characterization" in which the differentiability a.e. of all the functions in the class of primitives is implied by the main characteristic of the class. A full descriptive characterization of the Bap -integral is given in Theorem 15.4. The class of indefinite Bap -integrals coincides with the class of ap all functions F : [a, b] → R, F(a) = 0, generating absolutely continuous VF . This result can be easily obtained from [14], Theorem 5.1. In [61] this theorem was formulated as a corollary of a more general result. Making use of V. Ene’s result ([13], Theorem 3), we pointed out in [61] that even a weaker ap ap assumption than absolute continuity of VF , namely finiteness of VF on each nullset, implies F is almost everywhere approximately differentiable. Moreap over, in the last statement the assumption of finiteness of VF can be replaced ap by the assumption of σ -finiteness of VF on nullsets. So we have (see [53]) ap

ap

Theorem 15.5. Assume VF is σ -finite on each nullset. Then VF is σ -finite and F is approximately differentiable a.e. The proof is based on the important result (see [61]): ap

Theorem 15.6. If VF is σ -finite on a set E ⊂ R then F is VBG on E. Having VBG property we can use a Denjoy–Khintchine result ([50], chapter 7 (4.3) p.222): If a VBG-function F on a measurable set E is measurable, then it is approximately differentiable at almost every x ∈ E. So the problem of measurability is involved here. In [61], we used the following result due to Ene ([13], Theorem 3): A measurable F : [a, b] → R is VBG if and only if it is so on each nullset. The measurability assumption in this theorem is essential

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(we showed in [53] that there is a function F : [0, 1] → [0, 1] which is not VBG, but it is so on each null subset of [0, 1]). To obtain the required measurability ap in this context we proved in [53] that if VF is σ -finite on each nullset then F : R → R is measurable.

15.5 Composite–approximate Riemann-type integration The problem of defining the wide Denjoy integrals in terms of Riemann sums (thus generalizing the description of restricted Denjoy integral due to Kurzweil and Henstock) was raised by Henstock himself, yet in 1967, with a suggestion of solution, see [25]. Henstock suggested the use of what one may call a mixed derivation basis: integration ’along’ a given [a, b]-form should lead to ACG property of indefinite integrals, while integration with respect to the full integration basis (at separate, i.e., not all, points, so not to destroy the feature of the first component) should guarantee continuity of respective integrals. Henstock’s suggestion was considered later on in connection with approximate counterparts of wide Denjoy integrals (L1 - to L4 -integrals, see Section 3), [56] and [34], as well as in generalized setting [16], [60]. In all the cases (including the description of wide Denjoy integral) the problem was eventually settled only in a work of the 3rd author [60]. In our presentation, consequently, we restrict ourselves to the case of approximately continuous primitives. Given an E-form {Ei }i , we write Is {Ei }i for the set of all x such that for some i, x ∈ Ei and x is isolated from either side of Ei . Fix a density gauge ∆ on A ⊂ [a, b] and an [a, b]-form {Ei }i . On each Ei define a gauge δi , i.e., δi : Ei → (0, ∞). We call the sequence {δi }i related to {Ei }i . Consider a tagged interval (hx, yi, x). We say it is {δi }i -fine if for some i, x, y ∈ Ei and it is δi -fine. We say (hx, yi, x) is (β∆ , {δi }i )-fine if it is either {δi }i -fine or β∆ -fine (see page 235). We say a division is {δi }i -fine, or (β∆ , {δi }i )-fine if all its members are such. One can refer to pairs (β∆ , {δi }i ) as to composite–approximate gauges and (as for {δi }i ) call them related to {Ei }i . A result of Ene [16], Lemma 4.2 (see also [56], Theorem 3.1) says that for every {δi }i , related to a closed [a, b]-form {Ei }i , and any density gauge ∆ on Is {Ei }i there is a (β∆ , {δi }i )-fine partition of [a, b]. It is an extension of Henstock’s result [25], Exercise 43.9 (for the case of all ∆x being ordinary neighborhoods of respective x). Theorem 15.7. A function f : [a, b] → R is (Ridder’s) β -integrable if and only if the following property holds:

15. On approximately continuous integrals

(A1)

243

there is a closed [a, b]-form {Ei }i with the following property: for each ε > 0 there is a sequence {δi }i related to {Ei }i and, for each countable superset A ⊃ Is {Ei }i , a density gauge ∆ on A such that (15.1) is fulfilled R by each (β∆ , {δi }i )-fine partition {(Ii , xi )}ni=1 of [a, b], I = ab f .

Theorem 15.7 has been claimed in [34] (with the property (A1) called AHintegration), but finally settled only in [60] (LL-integration, Definition 7’ there). Theorem 15.8. A function f : [a, b] → R is Tap -integrable if and only if the following property holds: (A2)

to each ε > 0 we can find a closed [a, b]-form {Ei }i and a sequence of gauges {δi }i related to {Ei }i such that for any countable superset A ⊃ Is {Ei }i there is a density gauge ∆ on A such that (15.1) holds for each R (β∆ , {δi }i )-fine partition {(Ii , xi )}ni=1 of [a, b], I = ab f .

This result has been finally settled in [60] (see also [16]). The condition (A2) was called [S1 S2 R]- or [SR]-integrability [16] and E-integrability in [60], and is a reformulation of a one from [56] (AH-integral there). By the aforementioned Ene’s partitioning result [16], Lemma 4.2, both conditions (A1) and (A2) are nontrivial; i.e., the number I is unique. For more detailed discussion on the above reformulations consult [60]. As the partitioning result is no longer true for non-closed [a, b]-forms, a Riemann-type description of Kubota’s AD-integral of [28] (the one defined with L3 ) is an issue. One can provide instead, a variational characterization of this integral. Theorem 15.9 ([60], § 4.3). A function f : [a, b] → R is AD-integrable [28], with a primitive F : [a, b] → R, if and only if the following property holds: (A3)

there is an [a, b]-form {Ei }i with the following property: for each ε > 0 there is a sequence {δi }i related to {Ei }i and, for each countable superset A ⊃ Is {Ei }i , a density gauge ∆ on A such that n  ∑ f (xi )λ (Ii ) − ∆ F(Ii ) < ε i=1 is fulfilled by each (β∆ , {δi }i )-fine division {(Ii , xi )}ni=1 in [a, b].

Analogous variational descriptions of integrals defined with classes L1 and L2 easily follow from their Riemann-type characterizations (Theorems 15.7 and 15.8). It is interesting to note that in (A3) the [a, b]-form {Ei }i can be

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made dependent on ε, as in (A2), nevertheless still this condition characterizes the same integral. Therefore, it is an open question how to define in analogous terms the L4 -integral (Gordon’s AK(N) -integral). For Perron-type definitions of integrals defined with L1 to L4 see [11], [12].

15.6 Approximate symmetric Kurzweil–Henstock integral As we have already mentioned, the approximate symmetric basis Bap.s does not have the partitioning property. An extensive proof of D. Preiss and B. Thomson [43] (other proof was given in [18]), leads however to the following result which one may find as a weaker version of partitioning property: for any β∆s ∈ Bap.s there exists a set N ⊂ R of measure zero such that for every interval [c, d] with c, d ∈ R \ N there is a β∆s -fine partition of [c, d]. Definition 15.10 (periodic version, [43], § 11). A periodic function f : R → R is called ASH-integrable over a period T > 0, if there exists a number I ∈ R with the property that for every ε > 0 there is an element β∆s ∈ Bap.s such that for any β∆s -fine partition {([xi − hi , xi + hi ], xi )}ni=1 of any interval of the form [p, p + T ], p ∈ R, for which this partition exists, the inequality n ∑ 2 f (xi )hi − I < ε i=1 holds. The number I is called the approximate symmetric Henstock–Kurzweil R integral (ASH-integral) of f over T and is denoted as (ASH) T f . Due to the weak partitioning property, the integrability condition is not empty: for any β∆s there is a β∆s -fine partition of [p, p + T ] for almost all p ∈ R. Any function f : [a, b) → R can be considered for ASH-integrability: it is R enough to extend it periodically from [a, b) onto R and find ab f as the integral of the extension over the period b − a. Definition 15.11 (variational version, A-integral of [43], § 8). A function f : R → R is said to be ASH-integrable if there is a function F, an indefinite integral of f , defined almost everywhere on R with the property that for every ε > 0 there is an element β∆s ∈ Bap.s such that for any β∆s -fine division {([xi − hi , xi + hi ], xi )}ni=1 in R, n





∑ 2 f (xi )hi − F(xi + hi ) + F(xi − hi ) < ε.

i=1

15. On approximately continuous integrals

245

Such indefinite integral F is unique up to an additive constant and a nullset in R. One can prove, see [43], § 11, that for a periodic function, integrability conditions from Definitions 15.10 and 15.11 are equivalent, moreover, the periodic intergral over T is F(p + T ) − F(p) for almost all p ∈ R. An appropriate Perron type definition (ASP-integral) and descriptive definitions similar to the ones considered in Section 15.4 for the basis Bap , can be given for the basis Bap.s . Note that in definition of ASP-integral on an interval [a, b] the endpoints a, b should be treated in a special way. Namely, the approximate limit of the difference M(b − h) − M(a + h), while h → 0, should exist for all major functions, together with analogous assumption imposed on minor functions; for details see [43].

Approximate symmetric variation The classical Perron integral was defined using continuous major and minor functions. A remarkable theorem of J. Marcinkiewicz asserts that the integrability of a measurable f can be deduced from the existence of a single pair of continuous major and minor functions. This theorem has been extended, in terms of respective continuity, to some more general Perron-type integrals, including AP-integral (see [6]). It was shown in [54] that the corresponding integral with respect to the basis Bap.s does not have the Marcinkiewicz property. In fact, a stronger result is obtained in [54]: Theorem 15.12. There is a measurable function f that is not ASP-integrable on an interval [a, b] and yet f has a pair of continuous (in the usual sense) major and minor (defined with respect to Bap.s ) functions. The main tool used in construction of the above example is the approximate symmetric β∆s -variation and its continuity in the case it is generated by a continuous function. A similar method, based on continuity of β∆ -variation, can be used to prove that given continuous indefinite AH-integral, the corresponding Perron integral can be defined with continuous ap-major and ap-minor functions [2]. In the general case, as we have already mentioned, the question as to whether we can drop the requirement of approximate continuity of ap-major and ap-minor functions in Definition 15.2 is open. A similar open problem can be formulated for ASP-integral. It is known (see [20]) that ASL functions (see Section 15.4) satisfy Lusin’s condition (N) (image of a nullset is a nullset). But it is not the case for functions ap.s generating absolutely continuous variational measure VF .

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Theorem 15.13 ([51]). There exists a continuous function F on [0, 1] such that 0 (x) is finite everywhere on (0, 1), the approximate symmetric derivative Fap.s 0 and Fap.s (x) = 0 on a perfect set S of measure zero. Furthermore, F maps S onto [0, 1] and thereby does not satisfy (N). ap.s

It is easy to check that, with F of the above theorem, VF (S) = 0. At the same ap time the construction of F in [51] implies that VF (S) > 0. So the approximate and the approximate symmetric variational measure of the same function can disagree and we have ap

Theorem 15.14. There exists a continuous function with non-zero VF on some ap.s set of measure zero, yet with VF equal to zero on the same set. Another consequence of this construction is the following assertion. Theorem 15.15. The ASH-integral and Denjoy–Khintchine integral can contradict each other on the class of exact approximate symmetric derivatives, that is, there exists a D-integrable function f : [0, 1] → R having a continuous on 0 (x) = f (x) everywhere [0, 1] approximate symmetric antiderivative F with Fap.s on (0, 1), but for which F(1) − F(0) = (ASH)

Z 1 0

f 6= (D)

Z 1

f. 0

An application of ASH-integral Here we consider some application of approximate symmetric Kurzweil– Henstock integral to the theory of trigonometric series and trigonometric integrals. One of the principle questions concerning trigonometric series ∞



cn einx

(15.3)

n=−∞

is the question of recovering the coefficients of every convergent trigonometric series from its sum. To answer this question the so-called Riemann and Lebesgue theories of trigonometric series were developed, the key objects of which are twice (the Riemann function) and once formally integrated (the Lebesgue function) series, respectively: F(x) = c0

cn x2 − ∑ 2 einx , 2 n6=0 n

cn inx e . n6=0 in

l(x) = c0 x + ∑

15. On approximately continuous integrals

247

Riemann proved that F is uniformly smooth if cn → 0 as n → ∞, and D2 F(x0 ) = inx0 if additionally (15.3) is convergent at x . Thus any integration ∑∞ 0 n=−∞ cn e process recovering a function F from its second symmetric Riemann derivative D2 F will solve the problem of recovering the coefficients. As for the Lebesgue function, it is known (see [67]) that: if cn → 0 then the series defining l(x) is almost everywhere convergent, and if (15.3) is conver0 (x ) exists and equals s. (Moreover, Zygmund jointly gent at x0 to s, then lap.s 0 with Rajchman showed that if cn → 0, then l is everywhere approximately symmetrically continuous and approximately continuous at every point where it is finite.) Thus, analogously to the Riemann theory case, one should look for an integral recovering a function from its approximate symmetric derivative. The ASH-integral defined above, recovers a measurable function from its approximate symmetric derivative and thus handles the problem of recovery. More precisely, Theorem 15.16 ([43]). If a function F : R → R is measurable, approximately symmetrically continuous at each point of the line and has nearly every0 is ASHwhere approximate symmetric derivative f , then the function f = Fap.s integrable with F being an indefinite integral of f . The proof follows a standard argument with the only peculiarity related to measurability of the density gauges used. The measurability assumption on F is essential here [43]. Theorem 15.17 ([43]). If a trigonometric series (15.3) converges nearly everywhere to a finite function f , then functions f (x) and f (x)e−inx , n ∈ Z, are ASH-integrable and Z 1 f (x)e−inx dx, cn = 2π 2π where the integral is understood over the period 2π. A generalized version of trigonometric series are trigonometric integrals; i.e., integrals of the form Z ∞

iλ x

e −∞

Z ω

c(λ ) dλ = lim (L) ω→∞

eiλ x c(λ ) dλ .

−ω

The problem of recovery has a natural formulation for trigonometric integrals. It was shown in [57] that ASH-integral handles the recovery in this case as well. To this purpose an analogue of the Lebesgue theory was developed for trigonometric integrals. Consider the following assumption on a function c, called condition N0 :

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 lim

u→±∞

Z u+h  max c(λ ) dλ = 0, 06h61 u

which is an analogue of the cn → 0 condition for trigonometric series. Note, that this condition necessarily holds for any c with trigonometric integral convergent on a set of positive measure. R ∞ iλ x Formal integration of the integral −∞ e c(λ ) dλ leads to the function eiλ x − 1 c(λ ) dλ + iλ

Z

L(x) =

Z

eiλ x c(λ ) dλ . iλ

|λ |≥1

|λ | 0 there is g ∈ Θ 0 which maps K onto a set closer to f (K) than ε in Hausdorff metric. However, as Example 16.16 shows, there is a continuous mapping f such that f ([0, 1] × {0}) is not equal to g([0, 1] × {0}) for any g ∈ Ξ 0 . We don’t know if we can obtain any continuous image of [0, 1]2 by an element of Ξ 0 or Ξ.

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16.4.2 Borelity Theorem 16.19 ([11]). Every Borel function f : R2 → R2 is a composition of three axial Borel functions. We can also demand that the first axial function is horizontal. The axial functions above are not onto, the question if we can require them to be onto (provided that f is) is harder. The next theorem is an answer to Ulam’s question ([14], IV 2) and a question in [11]. Theorem 16.20 ([13]). Every Borel permutation f : R2 → R2 is a composition of eleven axial Borel permutations of R2 . We can demand that the first axial permutation is, say, horizontal. Number eleven is surely not minimal.

16.4.3 Measurability Theorems 16.19 and 16.20 hold for (Lebesgue) measurable functions and for functions with Baire property. Theorem 16.21 ([11]). Every function from R2 to R2 is a composition of three axial functions both measurable and with Baire property.

16.4.4 Slides Definition 16.22 ([1]). Function f : R2 → R2 is a slide if f (x, y) = (x, y + g(x)) or f (x, y) = (x + g(y), y) for some g : R → R. Slide is a very special case of axial function, note that it is a permutation of the plane (as a translation on horizontal or vertical lines). Very interesting (and surprising) result appeared in [1]. Theorem 16.23. Every permutation of R2 is a composition of 209(!) slides. It is possible to decrease the number 209 to even below 100 (private communication with authors of [1]). We may extend the definition of slide to any group.

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Theorem 16.24 ([6]). Let X be an infinite group and let B ⊂ X 2 such that |B| = |X 2 \ B| then using five slides we can map set B onto a fixed set D = {(x, x) : x ∈ X}. Thus using ten slides we can map any set A ⊂ X 2 to a set C ⊂ X 2 provided that |A| = |C| and |X 2 \ A| = |X 2 \C|. The question that comes first to mind is if we can present continuous (or Borel, or measurable) permutations as a composition of such slides. The answer is mostly negative as measurable slide is a measure preserving mapping ([1]) so composition of slides preserves measure of every set as well. As example 16.16 shows, even measure preserving homeomorphism cannot be a composition of slides (even axial functions). It is not known which measure preserving homeomorphisms (or Borel isomorphisms) are compositions of continuous (or Borel) slides.

16.5 Higher dimensions Definition 16.25 ([3]). Function f : X1 ×...×Xn → X1 ×...×Xn is axial if there exists i ∈ {1, ..., n} such that f (x1 , ..., xn ) = (x1 , .., xi−1 , g(x1 , ..., xn ), xi+1 , .., xn ) for some g : X1 × ... × Xn → Xi . Except [3] (and some questions in [14]) there is no literature about axial functions in higher dimensions. Virtually repeating the proof of Theorem 16.10 we obtain Theorem 16.26 ([3]). If at least one of the sets X1 , ..., Xn is infinite, then every function f : X1 × ... × Xn → X1 × ... × Xn can be represented as a composition of n + 1 axial functions f = fn+1 ◦ ... ◦ f1 . The choice of f1 is determined by which Xi is the biggest (in cardinality), in particular, if |X1 | = ... = |Xn | then f1 may change for example the first coordinate. Theorem 16.27 ([3]). For any sets X1 , ..., Xn (finite or infinite) and any permutation f : X1 × ... × Xn → X1 × ... × Xn there is k ∈ N with f = fk ◦ ... ◦ f1 , where all fi are axial permutations.

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16.5.1 Borelity The situation when we allow the axial functions to be not permutations is quite simple. Theorem 16.28. Every Borel function f : Rn → Rn is a composition of n + 1 axial Borel functions. The proof is almost identical to that of Theorem 16.19. Borel isomorphisms We prove a three dimensional analog of Theorem 16.20. Theorem 16.29. Any Borel isomorphism from R3 to R3 is a composition of 22 axial Borel isomorphisms. Although the proof follows the proof of Theorem 16.20, it is more complicated, we suggest that the reader looks at the proofs in [13] first. In order to prove Theorem 16.29 we list some useful facts, they are either well known or obvious. Fact 16.30. 1. ([5], rem.1 §1 chp.13) If f is a 1-1 Borel function then f −1 is also Borel. 2. (Borel isomorphism theorem) ([5], cor.1 §1 chp.13) Any two Borel subsets of R or R2 of the same cardinality are Borel isomorphic. 3. For any Borel sets A, B ⊂ R with |A| = |B| and |R \ A| = |R \ B| there is a Borel permutation f : R → R with f (A) = B. 4. Composition of Borel functions is Borel. 5. Function f = ( f1 , f2 ) : R2 → R2 , where f1 , f2 : R2 → R, is Borel if and only if both f1 and f2 are Borel. 6. If a function f is axial so is f −1 , if f is a composition of axial functions so is f −1 . We also list lemmas used to prove Theorem 16.20. In what follows set C ⊂ [0, 1] is a standard ternary Cantor set. Lemma 16.31 ([13]). There are three axial Borel isomorphisms F1 , F2 , F3 : R2 → R2 such that F3 ◦ F2 ◦ F1 (R2 \C × {0}) = C × {0} (thus F3 ◦ F2 ◦ F1 (C × {0}) = R2 \C × {0}). Theorem 16.32 ([13]). Let f : R2 → R2 be a Borel permutation satisfying f (C × {0}) = C × {0} (so f (R2 \C × {0}) = R2 \C × {0}) then f is a composition of eight axial Borel permutations of R2 .

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Lemma 16.33 ([13]). For every Borel permutation f : R2 → R2 there are four axial Borel permutations g3 , g2 , g1 , g0 : R2 → R2 such that g3 ◦ g2 ◦ g1 ◦ f ◦ g0 (C × {0}) = C × {0}. We prove now three-dimensional counterparts of the statements above. Lemma 16.34. There are seven axial Borel isomorphisms F1 , ..., F7 : R3 → R3 such that F7 ◦ ... ◦ F1 (R3 \ C × {(0, 0)}) = C × {(0, 0)} (thus F7 ◦ ... ◦ F1 (C × {(0, 0)}) = R3 \C × {(0, 0)}). Proof. The proof is very similar to the proof of Lemma 16.31. We may partition C into continuum many "subcantors" Ct for t ∈ R - see [13]. Sets Ct are labeled in a Borel way i.e. there is a Borel function c : C → R such that c−1 (t) = Ct , moreover all Ct are translations of each other, that is ∀t ∃mt Ct − mt = C0 (where mt = minCt ). We shift the sets Ct × {(0, 0)} on different ’z levels’, let ( (x, 0, z + c(x)) if x ∈ C, y = 0 F1 (x, y, z) = (x, y, z) otherwise (equivalently, we may write F1 (x, 0, z) = (x, 0, z + t) for x ∈ Ct ). F1 is a slide thus a bijection, it is Borel since the function c : C → R is Borel. Now we shift sets Ct × {(0,t)} ’one over other’ by a slide F2 (x, 0,t) = (x − mt , 0,t) for x ∈ Ct and identity elsewhere, this way F2 (Ct × {(0,t)}) = C0 × {(0,t)}. On every plane z = t we use Lemma 16.31 with the set C replaced with C0 to obtain three axial Borel permutations F3 , F4 , F5 satisfying F5 ◦ F4 ◦ F3 (C0 × {(0,t)}) = (R2 \ (C0 × {0})) × {t}. The sixth permutation is to ’shift back’ sets C0 to the place of Ct . Define F6 (x, 0, z) = F2−1 (x, 0, z) = (x+mt , 0,t) for x ∈ C0 and identity for other (x, y, z). The last permutation F7 = F1−1 i.e. ( (x, 0, z − c(x)) if x ∈ C, y = 0 F7 (x, y, z) = (x, y, z) otherwise (equivalently, we may write F7 (x, 0, z) = (x, 0, z − t) when x ∈ Ct ).

t u

Theorem 16.35. Let f : R3 → R3 be a Borel permutation satisfying f (C × {(0, 0)}) = C × {(0, 0)}. Then f is a composition of sixteen axial Borel permutations of R3 . Proof. The proof is almost a repetition of that of Theorem 16.32. The first seven functions F1 , ..., F7 are from Lemma above. We define F˜8 : C × {(0, 0)} → C × {(0, 0)} as follows

262

Marcin Szyszkowski

F˜8 = F7 ◦ ... ◦ F1 ◦ f ◦ (F7 ◦ ... ◦ F1 |C×{(0,0)} )−1 . It is easy to verify that F˜8 is a permutation of C × {(0, 0)} indeed. We extend F˜8 to F8 defined on the entire space R3 putting identity on R3 \C × {(0, 0)}. Functions F9 , ..., F15 are defined so that F15 ◦ ... ◦ F9 = (F7 ◦ ... ◦ F1 )−1 . We can verify that F15 ◦ ... ◦ F1 = f on R3 \C × {(0, 0)} and F15 ◦ ... ◦ F1 is identity on C × {(0, 0)}. To finish we set F16 = f on C × {(0, 0)} and F16 is identity on R3 \C × {(0, 0)}. t u Lemma 16.36. For every Borel permutation f : R3 → R3 there are six axial Borel permutations g5 , ..., g0 : R3 → R3 such that g5 ◦ ... ◦ g1 ◦ f ◦ g0 (C × {(0, 0)}) = C × {(0, 0)}. Proof. We again follow the proof of Lemma 16.33. By a perfect set we understand additionally a compact set. By Borelity there is a perfect set P ⊂ R × {(0, 0)} such that f |P is continuous (thus homeomorphism). The set f (P) ⊂ R3 is a perfect set (thus of size continuum). The projection of f (P) on XY -plane (z = 0) is a compact set, if it is of size continuum we set g1 as identity, if not then g1 (x, y, z) = (x + z, y, z) (planes perpendicular to the plane z = 0 become ’slant’), this way we assure that g1 ( f (P)) has projection at XY plane of size continuum and is still a compact set. Denote this projection by ΠXY g1 ( f (P)) ⊂ R2 × {0}. Take a function g˜2 : ΠXY g1 ( f (P)) → R defined by g˜2 (x, y) = min{z : (x, y, z) ∈ g1 ( f (P)), since g1 ( f (P)) is compact it is a lower semicontinuous function thus Borel (even first Baire class) [5, chpt.11 §2]. We define slide g2 (x, y, z) = (x, y, z − g˜2 (x, y)) for (x, y) ∈ ΠXY g1 ( f (P)) and g2 (x, y, z) = (x, y, z) for other (x, y). Using function g3 we may ensure that the projection of ΠXY g1 ( f (P)) on X-axis, denoted ΠX [ΠXY g1 ( f (P))], is compact and of size continuum. (take g2 (x, y, z) = (x + y, y, z) if necessary or g2 = identity). Again the function g˜4 : R → R defined on ΠX [ΠXY g1 ( f (P))] by g˜4 (x) = min{y : (x, y) ∈ ΠXY g1 ( f (P))} is lower semicontinuous and Borel. We define a slide g4 (x, y, z) = (x, y − g˜4 (x), z) when x ∈ ΠX [ΠXY g1 ( f (P))] and identity for other (x, y, z). Since ΠX [ΠXY g1 ( f (P))] is compact of cardinality continuum it contains a perfect set S, the set P0 = (g4 ◦ g3 ◦ g2 ◦ g1 ◦ f )−1 (S × {(0, 0)}) ⊂ P is perfect again (because functions gi restricted to proper compact sets are continuous and 1-1). Let g˜5 : R → R be a Borel permutation such that g˜0 (S) = C (where C is the Cantor set), such permutation exists by Borel isomorphism theorem ([5], Cor.1, paragraf 1, Chapter 13). Axial function g5 : R3 → R3 is defined by g5 (x, y, z) = (g˜5 (x), y, z).

16. Axial functions

263

Let g˜0 : R → R be a Borel permutation such that g˜0 (C) = P0 . Again we define g0 : R3 → R3 as g0 (x, y, z) = (g˜0 (x), y, z). We can verify that g5 ◦ ... ◦ g1 ◦ f ◦ g0 (C × {(0, 0)}) = C × {(0, 0)}. t u Proof of Theorem 16.29. Let f : R3 → R3 be a Borel permutation. Combining Lemma 16.36 and Theorem 16.35 (and using their notations) we obtain g5 ◦ ... ◦ g1 ◦ f ◦ g0 = F16 ◦ . . . ◦ F1 thus f = (g5 ◦ ... ◦ g1 )−1 ◦ F16 ◦ . . . ◦ F1 ◦ g−1 0 . Since g−1 and F are of the same type they change x-coordinate, we treat g−1 16 5 5 ◦ F16 as one permutation and conclude the proof. t u It is visible that applying the same method we obtain theorems for Rn . Theorem 16.37. Any Borel isomorphism of Rn is a composition of finitely many axial Borel isomorphisms.

16.5.2 Continuity The author conjectures that Eggleston’s Theorem 16.13 and 16.15 can be generalised to R3 , however, the proof for the plane can not be applied for R3 . As for Rn we do not dare to state any hypothesis.

References [1] M. Abert, T. Keleti, Shuffle the plane, Proc. Amer. Math. Soc. 130 (2) (2002), 549– 553. [2] H. G. Eggleston, A property of plane homeomophisms, Fund. Math. 42 (1955), 61–74. [3] A. Ehrenfeucht, E. Grzegorek, On axial maps of direct products I, Colloq. Math. 32 (1974), 1–11. [4] E. Grzegorek, On axial maps of direct products II, ibid. 34 (1976), 145–164. [5] K. Kuratowski, A. Mostowski, Set Theory, PWN Warszawa 1976. [6] P. Komjath, Five degrees of separation, Proc. Amer. Math. Soc. 130 (8) (2002), 2413– 2417. [7] M. Nosarzewska On a Banach’s problem of infinite matrices, Colloq. Math. 2 (1951), 192–197. [8] K. Płotka, Composition of axial functions of products of finite sets, Colloq. Math. 107 (2007), 15–20. [9] The Scottish Book, Edited by R. Mauldin, Birkhäuser, Boston 1981. [10] M. Szyszkowski, On axial maps of the direct product of finite sets, Colloq. Math. 75 (1998), 31–38. [11] M. Szyszkowski, A note on axial functions on the plane, Tatra Mt. Math. Publ. 40 (2008), 59–62.

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[12] M. Szyszkowski, Axial continuous functions, Topology Appl. 157 (2010), 559–562. [13] M. Szyszkowski, Axial Borel functions, Topology Appl. 160, no 15 (2013), 2049— 2052. [14] S. Ulam, A collection of mathematical problems, New York 1960.

M ARCIN S ZYSZKOWSKI Institute of Mathematics, University of Gda´nsk ul. Wita Stwosza 57, 80-952 Gda´nsk, Poland E-mail: [email protected]

Chapter 17

Measurability of multifunctions with the (J) property

˙ ´ GRAZYNA KWIECINSKA

2010 Mathematics Subject Classification: 54C60, 28C20, 26B05, 28A15, 58C20. Key words and phrases: multifunctions, measurability, sup-measurability, integrability, derivative.

In various problems, one encounters measurability of multifunctions (called also set-valued functions) of two variables. Obviously, each multifunction of two variables x ∈ X and y ∈ Y may be treated as a multifunction of the single variable (x, y) ∈ X ×Y . The essential difference is the possibility of formulating hypotheses concerning the multifunction in terms of its sectionwise properties. In this case, we can speak about product (sometimes called joint) measurability and superpositional measurability (sup-measurability for short), i.e., roughly speaking, measurability with respect to a product σ -algebra and measurability of Carathéodory type superposition F(x, G(x)), respectively, where F and G are multifunctions. The difference between sup-measurability and joint measurability is essential. In general, neither of the inclusions between the class of joint measurable multifunctions and the class of sup-measurable multifunctions is true. It is easy to define a joint Lebesgue measurable real function which is not supmeasurable [15]. On the other hand Z. Grande and J. S. Lipi´nski have given an example of a sup-measurable real function which is not measurable as a function of two variables [8].

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In the single valued version, the problem of product measurability and supmeasurability has been studied very extensively (an overview of some papers in this field can be found in [7]). An important contribution to this field, among others, has made J. S. Lipi´nski. Far less is known, however, in the multivalued case. There are various sufficient conditions on sections of f ensuring that f is product measurable (e.g. [2], [3] and [5]–[7]). The most important one (given by H. D. Ursell [14]) is measurability of f in the first and its continuity in the second variable. The measurability of f can be obtained from weaker assumptions. J. S. Lipi´nski [11] has shown that under an additional assumption one can obtain product measurability of f if it is a derivative in the second variable. In order to attain this result he introduced the (J) property of a real function of two real variables (intensively studied by Z. Grande in the case of real functions defined on more general spaces [7]). Our purpose is to consider this topic in the case of multifunctions. Let S and Z be nonempty sets and let Φ be a mapping which associates to each point s ∈ S a nonempty set Φ(s) ⊂ Z. Such a mapping is called a multifunction from S to Z and we write Φ : S Z. If Φ : S Z is a multifunction, then for a set A ⊂ Z two inverse images of A under Φ are defined as follows: Φ + (A) = {s ∈ S : Φ(s) ⊂ A} and Φ − (A) = {s ∈ S : Φ(s) ∩ A 6= 0}. / A function f : S → Z may be considered as a multifunction assigning to s ∈ S the singleton { f (s)}. It is clear that in this case for a set A ⊂ Z we have f + (A) = f − (A) = f −1 (A). Let us suppose that (Z, d) is a metric space. If z0 ∈ Z and M ⊂ Z, then in standard notation, d(z0 , M) = infz∈M {d(z0 , z)}. Let P(Z) be the power set of Z and let P0 (Z) = P(Z) \ {0}. / We put Cb (Z) = {A ∈ P0 (Z) : A is closed and bounded}, K(Z) = {A ∈ P0 (Z) : A is compact}. Let h be the Hausdorff metric in Cb (Z) generated by the metric d, i.e. for A, B ∈ Cb (Z) h(A, B) = max(supz∈B {d(z, A)}, supz∈A {d(z, B)}). There are several ways of defining convergence in P0 (Z) and in consequence its connections with continuity. Throughout the chapter, convergence in the space Cb (Z) will be convergence in the Hausdorff metric h.

17. Measurability of multifunctions with the (J) property

267

A sequence (Φn )n∈N of multifunctions Φn : S Z with values in Cb (Z) is called converging to a multifunction Φ : S Z if for each s ∈ S the sequence (Φn (s))n∈N converges to Φ(s) with respect to the Hausdorff metric h. We will write Φ = h-limn→∞ Φn . It is clear that (1) If s ∈ S and Φ(s) = h-limn→∞ Φn (s), then for each z ∈ Z d(z, Φ(s)) = limn→∞ d(z, Φn (s)). Now let (X, A) be a measurable space and (Z, T ) a topological space. We will say a multifunction Φ : X Z is A-measurable (weakly A-measurable) if Φ + (G) ∈ A (Φ − (G) ∈ A) for each G ∈ T . It is evident that in the case of a single valued function f : X → Z, the notions of A-measurability of f and weak A-measurablity of f coincide with the usual notion of measurability of f , i.e., f −1 (G) ∈ A for each G ∈ T . Excellent source of information on measurability properties of multifunctions with values in a metric space is the paper of Castaing and Valadier [1]. We now mention those properties which will be useful later on. Proposition 17.1. If (X, A) is a measurable space, (Z, d) is a metric space and Φ :X Z is a multifunction, then (i) A-measurability of Φ implies weak A-measurability of Φ. (ii) If Φ is compact valued, then A-measurability of Φ and weak A-measurability of Φ are equivalent. (iii) If the space (Z, d) is separable, then Φ is weakly A-measurable if and only if the function gz : X → R given by gz (x) = d(z, Φ(x)) is A-measurable for each z ∈ Z. (iv) If Φ is compact valued, then A-measurability and weak A-measurability of Φ are equivalent to A-measurability of the function Φ : X → (K(Z), h). Observe that, by (1) and Proposition 17.1 (iii), the following property is true. (2) If (Z, d) is separable and a sequence (Φn )n∈N converges to Φ, then Φ is weakly A-measurable whenever Φn is weakly A-measurable for each n ∈ N. There are several ways of defining continuity of multifunctions. Since we well consider multifunctions with values in a metric space we mention only continuity with respect to the Hausdorff metric h. Let (Y, ρ) be a metric space and let Φ : Y Z be a multifunction with values in Cb (Z). The statement that Φ is h-continuous will mean that Φ treated as a function from Y to the space (Cb (Z), h) is continuous.

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From now on, let (Z, || · ||) be a reflexive Banach space with a metric d generated by the norm; θ will denote the origin of Z, ||K|| = h(K, {θ }) when K ∈ Cb (Z); co(K) will denote the convex hull of K. If A ⊂ Z and B ⊂ Z and α ∈ R then, as usual, A + B = {a + b : a ∈ A ∧ b ∈ B}

and

αA = {αa : a ∈ A}.

It is known that ([4], Lem. 2.2 (ii)) (3) If Ai , Bi ∈ Cb (Z) for i = 1, 2, then h(A1 + A2 , B1 + B2 ) ≤ h(A1 , B1 ) + h(A2 , B2 ). We put Cbc (Z) = {A ∈ Cb (Z) : A is convex}. By reflexivity of (Z, || · ||), the space Cbc (Z) with the addition defined above is a commutative semigroup which satisfies the cancellation law (see [13]). The assumption that (Z, || · ||) is reflexive is used to show that (4) (5)

A + B ∈ Cbc (Z) whenever A, B ∈ Cbc (Z) ([13], Th. 2). If A, B,C ∈ Cbc (Z), then h(A, B) = h(A +C, B +C) ([13], Lem. 3).

The completeness of (Z, d) implies (Cb (Z), h) is complete. Therefore Price’s inequality h(co(A), co(B)) ≤ h(A, B) ([12], (2.9), p.4) implies that (6) If (Z, d) is complete, then a Cauchy sequence in Cbc (Z) must converge to an element of Cbc (Z). From now on, unless otherwise stated, we assume that all considered multifunctions have values in Cbc (Z). Let T ⊂ R be an L-measurable set and let Φ : T Z be an L-measurable multifunction. Suppose that Φ is bounded, i.e. there is a totally bounded set K ⊂ Z such that Φ(t) ⊂ K for each t ∈ T . We define an integral of Φ as follows (cf. [9], p. 218, in the case Z = Rk ). If Φ takes only a finite number of values B1 , B2 , ..., Bn , then we put Z E

n

Φ(t) dt = ∑ λ (Di ) · Bi , i=1

where E ⊂ T is a bounded L-measurable set and Di = {t ∈ E : Φ(t) = Bi } for i = 1, 2, ..., n. By (4), (7)

R

E Φ(t) dt

∈ Cbc (Z).

It is easy to see that

17. Measurability of multifunctions with the (J) property

(8)

If A, B ∈ L are non-overlapping and E = A ∪ B, then Z

Z

Φ(t) dt = E

Let Ψ : T obtains

Z

Φ(t) dt + A

Φ(t) dt. B

Z be an L-measurable and bounded multifunction. Using (3) one Z

(9)

269



Z

h E



Ψ (t) dt

Φ(t) dt, E

Z

h(Φ(t),Ψ (t)) dt E

whenever Φ and Ψ take a finite number of values. For a general case of an L-measurable and bounded multifunction the definition of its integral is based on the following lemma ([10], Lem. 1). Lemma 17.2. Let a totally bounded convex set K ⊂ Z and a number δ > 0 be given. Then there exists a finite family Fδ ⊂ Cbc (Z) such that if D ∈ Cbc (K), then there exists a smallest set B ∈ Fδ such that D ⊂ B ⊂ B(D, δ ). Now, take K in the lemma to be the totally bounded convex set containing all the values of Φ. Suppose t ∈ T . Let Fδ be the family corresponding to δ > 0, and let Φδ (t) be the smallest member of Fδ containing Φ(t). Then h(Φ(t), Φδ (t)) < δ and Φδ : T Z takes only a finite number of values. Moreover, if (δn )n∈N is a sequence of positive real numbers and limn→∞ δn = 0, then, by (7) and (9), Z  Φδn (t) dt E

n∈N

R

is a Cauchy sequence in Cbc (Z). Thus, by (6), the limit h − limδ →0 E Φδ (t) dt exists in Cbc (Z) and we take this limit to be the integral of Φ on E, i.e. Z E

Φ(t) dt := h − lim

Z

δ →0 E

Φδ (t) dt ∈ Cbc (Z).

Note that by a passage to a limit in (8) and (9) we see that (10) The properties (8) and (9) are true for each L-measurable and bounded multifunction. In particular, R R || E Φ(x) dx|| ≤ E ||Φ(x)|| dx. From now on we make the assumption that I ⊂ R is an interval. Lemma 17.3. Let I = [a, b]. If an L-measurable multifunction Φ : I Z is bounded and 0 < δ < b − a, then the multifunction Φδ : I Z given by

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270

(R

x+δ x

Φδ (x) =

Rb b−δ

Φ(t) dt

if a ≤ x < b − δ ,

Φ(t) dt

if b − δ ≤ x ≤ b,

is h-continuous. Proof. Let x0 ∈ I be fixed. Let us suppose that x0 < b − δ and x0 < x < b − δ . Then Z x +δ  Z x+δ 0 Φ(t) dt, Φ(t)dt = h(Φδ (x0 ), Φδ (x)) = h x0 x Z x  Z x0 +δ Z x0 +δ Z x+δ =h Φ(t) dt + Φ(t) dt, Φ(t) dt + Φ(t) dt = x0 x x x0 +δ  Z x Z x+δ Φ(t) dt, Φ(t)dt , by (5). =h x0 +δ

x0

Thus, by (10), Z

x

Z x+δ



h(Φδ (x0 ), Φδ (x)) = h Φ(t) dt, Φ(t) dt ≤ x0 x0 +δ Z x Z x+δ ≤ Φ(t) dt + Φ(t) dt → 0 as x → x0 . x0 +δ

x0

If x0 − δ < x < x0 , then Z

x0 +δ

Z x+δ



h(Φδ (x0 ), Φδ (x)) = h Φ(t) dt, Φ(t)dt = x0 x Z x+δ  Z x0 +δ Z x0 Z x+δ =h Φ(t) dt + Φ(t) dt, Φ(t) dt + Φ(t) dt = x0 x+δ x x0 Z x +δ  Z x0 0 =h Φ(t) dt, Φ(t) dt → 0 as x → x0 . x+δ

x

Now let us suppose that x0 ≥ b − δ . Since Φδ is constant for b − δ ≤ x ≤ b, it is enough to consider only the case x0 = b − δ and x0 − δ < x < x0 . Then Z b  Z x+δ h(Φδ (x0 ), Φδ (x)) = h Φ(t) dt, Φ(t)dt = x0 x Z x+δ  Z b Z x0 Z x+δ =h Φ(t) dt + Φ(t) dt, Φ(t) dt + Φ(t) dt = x0 x+δ x x0 Z x +δ  Z x0 0 =h Φ(t) dt, Φ(t) dt → 0 as x → x0 , x+δ

which proves Lemma 17.3.

x

t u

17. Measurability of multifunctions with the (J) property

271

Z be an L-measurable bounded multifunction and x0 ∈ I.

Let Φ : I

Definition 17.4. The statement that Φ is a derivative at x0 ∈ I means, that 1 x→x0 x − x0

Φ(x0 ) = h− lim

Z x

Φ(t) dt. x0

Φ is a derivative if it is a derivative at each point x ∈ I. Similarly to the case of real functions one can show: Proposition 17.5. Let x0 ∈ I. If a multifunction Φ : I x0 , then Φ is a derivative at x0 .

Z is h-continuous at

Now we present a different approach of defining integrability for multifunctions. It is based on the definition of Riemann integral. Moving from Hukuhara’s idea (cf. [9] in the case Z = Rk ) we define R-integrability of multifunctions in a more general case. Let Φ : I Z be a bounded multifunction. Let ∆ = {a0 , a1 , ...an } be a partition of I and let ν(∆ ) = max{ai+1 − ai } be the diameter of the partition. Let P denote the family of all pairs (∆ , τ), where τ = (t0 ,t1 , ...tn−1 ) is a sequence of points such that ti ∈ [ai , ai+1 ] for i = 0, ..., n − 1. We put n−1

CΦ (∆ , τ) =

∑ (ai+1 − ai )Φ(ti )

i=0

for (∆ , τ) ∈ P. Note that (4) implies CΦ (∆ , τ) ∈ Cbc (Z). We say that a multifunction Φ : I Z is R-integrable (on I) if there exists B ∈ Cbc (Z) such that ∀ε>0 ∃η>0 ∀(∆ ,τ)∈P [ν(∆ ) < η ⇒ h(CΦ (∆ , τ), B) < ε], R

and we define (R) I Φ(t) dt to be the set B. Note that, by (3), n−1

h(CΦ (∆ , τ),CΨ (∆ , τ)) ≤

∑ (ai+1 − ai ) h(Φ(ti ),Ψ (ti ))

i=0

whenever Ψ : I Z is a bounded multifunction. Thus Z  Z Z h Φ(t) dt, Ψ (t) dt ≤ h(Φ(t),Ψ (t)) dt ≤ (b − a) ε, I

I

I

provided that h(Φ(ti ),Ψ (ti )) ≤ ε for each t ∈ I. Therefore, similarly to the case of real functions, (11)

If Φ : I

Z is h-continuous, then Φ is R-integrable.

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Proposition 17.6. If a multifunction Φ : I Z is bounded and almost everywhere h-continuous, then Φ is R-integrable. Proof. Let K ∈ Cbc (Z) be such that Φ(t) ⊂ K for t ∈ I. Let DΦ denote the set of discontinuity points of Φ. By assumption, λ (DΦ ) = 0. Fix ε > 0. Let (In )n∈N S be a sequence of open intervals such that DΦ ⊂ n∈N In and Σn∈N λ (In ) < ε. Without loss of generality we can assume that In ∩ Im = 0/ for n 6= m. Let In = S (αn , βn ) for n ∈ N and Aε = [a, b] \ n∈N In . Then λ (Aε ) > b − a − ε. We define a multifunction Φε : I Z by ( Φ(t) if t ∈ Aε , Φε (t) = βn −t t−αn if t ∈ (αn , βn ) ∩ I, n ∈ N. βn −αn Φ(αn ) + βn −αn Φ(βn ) Note that Φε (t) ∈ Cbc (Z). Moreover, Φε is h-continuous and, by (11), also RR integrable. Let B ∈ Cbc (Z) be such that I Φε (t) dt = B. Let (∆ , τ) ∈ P and η > 0 be such that ν(∆ ) < η and h(CΦε (∆ , τ), B) < ε. Then h(CΦ (∆ , τ), B) ≤ h(CΦ (∆ , τ),CΦε (∆ , τ)) + h(CΦε (∆ , τ), B) = n−1 n−1 = h(Σi=0 (ai+1 − ai ) Φ(ti ), Σi=0 (ai+1 − ai )Φε (ti )) + h(CΦε (∆ , τ), B),

and then, by (3), n−1 h(CΦ (∆ , τ), B) ≤ Σi=0 (ai+1 − ai ) h(Φ(ti )), Φε (ti ))) + h(CΦε (∆ , τ), B).

For that reason h(CΦ (∆ , τ), B) ≤ 2 ε ||K|| + ε, since Φ(ti ) = Φε (ti ) for ti ∈ [ai−1 , ai ] ∩ Aε and h(Φ(ti ), Φε (ti )) ≤ 2 ||K|| for ti ∈ [ai−1 , ai ] \ Aε . This finishes the proof of Proposition 17.6. t u Followig Hukuhara [9], one can prove that (12) If a bounded L-measurable multifunction Φ : I R R then (R) I Φ(t) dt = I Φ(t) dt.

Z is R-integrable,

Now we pass to the multifunctions of two variables. If S = X ×Y , F : X ×Y Z is a multifunction and (x0 , y0 ) ∈ X ×Y , then the multifunction Fx0 : Y Z defined by Fx0 (y) = F(x0 , y) is called the x0 -section of F, and the multifunction F y0 : X Z defined by F y0 (x) = F(x, y0 ) is called the y0 -section of F. It is well known that if (X, A) is a measurable space, (Y, ρ) is a separable metric space and (Z, d) is a metric space, then a function f : X × Y → Z,

17. Measurability of multifunctions with the (J) property

273

A-measurable in the first and continuous in the second variable is measurable with respect to the product of A and the Borel σ -algebra of Y . Thus by Proposition 1 (iv) we have the following result (cf. [15], Th. 2) Proposition 17.7. If (X, A) is a measurable space, (Y, ρ) is a separable metric space and (Z, d) is a metric space, and if F : X ×Y Z is a compact valued multifunction such that each section Fx is h-continuous and each section F y is A-measurable, then F is A ⊗ Bor(Y )-measurable. The product measurability of multifunctions can be obtained from weaker assumptions. We introduce a concept of multifunctions with the (J) property, which may be considered as a multivalued counterpart of the (J) property given by J. S. Lipi´nski and we show that a multifunction with the (J) property which is a derivative in the second variable is product measurable and sup-measurable. Let (X, A, µ) be a measure space with µ σ -finite. Still let (Z|| · ||) be a reflexive Banach space with the metric d generated by the norm, and still we will consider multifunctions F : X × I Z with values in Cbc (Z). Let A ⊗ Bor be the σ -algebra generated by the family of sets A × B, where A ∈ A and B ∈ Bor. Definition 17.8. A bounded multifunction F : X × I Z has the (J) property if, for each y ∈ I, F y is weakly A-measurable, for each x ∈ X, Fx is weakly L-measurable and for each interval P ⊂ I, the multifunction ΦP : X Z given by Z

(13)

ΦP (x) =

F(x, y) dy P

is weakly A-measurable. A multifunction with the (J) property need not be product measurable. Example 17.9. Suppose CH. Let E ⊂ R2 be Sierpi´nski’s set such that E 6∈ L2 and each x-section of E, i.e. Ex = {y ∈ R : (x, y) ∈ E}, and each y-section of E, i.e. E y = {x ∈ R : (x, y) ∈ E}, have at most two elements. Let F : R2 R be given by  [0, 1], if (x, y) 6∈ E, F(x, y) = {0}, if (x, y) ∈ E. Then F is not L2 -measurable, but F has the (J) property. Proposition 17.10. If (Z, d) is separable and F : X ×I Z is a bounded multifunction such that each section Fx is R-integrable and each section F y is weakly A-measurable, then F has the (J) property.

Graz˙ yna Kwieci´nska

274

Proof. Let P = [c, d] ⊂ I be fixed. We only need to show that the multifunction ΦP , given by (13), is weakly A-measurable. Let yi = c + i d−c n for i = 0, 1, 2, ..., n and n ∈ N. If x ∈ X, then, by R-integrability of Fx , we have n

Z

(R)

1

n

1

lim ∑ F y (x), ∑ n Fx (yi ) = h− n→∞ n→∞ n

F(x, y) dy = h− lim P

i=1

i

i=1

and then, applying (12), we have 1 n yi ∑ F (x). n→∞ n i=1

ΦP (x) = h− lim

Let n ∈ N be fixed and let as define the multifunction Φn : X

Z by

n

Φn (x) = ∑ F yi (x). i=1

Then Φn (x) ∈ Cbc (Z) for x ∈ X (see (4)). Since the multifunction F yi is weakly A-measurable for i = 0, 1, ..., n, the multifunction Φn is weakly A-measurable, by Theorem III.40 in [1]. Thus ΦP is weakly A-measurable, by (2). t u Theorem 17.11. Suppose that (Z, d) is separable. If a bounded multifunction F : X × I Z has the (J) property and for each x ∈ X, Fx is a derivative, i.e., Fx (y) = h − lim∆ y→0

1 ∆y

Z y+∆ y

Fx (t) dt for y ∈ I,

y

then F is measurable with respect to the µ × λ -completion of A ⊗ Bor. Proof. Let n ∈ N be fixed and let ∆ = {y0,n , y1,n , ..., yn,n } be a partition of I into n equal intervals. Let us put ( R yi,n 1 if x ∈ X and y ∈ (yi−1,n , yi,n ), yi,n −yi−1,n yi−1,n F(x, y) dy Fn (x, y) = {θ } if x ∈ X and y = yi,n , i = 0, 1, ..., n. Z, for i = 1, 2, ..., n, be a multifunction given by

Next, let Φi,n : X

Z yi,n

Φi,n (x) =

F(x, y) dy. yi−1,n

By the (J) property of F, we see that (14) Φi,n is weakly A-measurable for each i = 1, 2, ..., n. Define Φn : X ×

Sn

i=1 (yi−1,n , yi,n )

Z by

17. Measurability of multifunctions with the (J) property

275

Φn (x, y) = Φi,n (x) for y ∈ (yi−1,n , yi,n ). If V is an open subset of Z, then, by (14), we have Φn− (V ) =

n [

− Φi,n (V ) × (yi−1,n , yi,n ) ∈ A ⊗ Bor.

i=1

Therefore Fn is weakly A ⊗ Bor-measurable and by (2) we only need to show that (15) h-limn→∞ Fn (x, y) = F(x, y) for every x ∈ X and for almost every y ∈ I. Fix (x0 , y0 ) ∈ X × I such that y0 6= yi,n for n ∈ N and i = 1, 2, ..., n, and choose a sequence (yin ,n ) such that yin −1 < y0 < yin . Since Fx0 is a derivative at y0 , it follows that Z 1 y0 +∆ y F(x0 , y) dy. F(x0 , y0 ) = h− lim ∆ y→0 ∆ y y0 Assume that An =

1 y0 − yin −1,n

Z y0 yin −1,n

F(x0 , y) dy and

Bn =

1 yin ,n − y0

Z yi ,n n y0

F(x0 , y) dy.

Then (16) limn→∞ h(An , F0 ) = 0 where F0 = F(x0 , y0 ).

limn→∞ h(Bn , F0 ) = 0,

and

Let us put zn = h(Fn (x0 , y0 ), F0 ). Note that   Z yi ,n Z yi ,n n n 1 1 zn = h F(x0 , y) dy, F0 dy = yin ,n − yin −1,n yin −1,n yin ,n − yin −1,n yin −1,n Z y  Z yi ,n in ,n n 1 h F(x0 , y) dy, = F0 dy . yin ,n − yin −1,n yin −1,n yin −1,n By (10), we have Z yi ,n n yin −1,n

and

Z y0

F(x0 , y) dy = Z yi ,n n yin −1,n

Next, (3) shows that

yin −1,n

F(x0 , y) dy +

Z y0

F0 dy =

yin −1,n

F0 dy +

Z yi ,n n y0

F(x0 , y) dy

Z yi ,n n y0

F0 dy.

Graz˙ yna Kwieci´nska

276 y0

Z h

yin −1,n

≤h

Z

F(x0 , y) dy +

Z yi ,n n y0

Z y0

y0

yin −1,n

F(x0 , y) dy,

yin −1,n

Z y0

F(x0 , y) dy,

yin −1,n

 Z F0 dy + h

yin ,n

y0

F0 dy +

Z yi ,n n y0

F(x0 , y) dy,

 F0 dy ≤

Z yi ,n n y0

 F0 dy .

Moreover 1 1 < yin ,n − yin −1,n y0 − yin −1,n

and

1 1 < . yin ,n − yin −1,n yin ,n − y0

Therefore,  Z y Z y0 0 1 F(x0 , y) dy, F0 dy + h y0 − yin −1 , n yin −1,n yin −1,n Z y  Z yin ,n in ,n 1 h F(x0 , y) dy, + F0 dy , yin ,n − y0 y0 y0

zn
0 : ∀x∈X ∀y∈Y ||T (x, y)|| ≤ K||x|| ||y||}. We will deal several times with Banach spaces of integrable functions. If (X, Σ , µ) is a measure space and p ∈ [1, ∞], then by L p (X) (L p in short) we denote the Banach space of all measurable real functions f (more formally, all equivalence classes of functions equal µ-a.e.) such that || f || p < ∞, where || f || p is given by: ( R 1 ( X | f | p dµ) p , if p ∈ [1, ∞); || f || p := ess sup | f |, if p = ∞, where ess sup | f | := inf{u > 0 : | f | ≤ u, µ-a.e.}.

18.1 Openness of bilinear mappings 18.1.1 The lack of the openness principle for bilinear mappings A mapping f between topological spaces X and Y is called open if the image of every open set in X is open in Y . The mapping f is called open at a point x ∈ X whenever f (x) is in the interior of the image f [U] for every open neighbourhood U of x. It follows that f is open if and only if f is open at every point of X. Easy examples show that a continuous mapping need not be open. However, there are important situations where continuous functions with additional properties are open or at least they have some local features of openness. The Banach openness principle in functional analysis and the openness principle for holomorphic complex-valued functions of several variables are

18. Bilinear mappings – selected properties and problems

283

well-known important instances of criteria of global openness of mappings. A local kind of openness plays a key role in the local inverse theorem and in the implicit function theorem used in classical real analysis. Through this section, X, Y , Z will usually denote Banach spaces. If they are more general spaces, this will be specified. The Banach openness principle states that a continuous linear surjection between Banach spaces is open. This can be extended to Fréchet spaces [37]. Some further extensions are known where the assumptions either on the operator or on the spaces have been relaxed. This series of results is cited in [33] where a new general criterion of openness for mappings is proved, with the assumed properties independent of continuity and linearity. The bilinear counterpart of the Banach openness principle is false even for finite-dimensional Banach spaces. The full description of spaces and continuous surjections for which it is true, seems difficult. We will show some known counterexamples and positive results concerning selected special cases. Let us start with a simple positive observation which should be known. Proposition 18.1. Given a Banach space X, the bilinear continuous functional T : X × X ∗ → R, defined by T (x, y) := y(x) for (x, y) ∈ X × X ∗ , is an open surjection. Proof. It is known that every nonzero functional z ∈ X ∗ is an open surjection; cf. [20], Exercise 2.28. Hence T is a surjection. Fix balls B(x, r) ⊂ X, B(y, R) ⊂ X ∗ and define U := T [B(x, r) × B(y, R)]. We have [

U=

z[B(x, r)],

z∈B(y,R)

so, if 0 ∈ / B(y, R) then U is open. Assume that 0 ∈ B(y, R). We have then U=

[

z[B(x, r)] ∪ {0}.

z∈B(y,R)\{0}

Hence it suffices to show that 0 ∈ intU. Since 0 ∈ B(y, R), we can pick ε > 0 such that B(0, ε) ⊂ B(y, R). Fix any y0 ∈ X ∗ \ {0} and pick x0 ∈ B(x, r) such that α := |y0 (x0 )| > 0. Then   [ εα εα 0 0 . U⊃ {(ty )(x )} = − 0 , 0 ||y || ||y || t∈(−ε/||y0 ||,ε/||y0 ||) t u

284

Marek Balcerzak, Filip Strobin, Artur Wachowicz

Examples witnessing that a bilinear continuous surjection need not be open at the origin were given by Cohen [17] and Horowitz [27] – they answered a question of Rudin. All these examples violate the necessary condition for the openness of a bilinear surjection at (0, 0), described in the following lemma. Lemma 18.2. Let X,Y, Z be normed spaces and let T : X ×Y → Z be a bilinear surjection open at (0, 0). Then for every bounded set E ⊂ Z there is a bounded set A ⊂ X ×Y such that T [A] = E. Proof. Since T is open at (0, 0) and T (0, 0) = 0, there is r > 0 such that BZ (0, r) ⊂ T [BX (0, 1) × BY (0, 1)].

(18.1)

Let E ⊂ Z be bounded and pick R > 0 such that E ⊂ BZ (0, R). Using (18.1) and the bilinearity of T , we have R R BZ (0, r) ⊂ T [BX (0, 1) × BY (0, 1)] ⊂ r r " ! r r !# R R ⊂ T BX 0, × BY 0, . r r

E ⊂ BZ (0, R) =

Then the set A := T

−1

[E] ∩ BX

r ! R 0, × BY r

r !! R 0, r

is as desired.

t u

Now, we sketch the counterexample from [27] where C was used instead of R (however, the reasoning is the same). Example 18.3. Let T : R3 × R3 → R4 be defined by T (x, y) := (x1 y1 , x1 y2 , x1 y3 + x3 y1 + x2 y2 , x3 y2 + x2 y1 ) where x := (x1 , x2 , x3 ), y := (y1 , y2 , y3 ). Obviously, T is bilinear and continuous. We omit the proof that T is surjective, given in [27]. Let    1 1 1 4 , , ,1 ∈ R : n ∈ N . E := n n n Then E is bounded. Suppose that T is open at (0, 0). By Lemma 18.2 pick a bounded set A ⊂ R3 × R3 such that T [A] = E. For each n ∈ N, choose (x(n) , y(n) ) ∈ A such that T (x(n) , y(n) ) = (1/n, 1/n, 1/n, 1). Let

18. Bilinear mappings – selected properties and problems (n)

(n)

(n)

(n)

285 (n)

(n)

wn := (x1 + x2 + x3 )(y1 + y2 + y3 ). By the calculations in [27], we have wn = (3/n) + 2 − n. Since A is bounded, so is {wn : n ∈ N} which yields the contradiction. A simple example of a continuous bilinear surjection T0 : R × R2 → R2 , non-open at (0, 1, 1), can be found in [37], Chapter 2, Exercise 11. Namely, let T0 (t, x, y) := (tx,ty). Then (0, 0) = T0 (0, 1, 1) ∈ / int T0 [(−1, 1) × (1/2, 3/2)2 ]. Moreover, it can be shown that T0 is not open at any point (0, x, y) with x2 + y2 > 0, and it is open at the remaining points. Note that Horowitz in [27] modified his example to obtain the respective infinite-dimensional case. Another infinite-dimensional example was given earlier in [17] with a more involved construction. An important infinitedimensional example, due to Fremlin, will be discussed in the next subsection. An interesting example dealing with the Banach algebra of operators on `∞ was given in [14]. Question 18.4. The Cauchy product S : `1 × `1 → `1 (which can be treated as a special case of convolution), given by ! n

S((xn )n≥0 , (yn )n≥0 ) =

,

∑ xi yn−i

i=0

n≥0

is a continuous bilinear surjection. Is it an open mapping? In [8], the following strengthened notion of openness for mappings was introduced. We say that a mapping f between metric spaces X and Y is uniformly open whenever ∀ε>0 ∃δ >0 ∀x∈X B( f (x), δ ) ⊂ f [B(x, ε)]. It was observed that arctan is an open function from R into R which is not uniformly open. In [8], some important examples of multiplication were found where the uniform openness holds. Namely, we have Theorem 18.5 ([8]). Multiplication Φ : X 2 → X is uniformly open in the following cases: (1) X := R; (2) X is the Banach algebra of real-valued bounded functions measurable with respect to a given σ -algebra of subsets of a fixed set E;

286

Marek Balcerzak, Filip Strobin, Artur Wachowicz

(3) X := bBorα is the Banach algebra of real-valued bounded functions on a fixed metrizable space that are Borel measurable of class α (where α is an ordinal, 0 < α < ω1 ). (In cases (2), (3), X is endowed with the supremum norm.) Note that, statement (1) holds with an analogous proof if R is replaced by C. Given a measure space (X, Σ , µ), consider the respective Banach spaces L p , p ∈ [1, ∞], of integrable functions. The main result of [8] is the following Theorem 18.6 ([8]). Multiplication from L p × Lq to L1 (for p ∈ [1, ∞] and 1/p + 1/q = 1) given by ( f , g) 7→ f g is an open mapping, being a continuous bilinear surjection. Recently, this result has been improved in [9] by showing that multiplication in Theorem 18.6 is a uniformly open mapping.

18.1.2 Openness of multiplication in spaces of continuous functions Let Z and T be topological spaces. We say that a function Φ : Z → T is weakly open if, for every nonempty open set U ⊂ Z, the interior of Φ(U) is nonempty in T . Weak openness was considered by many authors (see, e.g., Burke [15]). We will mainly focus on the operation Φ : C(X) × C(X) → C(X) of multiplication Φ( f , g) := f g where C(X) denotes the space of all continuous real functions defined on a topological space X, with the metric of uniform convergence d( f , g) := min{1, sup{| f (x) − g(x)| : x ∈ X}}. For U,V ⊂ C(X) we write U ·V instead of Φ[U ×V ]. If X is a compact Hausdorff space then C(X) forms a Banach algebra equipped with the supremum norm. It turns out that, for X := [0, 1], multiplication in C[0, 1] yields the following very simple example of the lack of openness for a continuous bilinear surjective mapping between infinite dimensional spaces (compare with the mentioned earlier examples by Cohen [17] and Horowitz [27]). Example 18.7 ([12]). (due to D.H. Fremlin) Let f (x) = x − 1/2, x ∈ [0, 1] and r = 1/2. Then f 2 ∈ / int(B( f , r) · B( f , r)). Indeed, it is easy to see that every

18. Bilinear mappings – selected properties and problems

287

function in B( f , r) possesses zeros, whereas every neighbourhood of f 2 contains functions f 2 + δ , δ > 0, without zeros. Consequently, multiplication is not an open mapping in C[0, 1] since it is not open at ( f , f ). Fremlin’s example was an impetus for Wachowicz to examine the weak openness of multiplication in C[0, 1]. In [44] he proved the following theorem. Theorem 18.8 ([44], [12]). Multiplication Φ : C[0, 1] × C[0, 1] → C[0, 1] is a weakly open mapping. The proof of the above theorem presented in [44] was complicated and it was based on the density of the set of all polygonal functions in C[0, 1], and the fact that in any two open balls in C[0, 1], one can find two polygonals (respectively) with finite disjoint sets of zeros. To this aim, the technique of “gluing of open balls" on adjacent closed intervals was used. This proof of Theorem 18.8 has been nowhere published except for the PhD thesis [44], although the technique of gluing of open balls seems to be interesting and its variant was used by Balcerzak, Wachowicz and Wilczy´nski in [12] where a much shorter and more transparent proof of Theorem 18.8 was presented. This technique was also used by Wachowicz [43] in the proof of an analogous result obtained in the space Cn [0, 1] of all functions f : [0, 1] → R with continuous n-th derivative (here also Fremlin’s example works). A next essential step was made by Komisarski in [29] who considered the operation of multiplication in C(X) where X is a compact Hausdorff space. Komisarski linked the concepts of openness and weak openness with the topological dimension of X, and he generalized Theorem 18.8 in the following way. Theorem 18.9 ([29]). Let X be a compact Hausdorff space. The following equivalences hold: (1) multiplication in C(X) is open iff dim X < 1, (2) multiplication in C(X) is weakly open and not open iff dim X = 1, (3) multiplication in C(X) is not weakly open iff dim X > 1, where dim X denotes the topological (covering) dimension of X. Another result in that matter was obtained by Kowalczyk in [30] for the multiplication map in C(X) where X = (0, 1). This multiplication is not continuous (cf. also [10], Prop. 3) which makes it different from the cases considered earlier. In spite of this, the following theorem holds. Theorem 18.10 ([30]). Multiplication in the space C(0, 1) is a weakly open mapping.

288

Marek Balcerzak, Filip Strobin, Artur Wachowicz

It is also interesting that a technique used in the proof of Theorem 18.10 gives (with the aid of the Tietze extension theorem) a new proof of Theorem 18.8. The result established by Kowalczyk was then strengthened by Balcerzak and Maliszewski in [10]. The authors introduced a notion of dense weak openness, a stronger version of weak openness, cf. [30]. If Y, Z are topological spaces then a function T : Y → Z is called densely weakly open whenever int(T [U]) is a dense set in T [U] for any nonempty open set U ⊂ Y . The following theorem was proved. Theorem 18.11 ([10]). If X ⊂ R is an interval, then multiplication in C(X) is densely weakly open. In particular, this yields a next proof of Theorem 18.8. Now, let us turn to the article [13] by Behrends who characterized the pairs ( f , g) at which multiplication in C[0, 1] is open. To formulate the main result of [13] we need some definitions. For every pair ( f , g) of functions in C[0, 1], we consider a map γ(t) = ( f (t), g(t)), t ∈ [0, 1], which generates the so called path in R2 associated with ( f , g). Denote by Q++ , Q+− , Q−+ , Q−− the four quadrants of the plane, i.e., the sets {(x, y) : x, y ≥ 0}, {(x, y) : x ≥ 0, y ≤ 0}, {(x, y) : x ≤ 0, y ≥ 0}, {(x, y) : x, y ≤ 0}, respectively. Definition 18.12 ([13]). Let t0 ∈ (0, 1). We say that a path γ = ( f , g) has a positive saddle point crossing at t0 if γ(t0 ) = 0 and there exists ε > 0 such that the following two conditions hold: (1) γ(t) ∈ Q++ ∪ Q−− for t ∈ [t0 − ε,t0 + ε]; (2) there are t1 ∈ [t0 − ε,t0 ] and t2 ∈ [t0 ,t0 + ε] such that γ(t1 ) ∈ Q++ \ {(0, 0)} and γ(t2 ) ∈ Q−− \ {(0, 0)} or vice versa. A negative saddle point is defined similarly: then γ moves from Q−+ to Q+− or from Q+− to Q−+ . Behrends obtained the following characterization. Theorem 18.13 ([13]). Let f , g ∈ C[0, 1] and let γ = ( f , g) be the associated path in R2 . Then the following assertions are equivalent: 1. Multiplication in C[0, 1] is open at ( f , g). 2. For every r > 0 there exists δ > 0 such that the functions f g + δ , f g − δ are in B( f , r) · B(g, r).

18. Bilinear mappings – selected properties and problems

289

3. γ has no positive and no negative saddle point crossings. Behrends described details of his proof using a nice geometrical interpretation, connected with “walks in a landscape" with hills and valleys where an accompanying dog can move in a certain prescribed way. Behrends proved also the following generalization of Theorem 18.13 in his next article [14]. Theorem 18.14 ([14]). Let f1 , . . . , fn ∈ C[0, 1] be given. The following assertions are equivalent: 1. f1 · f2 · · · fn lies in the interior of B( f1 , r) · B( f2 , r) · · · B( fn , r) for every r > 0. 2. The associated path γ : t 7→ ( f1 (t), . . . , fn (t)) is admissible. The notion of admissibility defined in [14] is much more complicated than the respective description in [13], and indeed the above theorem is not a simple generalization of Theorem 18.13. In [14] Behrends considered also multiplication in CC [0, 1], the Banach algebra of continuous complex-valued functions on [0, 1], endowed with the supremum norm. He proved the following theorem which makes a contrast to the real-valued case. Theorem 18.15 ([14]). If O1 , . . . , On are open subsets of CC [0, 1] then O1 · · · On is also open. Currently, we focus on extensions of Theorem 18.8 to the case where, instead of multiplication, we consider some other functions of two variables. This was investigated in general by Kowalczyk [31], and also by Wachowicz [44] who considered the operations of addition, minimum, maximum and composition. Kowalczyk in [31] proposed for C(X), where X is a compact Hausdorff space, studies of openness for the operation Φ : C(X) × C(X) → C(X) determined by a continuous function ϕ : R2 → R by the formula Φ( f , g)(x) := ϕ( f (x), g(x)), x ∈ X. Kowalczyk obtained a number of interesting results which generalize the above-mentioned theorems by Wachowicz and Komisarski. He introduced the following definition. Definition 18.16 ([31]). Let ϕ : R2 → R be a continuous function and let K be a nonempty subset of R2 . We say that a function α : K → R2 is ϕ-increasing (ϕ-decreasing, respectively) if it is continuous and for every v ∈ K the function ϕv : [0, 1] → R defined by ϕv (t) = ϕ((1 − t)v + tα(v)) is strictly increasing (strictly decreasing, respectively). If α is ϕ-increasing or ϕ-decreasing, it will be called ϕ-monotone.

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Let us observe (see [31], Example 1) that, for some ϕ and K ⊂ R2 , a ϕ-increasing (or ϕ-decreasing) function α does not exist (it is the case for ϕ(x, y) = x · y and K = [0, 1] × {0}). Theorem 18.17 ([31]). Let X be a compact Hausdorff space, ϕ : R2 → R be a continuous function, Φ : C(X) ×C(X) → C(X) be the operation determined by ϕ, and K ⊂ R2 be nonempty such that there exist a ϕ-increasing function and a ϕ-decreasing function defined on K. Then for each pair of continuous functions f , g ∈ C(X) such that {( f (x), g(x)) : x ∈ X} ⊂ K and for each r > 0, the image Φ[B( f , r) × B(g, r)] contains B(Φ( f , g), ε) for some ε > 0. Corollary 18.18 ([31]). Let ϕ : R2 → R be a continuous function. If there exist a ϕ-increasing function defined on the whole plane, and a ϕ-decreasing function defined on the whole plane, then for every Hausdorff compact topological space X, the operation Φ : C(X) ×C(X) → C(X) determined by ϕ is open. The next result is connected with a strengthening of assumptions on a function ϕ. Theorem 18.19 ([31], Theorem 4). Let X be a Hausdorff compact space. If ϕ : R2 → R is a C1 function such that ∇ϕ(v) 6= (0, 0) for v ∈ R2 , then the operation Φ : C(X) ×C(X) → C(X) determined by ϕ is open. On the other hand, the following result holds. Theorem 18.20 ([31], Theorem 5). Let ϕ : R2 → R be a continuous function and let X be a topological Hausdorff compact space. If ϕ has a local extremum, then the operation Φ : C(X) ×C(X) → C(X) determined by ϕ is not open. In further considerations, Kowalczyk observed that, if there are ϕ-increasing and ϕ-decreasing functions defined not on the whole plane but on a “big" set, then properties of Φ determined by ϕ, depend on the dimension of X. Kowalczyk showed a few facts on that topic which we join together in the following theorem. For this we need one else definition. Definition 18.21. We say that a function ϕ : R2 → R has a constant point x0 ∈ R if ϕ(x0 ,t) = ϕ(t, x0 ) = ϕ(x0 , x0 ) for t ∈ R. Theorem 18.22 ([31], Corollary 2). Let ϕ : R2 → R be a continuous function without local extremum and with a constant point x0 ∈ R, X be a Hausdorff compact space and Φ : C(X) × C(X) → C(X) be the operation determined by ϕ. Moreover, assume that for some boundary subsets A, B ⊂ R, there exist a ϕ-increasing function α : R2 \ (A × B) → R2 and a ϕ-decreasing function β : R2 \ (A × B) → R2 . Then

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(1) Φ is open iff dim X ≤ 0, (2) Φ is weakly open iff dim X ≤ 1, (3) Φ is not weakly open iff dim X ≥ 2. Theorem 18.9 by Komisarski is a particular case of the above theorem since it suffices to take ϕ(x, y) = x · y which fulfills the assumptions. However, if ϕ(x, y) = x2 − y2 , Theorem 18.22 is also applicable (for details, see [31], Theorem 11 and Example 2). In the last part of the paper [31] Kowalczyk gave some necessary conditions for the openness and the weak openness of the operation Φ : C(X) ×C(X) → C(X) determined by a function ϕ where X is a connected Hausdorff compact space.

18.1.3 Opennes of multiplication in other function spaces Consider the Banach algebra BV of all real-valued functions on [−1, 1] with the bounded variation (interval [−1, 1] is used only by technical reasons). The 1 ( f ) where V 1 ( f ) denotes the norm on BV is given by || f || := | f (0)| + V−1 −1 variation of f on [−1, 1]. The following proposition shows that the argument used in Example 18.7 does not work in BV. Proposition 18.23 ([16]). Let f0 (x) := x for x ∈ [−1, 1]. There exists δ > 0 such that f02 + δ can be expressed in the form f g where f , g ∈ B( f0 , 1) in BV. Proof. Fix c ∈ (0, 1/3) and choose δ such that 0 < δ < (c−c2 )/(5+3c). Then let  −c if −1 ≤ x < 0 α(x) := c if 0 ≤ x < 1. 1 (α) = c + 2c = 3c < 1. We set f := We have α ∈ BV and ||α|| = |α(0)| +V−1 f0 + α. Hence f ∈ B( f0 , 1) in BV. We want to obtain f02 + δ = f g for some g ∈ B( f0 , 1). This g will be of the form f0 + β with β ∈ BV and ||β || < 1. Thus x2 + δ = (x + α(x))(x + β (x)) for x ∈ [−1, 1]. Hence let

β (x) :=

δ − xα(x) for x ∈ [−1, 1]. x + α(x)

For 0 ≤ x ≤ 1, we have β (x) = (δ − xc)/(x + c) and β 0 (x) = −(c2 + δ )/(x + c)2 < 0. Hence V01 (β ) = β (0) − β (1) = (c2 + δ )/(c + c2 ).

(18.2)

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For x ∈ [−1, 0), we have β (x) = (δ + xc)(x − c) and β 0 (x) = −(c2 + δ )/(x − c)2 < 0. Hence 0 V−1 (β ) = (β (−1) − β (0− )) + |β (0) − β (0− )| =

c2 + δ 2δ + . c(1 + c) c

(18.3)

Now, by (18.2) and (18.3), we have β ∈ BV and 1 ||β || = |β (0)| +V−1 (β ) = 2

δ 2c2 + 5δ + 3δ c c2 + δ +3 = . c(1 + c) c c(1 + c)

Then we calculate that ||β || < 1 is equivalent to δ < (c − c2 )/(5 + 3c) which is true by our choice of δ . t u Question 18.24. Is multiplication in BV open? Let us finish with some other questions and remarks. Question 18.25. Consider the subspace CBV of BV consisting of continuous functions with a bounded variation. Note that multiplication in CBV is not open since Fremlin’s example works. Is multiplication in CBV weakly open? It might be interesting to consider the question on the weak openness of multiplication in CBV when the Adams metric is considered in these spaces: Z 1

d( f , g) :=

| f − g| + |V ( f ) −V (g)|.

−1

Observe that Fremlin’s example, in spite of changing a metric, is still valid. Indeed, let id be the identity function on [−1, 1]. If we suppose that id2 ∈ int(B(id, 1/2)·B(id, 1/2)), then for some ε > 0 we have id2 +ε ∈ int(B(id, 1/2)· B(id, 1/2)). So, there are functions f , g ∈ B(id, 1/2) such that f g = id2 +ε. However, note that since f , g are continuous, both these functions are either positive or negative on [−1, 1]. Assume for instance that both f and g are posiR0 R0 tive. Then d( f , id) ≥ −1 | f − id | ≥ −1 | id | = 1/2 which yields f ∈ / B(id, 1/2), a contradiction. Observe that the same example works for the space of all continuous functions on [−1, 1] with the integral norm, so we can also ask about the weak openness of multiplication in this space. Recall that multiplication is not continuous in CBV under the Adams metric – this is a consequence of the example given by Adams and Clarkson in [3] where it is shown that even the operation of addition in CBV is not continuous. Using the composition with the exponential function one can prove that it is the case for multiplication, too. For more details on the Adams metric, see also [2] and [4].

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18.2 Dichotomies for bilinear mappings 18.2.1 Introduction Let (X, Σ , µ) be a measure space. If p, q, r ∈ (0, ∞] satisfy 1/p + 1/q = 1/r, then by a general version of the Hölder inequality [21], Exercise 1.1.2, we see that Φ( f , g) = f g ∈ Lr for any f ∈ L p and g ∈ Lq (later we will write f g instead of Φ( f , g)). Hence in this case, {( f , g) ∈ L p × Lq : f g ∈ Lr } = L p × Lq . On the other hand, Balcerzak and Wachowicz in [11] proved the following theorem (we consider the Lebesgue measure on the interval [0, 1] and c0 denotes the space of all real sequences which converge to 0, endowed with the supremum norm; we abbreviate x = (xk ), y = (yk ), etc.). Theorem 18.26.  (i) The set ( f , g) ∈ L1 [0, 1] × L1 [0, 1] : f g ∈ L1 [0, 1] is a meager subset of L1 [0, 1] ×L1 [0, 1]. (ii) The set (x, y) ∈ c0 × c0 : (∑nk=1 xk yk )∞ n=1 is bounded is a meager subset of c0 × c0 . Note that Wachowicz in [44] extended part (i) by considering L p [0, 1] space for p ∈ [1, ∞). This result suggests the following general problem: given spaces of real functions X,Y, Z and a bilinear map T defined on X ×Y , investigate the size of the set {( f , g) ∈ X ×Y : T ( f , g) ∈ Z}. In this section we present several results which deal with such a problem, mainly in the case when T is multiplication or convolution. In most situations an interesting phenomenon is observed – either such sets are equal X × Y , or they are very small (σ -porous and, in particular, meager). Now we describe the notions of smallness that our study relies on. If (X, τ) is a Baire topological space, then it is known that meager sets (or sets of the first category, that is sets which are countable unions of nowhere dense sets) can be considered as small sets. It turns out that within complete metric spaces we can define a notion of smallness more restrictive than meagerness – the σ -porosity. The main idea is that we can modify the “ball" definition of nowhere density by the requirement that the sizes of “pores" are somehow

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estimated. This idea can be formalized in many ways, so there are many notions of porosity (cf. the survey papers of Zajíˇcek [45] and [46]). Note that σ -porous sets (that is, countable unions of porous sets) are meager, but the really interesting fact is that in every “reasonable" complete metric space there are sets which are meager but not σ -porous. Hence the fact that a particular set is not only meager but also σ -porous, means that it is even smaller. Let X be a metric space and M ⊂ X. We say that M is α-lower porous (for α > 0), if ∀x∈M ∀β ∈(0, α ) ∃R0 >0 ∀R∈(0,R0 ) ∃z∈X B(z, β R) ⊂ B(x, R)\M. 2 If M is a countable union of α-lower porous sets (for the same constant α > 0), then we say that M is σ -α-lower porous. The notion of α-lower porosity is commonly known – it appears at the beginning of the mentioned survey [46] (in fact, this notion is defined in [46] in a bit different but equivalent way). It turns out that there are sets which are meager and are not σ -α-lower porous for any α > 0. For more information on porosity, we refer the reader to [45] and [46].

18.2.2 Bilinear mappings and the Banach-Steinhaus principle Recall the well known Banach-Steinhaus principle: Theorem 18.27. Let X, Z be Banach spaces and let {Tn : n ∈ N} ⊂ L(X, Z). Put E := {x ∈ X : (Tn (x)) is bounded}. The following conditions are equivalent: (i) E is meager in X; (ii) E 6= X; (iii) sup{||Tn || : n ∈ N} = ∞. In fact, implication (iii)⇒(i) is a formulation of the Banach-Steinhaus principle – the rest are trivial. Jachymski showed in [28] that an analogous fact holds for bilinear mappings: Theorem 18.28 ([28]). Let X,Y, Z be Banach spaces and let {Tn : n ∈ N} ⊂ B(X ×Y, Z). Put E := {(x, y) ∈ X ×Y : (Tn (x, y)) is bounded}.

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The following conditions are equivalent: (i) E is meager in X ×Y ; (ii) E 6= X ×Y ; (iii) sup{||Tn || : n ∈ N} = ∞. Proof. Implications (i)⇒(ii) and (ii)⇒(iii) are trivial. We prove (iii)⇒(i). We will show that for any M > 0, the set EM := {(x, y) ∈ X ×Y : ||Tn (x, y)|| ≤ M for every n ∈ N} is nowhere dense. Assume on the contrary that EM is not nowhere dense. Since it is also closed, EM has nonempty interior. Then for some (x0 , y0 ) ∈ X ×Y and r > 0, B((x0 , y0 ), r) ⊂ EM . Now let x ∈ X and y ∈ Y be such that ||x|| ≤ 1 and ||y|| ≤ 1. Then for every n ∈ N, we have r  4  r r  4  r r  x, y = 2 Tn x, y0 + y − Tn x, y0 = ||Tn (x, y)|| = 2 Tn r 2 2 r 2 2 2    4  r r  r  r = 2 Tn x0 + x, y0 + y − Tn x0 , y0 + y − Tn x0 + x, y0 + Tn (x0 , y0 ) ≤ r 2 2 2 2  4  r r  4  r r  4  ≤ 2 Tn x0 + x, y0 + y + 2 Tn x0 , y0 + y + 2 Tn x0 + x, y0 + r 2 2 r 2 r 2 4 4 16M + 2 ||Tn (x0 , y0 )|| ≤ 2 (M + M + M + M) = 2 . r r r

Hence ||Tn || ≤

16M r2

for any n ∈ N. This is a contradiction.

t u

As immediate corollaries, we get the following results. The first one is an extension of Theorem 18.26, part (i), proved in [28]. We will give a proof of the second one. Theorem 18.29 ([28]). Assume that z is any sequence of reals and let !∞ ) ( n

Ez :=

(x, y) ∈ c0 × c0 :

is bounded .

∑ zk xk yk k=1

n=1

Then the following statements are equivalent: (i) Ez is meager in c0 × c0 ; (ii) Ez 6= c0 × c0 ; (iii) z ∈ / l 1 , that is ∑∞ n=1 |zn | = ∞. Theorem 18.30 ([28]). Assume that z is any sequence of reals, p ∈ [1, ∞], and let !∞ ( ) n

Ez :=

(x, y) ∈ c0 × l p :

is bounded .

∑ zk xk yk k=1

n=1

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Then the following statements are equivalent: (i) Ez is meager in c0 × l p ; (ii) Ez 6= c0 × l p ; (iii) z ∈ / l q , where q is such that

1 p

+ q1 = 1.

Note that in the above formulation we use the abbreviation

1 ∞

:= 0.

Proof. For every n ∈ N, set Tn (x, y) := ∑nk=1 xk yk zk . Then every Tn is bilinear, continuous and Ez = {(x, y) ∈ c0 × l p : (Tn (x, y)) is bounded}. 1

Using the fact that (l p )∗ = l q , we get ||Tn || = (∑nk=1 |zk |q ) q (in the case q < ∞) or ||Tn || = max{|z1 |, . . . , |zn |} (in the case q = ∞). Hence sup{||Tn || : n ∈ N} = ∞ if and only if z ∈ / l q . The result follows then from Theorem 18.28. t u Using Theorem 18.28, we could try to strengthen, in the same way, part (ii) of Theorem 18.26. However, the problem would appear with the bilinearity of an appropriate functional. To overcome these difficulties, Jachymski proved a nonlinear version of the Banach-Steinhaus principle which is an extension of Theorem 18.28. We will present its particular version. A function ϕ : X → R+ is L-subadditive for some L ≥ 1, if ϕ(x + y) ≤ L(ϕ(x) + ϕ(y)) for x, y ∈ X, and is even, if ϕ(−x) = ϕ(x) for x ∈ X. Theorem 18.31 ([28]). Given k ∈ N, let X1 , . . . , Xk be Banach spaces, X = X1 × · · · × Xk . Assume that L ≥ 1, Fn : X → R+ (n ∈ N) are lower semicontinuous and such that all functions xi 7→ Fn (x1 , . . . , xk ) (i = 1, . . . , k) are L-subadditive and even. Let E := {x ∈ X : (Fn (x))∞ n=1 is bounded}. Then the following statements are equivalent: (i) E is meager; (ii) E 6= X; (iii) sup{Fn (x) : n ∈ N, ||x|| ≤ 1} = ∞. Note that Theorem 18.28 is given in [28] as a corollary of the above theorem; we gave here an alternative, straightforward proof. As a corollary, Jachymski obtained a strengthening of part (ii) of Theorem 18.26. Put Σ+ := {A ∈ Σ : 0 < µ(A) < ∞}. Theorem 18.32. Assume that p ∈ [1, ∞) and (X, Σ , µ) is a measure space such that Σ+ 6= 0. / Let

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E p := {( f , g) ∈ L p × L p : f g ∈ L p }. Then the following statements are equivalent: (i) E p is meager in L p × L p ; (ii) E p 6= L p × L p ; (iii) inf{µ(A) : A ∈ Σ+ } = 0. Proof. The equivalence of (i) and (ii) is stated in [28], Proposition 2. The equivalence of (ii) and (iii) follows from [28, Proposition 3, (i)⇔(ii)] and [28, Lemma 4, (i)⇔(iii)]. t u Note that the condition inf{µ(A) : A ∈ Σ+ } = 0 means that there are sets with positive measure which is as small as we want. In particular, the Lebesgue measure on [0, 1] satisfies this condition. Theorem 18.31 is a nice and deep tool, and it seems that it could be used to study similar problems for other bilinear mappings. A question arises whether, in this result, we can replace meagerness by (any known notion of) σ -porosity. It turns out that in general this is not possible, cf. [22], p. 2. However, the following question is open: Question 18.33. Can we replace meagerness by (any notion of) σ -porosity in the condition (i) of Theorem 18.28? Note that in the case of classical BanachSteinhaus principle the answer is positive, cf. [34] and [40]. In the next subsections we will prove, however, that Theorem 18.32 can be further strengthened. It is possible to replace meagerness by the appropriate notion of σ -porosity but also it is possible to consider more general spaces (not necessarily Banach spaces). Note that in [23] and [42] there are results which generalize and strengthen Theorem 18.29.

18.2.3 Dichotomies for multiplication in L p spaces Let (X, Σ , µ) be a measure space. We will consider here L p spaces also for p ∈ (0, 1). In this case, L p is the space of all measurable real functions such that Z || f || p := | f | p dµ < ∞. X

It is well known that for p ∈ (0, 1), || · || p is not a norm, but the function d defined by d( f , g) := || f − g|| p is a complete metric. For any p, q, r ∈ (0, ∞], define

298

Marek Balcerzak, Filip Strobin, Artur Wachowicz r E p,q := {( f , g) ∈ L p × Lq : f g ∈ Lr }.

r = L p × Lq if p, q, r ∈ (0, ∞] and 1 + By the mentioned Hölder inequality, E p,q p p 1 1 . In this section we = . Theorem 18.32 discusses the size of the set E p,p q r r . present results which solve completely the problem of the size of sets E p,q The next two results follow from the main results of [22]. The first one is an extension of Theorem 18.32.

Theorem 18.34. Let (X, Σ , µ) be such that Σ+ 6= 0/ and p, q, r ∈ (0, ∞] be such that 1p + 1q > 1r . Then the following conditions are equivalent: r is a σ - 2 -lower porous subset of L p × Lq ; (i) E p,q 3 r 6= L p × Lq ; (ii) E p,q (iii) inf{µ(A) : A ∈ Σ+ } = 0.

Proof. Implication (i)⇒(ii) is trivial, implication (ii)⇒(iii) follows from [22], Proposition 8, and implication (iii)⇒(i) is an immediate consequence of [22], Theorem 6. t u The next result shows what happens if 1p + q1 < 1r . Note that the condition sup{µ(A) : A ∈ Σ+ } = ∞ means that there are sets with finite measure which is as big as we want. In particular, the counting measure on N satisfies this condition, hence the space l p is an example of a space which satisfies the assumption (iii) in the next result. The proof goes the same way as for Theorem 18.34, so we omit it. Theorem 18.35. Let (X, Σ , µ) be such that Σ+ 6= 0/ and p, q, r ∈ (0, ∞] be such that 1p + 1q < 1r and p < ∞ or q < ∞. Then the following conditions are equivalent: r is a σ - 2 -lower porous subset of L p × Lq ; (i) E p,q 3 r 6= L p × Lq ; (ii) E p,q (iii) sup{µ(A) : A ∈ Σ+ } = ∞.

The implication (iii)⇒(i) in Theorems 18.34 and 18.35 can be strengthened in classical cases as follows: Corollary 18.36. Let p, q ∈ (0, ∞] be such that p < ∞ or q < ∞, and put r p,q := 1 1/p+1/q . Then the sets  A := (x, y) ∈ l p × l q : ∃0 0; (ii) E p,q 6= L p × Lq ; (iii) G is not unimodular. Proof. Implication (i)⇒(ii) is trivial. Implication (ii)⇒(iii) follows from the Young inequality (note that 1/p + 1/q ≤ 2), and implication (iii)⇒(i) follows immediately from [5], Theorem 2.4 (in the same way as this implication in the proof of Theorem 18.41). t u The next result follows from the main result of the paper [7] of Akbarbaglu and Maghsoudi. Theorem 18.43. Let G be a locally compact group, p ∈ (0, 1) and q ∈ (0, ∞]. Then the following conditions are equivalent:

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(i) E p,q is σ -α-lower porous for some α > 0; (ii) E p,q 6= L p × Lq ; (iii) G is not discrete. 0

Proof. The implication (i)⇒(ii) is trivial. If G is discrete, then Lr ⊂ Lr for r, r0 such that 0 < r ≤ r0 ≤ ∞. Hence if ( f , g) ∈ L p × Lq , then ( f , g) ∈ L1 × Lq+1 . Thus the Minkowski inequality implies the implication (ii)⇒(iii). The remaining implication follows from [7], Theorem 2.1. t u Question 18.44. It is natural to ask what happens in the remaining cases, i.e., * p ∈ (1, ∞) and q ∈ (0, 1) ∪ {∞}; * p = 1 and q ∈ (0, 1); * p = ∞. It seems that the proof of [5], Theorem 2.4, can easily be rewritten so that the assertion of Theorem 18.42 holds also for the case p ∈ (1, ∞) and q ∈ (0, 1). Also, the case q = ∞ seems to be easy to handle. Question 18.45. We can ask similar (but it seems that more difficult) questions concerning sets r E p,q := {( f , g) ∈ L p × Lq : f ? g ∈ Lr }. r ⊂E Clearly, E p,q p,q for every r. It is easy to observe that using the Young and the Minkowski inequalities we can strengthen a bit Theorems 18.41, 18.42 and 18.43 by adding another condition r 6= L p × Lq , (iv) E p,q

for r := 1 in Theorem 18.41, r := r(p,q) in Theorem 18.42, and r := q + 1 in Theorem 18.43. Some results and questions can also be found in [7], [35] and [38]. In particular, it is natural to ask if the L p -conjecture can be strengthened by proving p that in the case p > 1 and G noncompact, the set E p,p is meager (or σ -porous) p p in L × L (cf. [38], Problem 2). Acknowledgement. The work of the first and the third authors has been supported by the (Polish) National Center of Science Grant No. N N201 414939 (2010-13). The first author thanks V.V. Chistyakov for the cooperation during his visit in Łód´z in spring 2011.

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References [1] F. Abtahi, R. Nasr–Isfahani, A. Rejali, On the L p –conjecture for locally compact groups, Arch. Math. (Basel) 89 (2007), 237–242. [2] C. R. Adams, The space of functions of bounded variation and certain general spaces, Trans. Amer. Math. Soc. 40 (1936), 421–438. [3] C. R. Adams, J. A. Clarkson, On convergence in variation, Bull. Amer. Math. Soc. 40 (1934), 413–417. [4] C. R. Adams, A. P. Morse, On the space (BV), Trans. Amer. Math. Soc. 42 (1937), 194–205. [5] I. Akbarbaglu, S. Maghsoudi, An answer to a question on the convolution of functions, Arch. Math. (Basel) 98 (2012), 545–553. [6] I. Akbarbaglu, S. Maghsoudi, Porosity and products of Orlicz spaces, Positivity, to appear, DOI: 10.1007/s11117-0120218-0. [7] I. Akbarbaglu, S. Maghsoudi, Porosity of certain subsets of Lebesgue spaces on locally compact groups, Bull. Austral. Math. Soc., to appear, DOI:10.1017/S0004972712000949. [8] M. Balcerzak, A. Majchrzycki, A. Wachowicz, Openness of multiplication in some function spaces, Taiwanese J. Math. 17 (2013), 1115–1126. [9] M. Balcerzak, A. Majchrzycki, F. Strobin, Uniform openness of multiplication in Banach spaces L p , preprint available at: http://arxiv.org/abs/1309.3433. [10] M. Balcerzak, A. Maliszewski, On multiplication in spaces of continuous functions, Colloq. Math. 122 (2011), 247–253. [11] M. Balcerzak, A. Wachowicz, Some examples of meager sets in Banach spaces, Real Anal. Exchange 26 (2000/01), 877–884. [12] M. Balcerzak, A. Wachowicz, W. Wilczy´nski, Multiplying balls in C[0, 1], Studia Math. 170 (2005), 203–209. [13] E. Behrends, Walk the dog, or: products of open balls in the space of continuous functions, Func. Approx. Comment. Math. 44 (2011), 153–164. [14] E. Behrends, Products of n open subsets in the space of continuous functions on [0, 1], Studia Math. 204 (2011), 73–95. [15] M. Burke, Continuous functions which take a somewhere dense set of values on every open set, Topology Appl. 103 (2000), 95–110. [16] V. V. Chistyakov, unpublished notes, 2011. [17] P. J. Cohen, A counterexample to the closed graph theorem for bilinear maps, J. Funct. Anal. 16 (1974), 235–239. [18] P. G. Dixon, Generalized open mapping theorems for bilinear maps, with an application to operator algebras, Proc. Amer. Math. Soc. 104 (1988), 106–110. [19] R. Engelking, General topology, second edition, Sigma Series in Pure Mathematics 6. Heldermann Verlag, Berlin, 1989. [20] M. Fabian, P. Hájek, V. Montesinos Santalucia, J. Pelant, V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, Springer, New York, 2001. [21] L. Grafakos, Classical Fourier Analysis, second edition, Graduate Texts in Mathematics 249 Springer, New York, 2008. [22] S. Głab, ˛ F. Strobin, Dichotomies for L p spaces, J. Math. Anal. Appl. 368 (2010), 382–390.

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[23] S. Głab, ˛ F. Strobin, Dichotomies for C0 (X) and Cb (X) spaces, Czechoslovak Math. J. 63 (2013), 91-105. [24] S. Głab, ˛ F. Strobin, Porosity and the L p -conjecture, Arch. Math. (Basel) 95 (2010), 583-592. [25] S. Głab, ˛ Chan Woo Yang, F. Strobin, Dichotomies for Lorentz spaces, Central European J. Math. 11 (2013), 1228–1242. [26] P. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, 1950. [27] C. Horowitz, An elementary counterexample to the open mapping principle for bilinear maps, Proc. Amer. Math. Soc. 53 (1975), 293–294. [28] J. Jachymski, A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces, Studia Math. 170 (2005), 303–320. [29] A. Komisarski, A connection between multiplication in C(X) and the dimension of X, Fund. Math. 189 (2006), 149–154. [30] S. Kowalczyk, Weak openness of multiplication in C(0, 1), Real Anal. Exchange 35 (2010), 235–241. [31] S. Kowalczyk, On operations in C(X) determined by continuous functions, Acta Math. Hungar., to appear. [32] R. Kunze, E. Stein, Uniformly bounded representations and harmonic analysis of the 2 × 2 real unimodular group, Amer. J. Math. 82 (1960), 1–62. [33] Li Ronglu, Zhong Shuhui, C. Swartz, An open mapping theorem without continuity and linearity, Topology Appl. 157 (2010), 2086–2093. [34] V. Olevskii, A note on the Banach-Steinhaus theorem, Real Anal. Exchange 17 (1991/92), 399–401. [35] T. Quek, L. Yap, Sharpness of Young’s inequality for convolution, Math. Scand. 53 (1983), 221–237. [36] N. Rickert, Convolution of L p functions, Proc. Amer. Math. Soc., 18 (1967), 762– 763. [37] W. Rudin, Functional Analysis, second edition, McGraw-Hill Inc. 1991. [38] S. Saeki, The L p –conjecture and Young’s inequality, Illinois. J. Math. 34 (1990), 615–627. [39] S.M. Srivastava, A Course on Borel Sets, Springer, New York, 1998. [40] F. Strobin, The porosity of convex nowhere dense subsets of normed linear spaces, Abstr. Appl. Anal. (2009), Article ID 243604, 11 pp. [41] F. Strobin, Dichotomies for Orlicz spaces, accepted in Mathematica Slovaca. [42] F. Strobin, Genericity and porosity of some subsets of function spaces, PhD thesis, Institute of Mathematics of the Polish Academy of Sciences, 2011, available at: http://im0.p.lodz.pl/˜filip [43] A. Wachowicz, Multiplying balls in C(n) [0, 1], Real Anal. Exchange 34 (2009), 445– 450. [44] A. Wachowicz, On some residual subsets, PhD thesis, Łód´z University of Technology, 2004 (in Polish). [45] L. Zajíˇcek, Porosity and σ -porosity, Real Anal. Exchange 13 (1987/88), 314–350. [46] L. Zajíˇcek, On σ -porous sets in abstract spaces, Abstr. Appl. Anal. 5 (2005), 509– 534.

18. Bilinear mappings – selected properties and problems M AREK BALCERZAK Institute of Mathematics, Łód´z University of Technology ul. Wólcza´nska 215, 93-005 Łód´z, Poland E-mail: [email protected]

F ILIP S TROBIN Institute of Mathematics, Łód´z University of Technology ul. Wólcza´nska 215, 93-005 Łód´z, Poland E-mail: [email protected]

A RTUR WACHOWICZ Institute of Mathematics, Łód´z University of Technology ul. Wólcza´nska 215, 93-005 Łód´z, Poland E-mail: [email protected]

305

Chapter 19

Stability Aspects of the Jensen-Hosszú Equation

ZYGFRYD KOMINEK

2010 Mathematics Subject Classification: 39B22, 39B52. Key words and phrases: general solutions of functional equations, Jensen and Hosszú functional equations, Hyers-Ulam stability.

19.1 Introduction The main motivation for the study of the subject of the stability is due to Ulam (cf. [13]). In 1940, Ulam presented some unsolved problems and among them posed the following question. Let G1 be a group and G2 be a group with a metric d. Given a real number δ > 0, does there exist ε > 0 such that if a map ϕ : G1 → G2 satisfies d(ϕ(xy), ϕ(x)ϕ(y)) ≤ ε for all x, y ∈ G1 , then there exists a homomorphism Φ : G1 → G2 such that d(ϕ(x), Φ(x)) ≤ δ for all x ∈ G1 ? If the answer to this question is "yes" then we say that the equation d(ϕ(xy), ϕ(x)ϕ(y)) = 0, x, y ∈ G1 is stable. Since then several, not necessarily equivalent, definitions of the stability have appeared. An extensive survey concerning this topic may be found

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in an excellent paper [10] by Z. Moszner. In this article our consideration are devoted to the following functional equation   x+y , x, y ∈ E (19.1) g(x + y − xy) + h(xy) = 2 f 2 in the class of functions f , g, h : E → X, where E is the set of all reals or the closed unit real interval and X is a real Banach space. We choose the following definition of the stability: given ε ≥ 0 and let F, G, H : E → X be functions satisfying the inequality

 

2F x + y − G(x + y − xy) − H(xy) ≤ ε, x, y ∈ E;

2 we say that functional equation 19.1 is stable if and only if there exist functions f , g, h : E → X fulfilling equation 19.1 and δ1 , δ2 , δ3 : [0, ∞) → [0, ∞), vanishing at zero and continuous, such that kF(x) − f (x)k ≤ δ1 (ε); kG(x) − g(x)k ≤ δ2 (ε); kH(x) − h(x)k ≤ δ3 (ε) for every x ∈ E.

19.2 Stability of Cauchy, Jensen and Hosszú functional equations We start with a result of D. Hyers [2] which contains a positive answer to the Ulam’s problem. Theorem 19.1. Given ε ≥ 0. Let X,Y be real Banach spaces and let f : X → Y satisfy the inequality k f (x + y) − f (x) − f (y)k ≤ ε

(19.2)

for all x, y ∈ X. Then there exists a unique additive function L : X → Y (i.e. L satisfies Cauchy functional equation) L(x + y) = L(x) + L(y), and the following estimation

x, y ∈ X,

19. Stability Aspects of the Jensen-Hosszú Equation

k f (x) − L(x)k ≤ ε,

309

x ∈ X.

Proof. (Sketch) Setting in (19.2) y = x we obtain k f (2x) − 2 f (x)k ≤ ε,

x ∈ X.

By induction we show that for every positive integer n we have





f (2n x) 1

2n − f (x) ≤ 1 − 2n ε. Defining

f (2n x) , x ∈ X, 2n one can check that (Ln (x))n∈N is a Cauchy sequence for each x ∈ X and its limit L(x) satisfies required conditions. t u Ln (x) :=

Assume, as previously, that X,Y are Banach spaces, ε ≥ 0 is a given constant and h : X → Y satisfies the following inequality





x+y h(x) + h(y)

h

≤ ε, (19.3) −

2 2 for all x, y ∈ X. Then function H given by the formula H(x) := h(x) − h(0), x ∈ X, also satisfies this inequality, i.e.,





H x + y − H(x) + H(y) ≤ ε, x, y ∈ X.

2 2 Moreover, observe that H(0) = 0 and kH(x + y) − H(x) − H(y)k ≤ 4ε,

x, y ∈ X.

On account of Theorem 19.1 there exists an additive function L : X → Y such that kH(x) − L(x)k ≤ 4ε, x ∈ X. Note that function J : X → Y defined as J(x) := L(x) + h(0), x ∈ X, satisfies the following Jensen functional equation of the form   x+y J(x) + J(y) = 2 J , (19.4) 2 x, y ∈ X, and estimation kh(x) − J(x)k ≤ 4ε,

(19.5)

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for every x ∈ X. Thus we have proved the following corollaries. Corollary 19.2. Given ε ≥ 0, let X,Y be Banach spaces and let h : X → Y satisfy (19.3) for all x, y ∈ X. Then there exists a function J : X → Y fulfilling Jensen equation (19.4) for all x, y ∈ X and estimation (19.5) for every x ∈ X. In other words, Jensen functional equation is stable in the class of functions transforming X into Y . Corollary 19.3. Let X,Y be Banach spaces. Then J : X → Y is a Jensen function if and only if J(x) = L(x) + c, x ∈ X, where c ∈ Y is a constant, and L : X → Y is additive. Observe that Jensen functional equation may be considered in the class of functions defined on a convex subset of X. But if the domain of J is a bounded subset of X then the method used above cannot be applied. The first step to solve the problem of the stability of Cauchy functional equation in the class of functions defined on a real interval was set out by F. Skof in [11]. She has proven that if f : [0, a) → Y, a > 0, (Y - a Banach space) satisfies inequality (19.2) for all x, y ∈ [0, a) such that x + y ∈ [0, a), then there exists an additive function L : R → Y such that k f (x) − L(x)k ≤ 3ε for every x ∈ [0, a). In [3], using this theorem we showed that if E ⊂ Rn is a bounded set and x0 is a point of its interior, 21 (E −x0 ) ⊂ (E −x0 ), and h : E → Y satisfies inequality (19.3) for all x, y ∈ E then there exist a Jensen function J : E → Y and a constant K (depending on the set E ) such that kh(x) − J(x)k ≤ Kε. for every x ∈ E. Applying this theorem M. Laczkovich has proved that in the case where X = R the assumption of the boundedness of E is superfluous. Theorem 19.4. [7] Given ε ≥ 0. If E ⊂ Rn is convex and h : E → R satisfies inequality (19.3), i.e.,   x+y h(x) + h(y) h ≤ ε, − 2 2 for all x, y ∈ E, then there exists a Jensen function J : E → R such that |h(x) − J(x)| ≤ Cε, for every x ∈ E, where C is a constant only depending on E.

19. Stability Aspects of the Jensen-Hosszú Equation

311

We omit sketch of the proof of this theorem because it is based also on some other results concerning local stability of convexity. In the class of real functions defined on R (or on the interval [0, 1]) the Jensen equation (19.4) is equivalent to the equation of the form ψ(x + y − xy) + ψ(xy) = ψ(x) + ψ(y), for all x, y ∈ R (or x, y ∈ [0, 1]) which is referred to as the Hosszú functional equation. This equation was mentioned for the first time by M. Hosszú [1] at the International Symposium on Functional Equations held in Zakopane (Poland). The proof of the equivalence of Jensen and Hosszú functional equations may be found in [6] in the case of functions transforming the set of all reals into itself. In the case of real functions defined on a unit interval it was proven by K. Lajkó [8]. In 1996 L. Losonczi [9] proved the stability of the Hosszú equation in the class of real functions defined on the set of all reals and posed the problem of the stability of this equation in the class of real functions defined on the unit interval [0, 1]. Surprisingly, in this case the Hosszú equation is not stable. J. Tabor (Jr.) [12] proved that for every ε > 0 one can find a function fε : [0, 1] → R such that | fε (x + y − xy) + fε (xy) − fε (x) − fε (y)| ≤ ε,

for all x, y ∈ [0, 1], (19.6)

and, simultaneously, for every Jensen function J : [0, 1] → R (or function satisfying the Hosszú functional equation) sup{| fε (x) − J(x)|; x ∈ [0, 1]} = ∞.

(19.7)

In fact, J. Tabor (Jr.) gave a very general method of constructions of such type examples. Using his ideas we give here an example of a function fε : [0.1] → R with ε = ln 4 fulfilling conditions (19.6) and (19.7). We define function f : [0.1] → R in the following way: ( 0, if x ∈ {0, 1};    f (x) := 1 1 , 21n ∪ 1 − 21n , 1 − 2n+1 , ln n, if x ∈ 2n+1 for arbitrary positive integer  n. For x1 = 0 or y = 0 condition (19.6) is trivially 1 1 1 fulfilled. Let x ∈ 2n+1 , 2n , y ∈ 2n+k+1 , 2n+k where n is an arbitrary positive and k is a non-negative integer. Then f (x) = ln n, f (y) = ln (n + k), f (xy) ≤ ln (2n + k + 1) and f (x + y − xy)  ≤1 ln n. It is not hard to check that condition 1 (19.6) is fulfilled for all x,  y ∈ 0, 2 with ε = ln 3. Assume that x, y ∈ [ 2 , 1). 1 Then 1 − x, 1 − y ∈ 0, 2 and hence

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| f (1−x+1−y−(1−x)(1−y))+ f ((1−x)(1−y))− f (1−x)− f (1−y)| ≤ ln 3. Consequently, | f (1 − xy) + f (1 − (x + y − xy)) − f (1 − x) − f (1 − y)| ≤ ln 3,  1 , 21n and y ∈ from which (19.6) easy follows. Assume now that x ∈ 2n+1   1 1 − 21m , 1 − 2m+1 for arbitrary positive integers n, m. Then f (x) = ln n, f (y) = ln m, f (xy) ≤ ln (n+1) and f (x+y−xy) ≤ ln (m+1). Thus in this case condition (19.6) is fulfilled with ε = ln 4. Assume that there exists a Jensen function J : [0, 1] → R such that sup{| f (x) − J(x)|; x ∈ [0, 1]} < ∞. Then J has to be a continuous function because it is locally bounded at 21 (for example [6]). Thus J is of the form J(x) = kx + c, x ∈ [0, 1], where k and c are real constants and hence it satisfies condition (19.7), because f is unbounded.

19.3 Stability of the Jensen-Hosszú functional equation In the previous part of the paper we gave an example of two equivalent functional equations (the Jensen and the Hosszú equations) one of which was stable, but one non-stable. The problem of the stability of the following JensenHosszú equation   x+y f (x + y − xy) + f (xy) = 2 f 2 now appears in a natural way. In this equation left-hand side coincides with the left-hand side of the Hosszú equation and the right-hand side coincides with the right-hand side of the Jensen equation. We will show that in the class of real functions defined on the whole set R, as well as on the unit interval, the Jensen-Hosszú equation is stable. In fact, we shall prove much more. Namely, in the case when the domain of functions f , g, h coincides with the set of all reals then equation (19.1) is stable which is not the case when the domain of f , g and h is the unit interval [0, 1]. In the former case, we obtain stability of our equation only if two among three functions f , g, h are equal. We start with the case when the domain of f , g, h is the set of all real numbers. The following theorem was proven in [4]. Theorem 19.5. [4] Let ε ≥ 0 be a fixed real number and let f , g, h : R → R be functions satisfying the following condition

19. Stability Aspects of the Jensen-Hosszú Equation

313

  2 f x + y − g(x + y − xy) − h(xy) ≤ ε, 2

x, y ∈ R.

(19.8)

Then there exist functions f1 , g1 , h1 : R → R fulfilling equation (19.1) and the following estimations | f (x) − f1 (x)| ≤ 7ε, |g(x) − g1 (x)| ≤ 11ε, and |h(x) − h1 (x)| ≤ 24ε, x ∈ R. Proof. Putting in (19.8) x = y = 0 we get |2 f (0) − g(0) − h(0)| ≤ ε. If F(x) = f (x) − f (0), G(x) = g(x) − g(0), H(x) = h(x) − h(0), x ∈ R, then the triple {F, G, H} satisfies the analogue condition, i.e.,   2F x + y − G(x + y − xy) − H(xy) ≤ 2ε, x, y ∈ R, (19.9) 2 and, moreover, F(0) = G(0) = H(0) = 0. Setting y = 0 in (19.9) we obtain x − G(x) ≤ 2ε, 2F 2

x ∈ R.

(19.10)

For arbitrary u ∈ R and v ≤ 0 the equation z2 − (u + v)z + v = 0 has two solutions x and y fulfilling the following equalities u+v = x+y

and

Consequently,   2F u + v − G(u) − H(v) ≤ 2ε, 2 Setting u = 0 in (19.11) we obtain v − H(v) ≤ 2ε, 2F 2

v = xy.

u ∈ R, v ≤ 0.

v ≤ 0.

(19.11)

(19.12)

By virtue of (19.10), (19.11) and (19.12), for all u ∈ R and each v ≤ 0, we have

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Zygfryd Kominek

        2F u + v − 2F u − 2F v ≤ 2F u + v − G(u) − H(v) + 2 2 2 2 u v + 2F − G(u) + 2F − H(v) ≤ 6ε, 2 2 which can be rewritten in the following equivalent form |F(u + v) − F(u) − F(v)| ≤ 3ε,

u ∈ R, v ≤ 0.

According to a well-known theorem ([11], see also remarks below Corollary 19.3 there exists a uniquely determined additive function A : R → R such that |F(v) − A(v)| ≤ 3ε,

v ≤ 0.

Using also (19.10) we obtain for v ≤ 0  v   v   v  − 2A |G(v) − A(v)| ≤ G(v) − 2F + 2F ≤ 8ε, , 2 2 2

(19.13)

and, similarly, using (19.12) instead of (19.10) |H(v) − A(v)| ≤ 8ε,

v ≤ 0.

It follows from (19.11) (by putting u = −v) that |G(−v) + H(v)| ≤ 2ε,

v ≤ 0.

For arbitrary v > 0 we have |G(v) − A(v)| ≤ |G(v) + H(−v)| + |A(−v) − H(−v)| ≤ 2ε + 8ε = 10ε, which together with (19.13) imply that |G(u) − A(u)| ≤ 10ε,

u ∈ R.

(19.14)

According to (19.14) and (19.10) 1 1 |F(u) − A(u)| ≤ |2F(u) − G(2u)| + |G(2u) − A(2u)| ≤ 6ε, 2 2 Putting x = v > 0 and y = 1 in (19.9) we get   v + 1 2F − G(1) − H(v) ≤ 2ε, 2 and, consequently,

u ∈ R.

19. Stability Aspects of the Jensen-Hosszú Equation

315

      v + 1 v+1 v + 1 |H(v) − A(v)| ≤ H(v) + G(1) − 2F −A + 2 F + 2 2 2 + |G(1) − A(1)| ≤ 24ε. Therefore, |F(x)−A(x)| ≤ 6ε, |G(x)−A(x)| ≤ 10ε

and

|H(x)−A(x)| ≤ 24ε,

x ∈ R.

Let us put d := 2 f (0) − g(0) − h(0), f1 (x) := A(x) + f (0) − d, g1 (x) := A(x) + g(0) − d, h(x) := A(x) + h(0), x ∈ R. We observe that |d| ≤ ε and   x+y − g1 (x + y − xy) − h1 (xy) = 0, x, y ∈ R. 2 f1 2 Moreover, | f (x)− f1 (x)| ≤ 7ε, |g(x)−g1 (x)| ≤ 11ε, and |h(x)−h1 (x)| ≤ 24ε,

x ∈ R. t u

This completes the proof.

In the next step we will show that the general solution of the equation (19.1) are Jensen functions in (0, 1). Assume (like in (19.8)) that   2 f x + y − g(x + y − xy) − h(xy) ≤ ε, x, y ∈ [0, 1]. 2 Setting F(x) = f (x) − f (0), G(x) = g(x) − g(0), H(x) = h(x) − h(0), x ∈ [0, 1], we obtain   2F x + y − G(x + y − xy) − H(xy) ≤ 2ε, x, y ∈ [0, 1], (19.15) 2 and, moreover, F(0) = G(0) = H(0) = 0. As an easy consequence (y = 0) we get x − G(x) ≤ 2ε, 2F 2

x ∈ [0, 1].

(19.16)

We define a subset D of [0, 1]2 in the following way D = {(u, v) ∈ [0, 1]2 ; (u + v)2 − 4v ≥ 0}. For every (u, v) ∈ D the equation x2 − (u + v)x + v = 0 has solutions x1 , x2 fulfilling conditions

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Zygfryd Kominek

x1 + x2 = u + v,

x1 x2 = v.

It is not hard to check that x1 , x2 ∈ [0, 1]. Hence and by virtue of (19.15) we infer that   2F u + v − G(u) − H(v) ≤ 2ε, (u, v) ∈ D. (19.17) 2 Observe that if (u, v) ∈ D and u + v ≤ 1 then (u + v, v) ∈ D whence   u + 2v 2F − G(u + v) − H(v) ≤ 2ε, (u, v) ∈ D0 := {(u, v) ∈ D; u+v ≤ 1}. 2 This together with (19.17) yields     2F u + 2v − G(u + v) − 2F u + v + G(u) ≤ 4ε, 2 2 and using (24) we get      u  u + 2v u + v F ≤ 4ε, − 2F +F 2 2 2

(u, v) ∈ D0

(u, v) ∈ D0 .

Putting u + 2v u , t= , (u, v) ∈ D0 , 2 2 after simple calculation we have   2F s + t − F(s) − F(t) ≤ 4ε, (s,t) ∈ ∆ , 2 s=

(19.18)

where ∆ is the region bounded by the curves     3 5 5 2 s + t = 1, s ∈ , , (s + t) − 4(s − t) = 0, s ∈ 0, 8 8 8 and   3 (s + t) − 4(t − s) = 0, s ∈ 0, . 8 2

More precisely, √ (s,t) ∈ ∆ if and only if s + t ≤ 1, (s + t)2 ≥ 4(s − t). Evidently, [2 2 − 25 , 12 ]2 ⊂ ∆ . On account of a theorem of Laczkovich [7] (see Theorem 19.4) there exists an additive function A : R → R and a real constant c such that

19. Stability Aspects of the Jensen-Hosszú Equation

317

  √ 5 1 |F(u) − A(u) − c| ≤ µ1 ε, u ∈ 2 2− , , 2 2 √ where µ1 does not depend on F. Let us put s0 = 12 , s1 = 2 2 − 52 and sn+1 = sn (1 − 21 sn ) for arbitrary positive integer n. Note that (sn )n∈N is a decreasing sequence converging to zero and, moreover, sn − sn+1 ≤ sn−1 − sn , n ∈ N0 . We will show that   1 (19.19) |F(u) − A(u) − c| ≤ µn ε, u ∈ sn , 2 where µn does not depend on F. It is true for n = 1. Observe that [sn , sn−1 ]2 ⊂ ∆ and if u ∈ [sn+1 , sn ] and v = 2sn −u, then u+v 2 , v ∈ [sn , sn−1 ], n ∈ N. For induction method assume (19.19) for a positive integer n. Take an arbitrary u ∈ [sn+1 , sn ]. According to (19.18) and (19.19) we obtain    |F(u) − A(u) − c| ≤ F(u) + F(v) − 2F u+v + 2 F u+v − A u+v − c 2

2

2

+|A(v) + c − F(v)| ≤ 4ε + 2µn ε + µn ε =: µn+1 ε, which ends the proof of (19.19). The sequence (sn ) tends to zero if n tends to infinity thus there exists a positive integer N such that     1 1 1 ⊃ , . (19.20) |F(u) − A(u) − c| ≤ µN ε, u ∈ sN , 2 4 2 It follows from (19.20) and (19.16) that u  u  u |G(u) − A(u) − 2c| ≤ G(u) − 2F −A − c + 2 F 2 2  2  1 ≤ 2ε + 2µN ε = 2(µN + 1)ε, u ∈ [2sN , 1] ⊃ , 1 . 2 Setting y = 21 in (19.15) we have       1 1 1 1 1 2F + x −G + x −H x ≤ 2ε, 4 2 2 2 2

(19.21)

x ∈ [0, 1].

  Therefore, according to (19.21) and (19.20), for every x ∈ 0, 12 , we get

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Zygfryd Kominek

          H 1 x − A 1 x ≤ H 1 x + G 1 + 1 x − 2F 1 + 1 x + 2 2 2 2 2 4 2         −G 1 + 1 x + A 1 + 1 x + 2c + 2F 1 + 1 x − 2A 1 + 1 x − 2c ≤ 2 2 2 2 4 2 4 2 ≤ 2ε + 2(µN + 1)ε + 2µN ε = 4(µN + 1)ε, whence |H(x) − A(x)| ≤ µε, where µ = 4(µN + 1). Consequently,   |F(u) − A(u) − c| ≤ µn ε, |G(u) − A(u) − 2c| ≤ µn0 ε,   |H(u) − A(u)| ≤ µε,

  1 x ∈ 0, , 4

  u ∈ sn , 21 ; u ∈ [2sn , 1];   u ∈ 0, 41 .

(19.22)

Thus we are in a position to prove the following theorem. Theorem 19.6. [5] Let f , g, h : [0, 1] → R be functions satisfying functional equation (19.1). Then there exists an additive function A : R → R and a constant c ∈ R such that f (x) = f (0) + A(x) + c,

x ∈ (0, 1);

g(x) = g(0) + A(x) + 2c,

x ∈ (0, 1];

h(x) = h(0) + A(x),

x ∈ [0, 1).

(19.23)

Conversely, if functions f , g, h are defined by (19.23) and 2 f (w) = g(w) + h(w) for w ∈ {0, 1}, then functional equation (19.1) is fulfilled. Proof. Putting F(x) = f (x)− f (0), G(x) = g(x)−g(0), H(x) = h(x)−h(0), x ∈ [0, 1], we observe that the triple (F, G, H) satisfies the following functional equation   x+y 2F = G(x + y − xy) + H(xy), x, y ∈ [0, 1]. 2 It follows from (19.22) (with ε = 0) that    x ∈ sn , 12 ;  F(x) = A(x) + c, G(x) = A(x) + 2c, x ∈ [2sn , 1];     H(x) = A(x), x ∈ 0, 14 . Taking n → ∞ we obtain

19. Stability Aspects of the Jensen-Hosszú Equation

  x ∈ 0, 12 ;  F(x) = A(x) + c, G(x) = A(x) + 2c, x ∈ (0, 1];     H(x) = A(x), x ∈ 0, 14 .

319

(19.24)

Let us put ρ0 = 12 , ρn+1 = 21 (1 + ρn2 ), n ∈ N0 . Evidently, sequence (ρn )n∈N is increasing and converging to 1. Using induction method we will prove that ( F(u) = A(u) + c, u ∈ (0, ρn ]; (19.25) H(u) = A(u) u ∈ [0, ρn2 ]. By virtue of (19.24) it is true for n = 0. Assume (19.25) for an n ∈ N0 . For 2 2 n each u ∈ [ρn , ρn+1 ] there exists an x ∈ [0, 1] such that u = x+ρ 2 . Setting y = ρn in (32) and applying the induction assumption we have   x + ρn2 2F = G(x + ρn2 − ρn2 x) + H(ρn2 x) 2   x + ρn2 2 2 2 = A(x + ρn − ρn x) + 2c + A(ρn x) = 2 A( )+c , 2 2 ] then there which proves the first equality of (19.25) for n + 1. If u ∈ [ρn2 , ρn+1 exists an x ∈ (0, 2ρn+1 ] such that x(2ρn+1 − x) = u. Therefore

H(u) = H(x(2ρn+1 − x)) = 2F(ρn+1 ) − G(x + 2ρn+1 − x − u)) = 2[A(ρn+1 ) + c] − A(x + 2ρn+1 − x − u) − 2c = A(u), which proves the second equality of (19.25) for n + 1 and ends the proof of (19.25). Since the part "conversely" is obvious, our assertion follows now from (19.24) and (19.25), because (34) hold for every non-negative integer. t u

19.3.1 The case h = f We have the following theorem. Theorem 19.7. [5] Let ε be a nonnegative number and let f , g : [0, 1] → R be functions satisfying the following condition   2 f x + y − g(x + y − xy) − f (xy) ≤ ε, x, y ∈ [0, 1]. 2

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Then there exists an additive function A : R → R and constants c, ρ1 , ρ2 ∈ R (ρ1 , ρ2 - not depending on f , g) such that | f (x) − f (0) − A(x) − c| ≤ ρ1 ε,

x ∈ [0, 1];

|g(x) − g(0) − A(x) − 2c| ≤ ρ2 ε,

x ∈ [0, 1].

(19.26)

Proof. Let F(x) = f (x) − f (0), G(x) = g(x) − g(0), x ∈ [0, 1]. Then also   2F x + y − G(x + y − xy) − F(xy) ≤ 2ε, x, y ∈ [0, 1]. (19.27) 2 Setting here y = 21 we have       2F 1 + 1 x − G 1 + 1 x − F 1 x ≤ 2ε, 4 2 2 2 2

x ∈ [0, 1].

 1 We  1 1may  use conditions (19.16) and (19.22). Taking N such that sN , 2 ⊃ 4 , 2 we infer that |c| ≤ (µ + µN )ε. Now, it follows from (19.22) and (19.16) that   1 |F(u) − A(u)| ≤ 2(µ + µN )ε = k ε, u ∈ 0, ; 2  u   u   u  |G(u) − A(u)| ≤ G(u) − 2F − 2A + 2F 2 2 2 ≤ 2(1 + k) ε = ρ2 ε, u ∈ [0, 1]. Hence           1 1 1 1 ≤ 2F 1 + 1 x − G 1 + 1 x − F 1 x 2F + x − 2A + x 4 2 4 2 4 2 2 2 2         1 1 1 1 1 1 + G + x −A + x + F x −A x ≤ (2 + ρ2 + k) ε, 2 2 2 2 2 2 whence

  k + ρ2 ε, |F(u) − A(u)| ≤ 1 + 2

Similarly, setting y =

  3 u ∈ 0, . 4

3 4

in (19.27), we get   k + ρ2 k + ρ2 |F(u) − A(u)| ≤ 1 + + ε, 2 4

  7 u ∈ 0, . 8

19. Stability Aspects of the Jensen-Hosszú Equation

Using induction method one can prove that   k + ρ2 k + ρ2 ε, |F(u)−A(u)| ≤ 1 + +···+ 2 2p

321

  1 u ∈ 0, 1 − p+1 , p ∈ N, 2

and therefore |F(u) − A(u)| ≤ (1 + k + ρ2 ) ε = ρ1 ε,

u ∈ [0, 1).

Since |2F(1) − G(1) − F(1)| ≤ 2ε then |F(1) − A(1)| ≤ |F(1) − G(1)| + |G(1) − A(1)| ≤ (2 + ρ2 ) ε, t u

which ends the proof of estimations (19.26).

19.3.2 The cases f = g and g = h The following two theorems easily follow from (19.22). Theorem 19.8. [5] Let ε ≥ 0 be fixed and let f , h : [0, 1] → R be functions satisfying the following condition   2 f x + y − f (x + y − xy) − h(xy) ≤ ε, x, y ∈ [0, 1]. 2 Then there exist an additive function A : R → R and constants ρ1 , ρ2 ∈ R (not depending on f , g) such that | f (x) − A(x) − f (0)| ≤ ρ1 ε,

x ∈ [0, 1];

|h(x) − A(x) − h(0)| ≤ ρ2 ε,

x ∈ [0, 1].

(19.28)

Proof. As in the proof of Theorem 19.6, we define functions F and H. It follows from (19.22) that |c| ≤ (µN + µN0 ) ε, where N ∈ N is so chosen that sN ≤ 14 , and |F(u) − A(u)| ≤ 2(µN + µN0 ) ε, u ∈ [sN , 1]. By (19.4) we have also |2F( 2x ) − F(x)| ≤ 2 ε, x ∈ [0, 1]. The last two inequal 1 ities imply that for every x ∈ 0, 4 we have 1 1 |F(x)−A(x)| ≤ [|2F(x)−F(2x)|+|F(2x)−A(2x)| ] ≤ (2+2(µN + µN0 )) ε. 2 2 This ends the proof of the first estimation of (19.28). If x ∈ [0, 1] then

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Zygfryd Kominek

  1 + x |H(x) − A(x)| ≤ H(x) + F(1) − 2F + |F(1) − A(1)| 2     1+x 1 + x + 2F − 2A ≤ (2 + 3 ρ1 ) ε. 2 2 t u In the same way one can prove the following theorem. Theorem 19.9. [5] Let ε ≥ 0 be fixed and let f , g : [0, 1] → R be functions satisfying the following condition   x + y 2 f − g(x + y − xy) − g(xy) ≤ ε, x, y ∈ [0, 1]. 2 Then there exist an additive function A : R → R and constants ρ1 , ρ2 ∈ R (not depending on f , g) such that | f (x) − A(x) − f (0)| ≤ ρ1 ε,

x ∈ [0, 1];

|g(x) − A(x) − g(0)| ≤ ρ2 ε,

x ∈ [0, 1].

Proof. Putting F(x) = f (x) − f (0) and G(x) = g(x) − g(0), x ∈ I, we observe that   2F x + y − G(x + y − xy) − G(xy) ≤ 2 ε, x, y ∈ [0, 1]. 2 According to (19.22) we get |c| ≤ 12 (µN0 + µ), where N ∈ N is so chosen that sN ≤ 81 . Therefore 3 |G(u) − A(u)| ≤ (µN0 + µ) ε = ρ2 ε, 2

u ∈ [0, 1].

Now, for all x, y ∈ [0, 1] we obtain       F x + y − A x + y ≤ F x + y − 1 G(x + y − xy) − 1 G(xy) + 2 2 2 2 2 1 1 + |G(x + y − xy) − A(x + y − xy)| + |G(xy) − A(xy)| ≤ (1 + ρ2 ) ε. 2 2 t u

19. Stability Aspects of the Jensen-Hosszú Equation

323

19.3.3 Counterexample The following example shows that in the general case of the inequality (19.8) an analogous assertion does not hold. Example 19.10. For arbitrary n ∈ N, we put r0 = 0, r1 = 12 , rn+1 =

g(u) = 0,

x+y = u ∈ [rn−1 , rn ), f (1) = 0; 2 u ∈ [0, 1];

h(u) = 2n + 1,

2 iff xy = u ∈ [rn−1 , rn2 ), h(1) = 0.

f (u) = n,

1+rn2 2 ,

and

iff

Functions f , g, h are well defined on the unit interval I. Note that if x+y 2 ∈ x+y 2 2 2 2 ) [r0 , r1 ) then xy ∈ [r0 , r1 ) and, moreover, if 2 ∈ [rn , rn+1 ) then xy ∈ [rn−1 , rn+1 for each positive integer n. Therefore for all x, y ∈ [0, 1] we have   2 f x + y − g(x + y − xy) − h(xy) ≤ 1, 2 which means that estimation (19.8) is fulfilled. On the other hand, every additive function A : R → R for which set {g(x) − A(x); x ∈ I} is bounded is of the form A(x) = kx, with a real constant k. But then sets { f (x) − A(x); x ∈ [0, 1]} as well as {h(x) − A(x); x ∈ [0, 1]} are unbounded. 

References [1] M. Hosszú, A remark on the dependence of functions, Zeszyty Naukowe Uniwersytetu Jagiello´nskiego, Prace Matematyczne 14 (1970), 127–129. [2] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-–224. [3] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 23, no 2, (1989), 499–507. [4] Z. Kominek, On the Hyers-Ulam stability of Pexider – type extension of the JensenHosszú equation, Bulletin of the International Mathematical Virtual Institute 1 (2011), 53–57. [5] Z. Kominek, On a pexidered Jensen-Hosszú functional equation on the unit interval, submitted. [6] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen Inequality, Polish Scientific Publishers and Silesian University Press, Warszawa-Kraków-Katowice, 1985. [7] M. Laczkovich, The local stability of convexity, affinity and of the Jensen equation, Aequationes Math. 58, no 1-2 (1999), 135–142.

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[8] K. Lajkó, Applications of Extensions of Additive Functions, Aequationes Math. 11 (1974), 68–76. [9] L. Losonczi, On the stability of Hosszú’s functional equation, Results Math. 29, no. 3–4 (1996), 305–310. [10] Z. Moszner, Sur les définitions différentes de la stabilité des équations fonctionnelles, Aequationes Math. 68, issue 3 (2004), 260–274. [11] F. Skof, Proprieta Locali e Approssimazione di Operatori, Rend. Semin. Mat. Fis. Milano 53 (1983), 113–129. [12] J. Tabor, Jr. Hosszú functional equation on the unit interval is not stable, Publ. Math (Debrecen) 49, fasc. 3–4 (1996), 335–340. [13] S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960.

Z YGFRYD KOMINEK Institute of Mathematics, Silesian University ul. Bankowa 14 , PL-40-007 Katowice, Poland E-mail: [email protected]

Chapter 20

Properties of the σ - ideal of microscopic sets

˙ ´ GRAZYNA HORBACZEWSKA, ALEKSANDRA KARASINSKA, ˙ ELZBIETA WAGNER-BOJAKOWSKA

2010 Mathematics Subject Classification: 28A05, 54C10, 54E52, 54H05, 02E15. Key words and phrases: microscopic sets, null sets, negligible sets, Cantor-type sets, comparison of σ -ideals, comparison of σ -fields.

The investigation of σ -ideals of subsets of the real line has a long tradition. The main motivation for the study of the collection of all microscopic sets, which constitutes a σ -ideal, stems from the fact that whenever one has to prove that a certain property in functional analysis or measure theory is fulfilled for "almost all" elements, the concept of "smallness" of the set of "exceptional points" should be described. The most classical of these concepts are related to Lebesgue nullsets and the sets of first Baire category. In certain applications, the ideals of measure and category turn out not to be suitable. In these situations it is useful to consider another σ -ideal having some good set-theoretic, algebraic and geometric properties. What makes microscopic sets interesting is the property that the collection of all microscopic sets constitutes a σ -ideal strictly smaller then the σ -ideal of sets of Lebesgue measure zero and orthogonal to the σ -ideal of sets of first Baire category. Therefore, in cases where it is well-known that a certain property holds everywhere except for a set of Lebesgue measure zero, it is important to check if the set of exceptional points is microscopic. If the answer is positive we get a stronger version of the property in question. In the classi-

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cal function theory on Rn the examples of such theorems are given by Fubini’s theorem, Kuratowski - Ulam theorem, Steinhaus theorem, Piccard theorem, results of Oxtoby and Ulam concerning homeomorphisms or Sierpi´nski - Erdös Duality Principle and many others. These results are of importance for various branches of mathematics, such as functional analysis, measure theory, geometric measure theory and descriptive set theory. Their stronger versions will provide a useful subtle tool for mathematicians working on different topics.

20.1 Microscopic sets on the real line The notion of microscopic set on the real line was introduced at the beginning of 21-st century in the paper [1] by J. Appell. Definition 20.1. A set E ⊂ R is microscopic if for each ε > 0 there exists S a sequence of intervals {In }n∈N such that E ⊂ n∈N In , and λ (In ) ≤ ε n for each n ∈ N. The family of all microscopic sets will be denoted by M. Deeper studies of microscopic sets were done by J. Appell, E. D’Aniello and M. Väth in the paper [3] from 2001. They showed that the collection of all microscopic sets constitutes a σ -ideal. It is not trivial to verify that a union of microscopic sets is microscopic, so we repeat it here. Let {Ak }k∈N be a sequence of microscopic sets. Let ε ∈ (0, 1) and define k εk := ε 2 for every k ∈ N. Then for any k ∈ N there exists a sequence of intervals S {Ik,n }n∈N such that Ak ⊂ n∈N Ik,n , and λ (Ik,n ) ≤ εkn , for each n ∈ N. Define a map φ : N × N → N by φ (k, n) = 2k−1 (2n − 1).

(20.1)

Obviously, it is a bijection. Now consider the sequence of intervals {Im }m∈N , where Im := Ik,n with m = φ (k, n). S S k k Then k∈N Ak ⊂ m∈N Im and λ (Im ) = λ (Ik,n ) ≤ εkn = (ε 2 )n = ε 2 n ≤ ε φ (k,n) = ε m , for every m ∈ N. S Therefore k∈N Ak ∈ M. Some properties of the considered σ -ideal are straightforward. If A ∈ M and α ∈ R, then 1. A + α = {x + α : x ∈ A} ∈ M,

20. Properties of the σ - ideal of microscopic sets

327

2. −A = {−x : x ∈ A} ∈ M, 3. α · A = {α · x : x ∈ A} ∈ M, 4. if 0 6∈ A, then A−1 = { 1x : x ∈ A} ∈ M. On account of the first two statements we can say that there is no "model set" for this ideal (see [13]). It means there is no set A ∈ M such that for each B ∈ M there exists x ∈ R such that B ⊂ A + x. Analogously as the σ -ideal of Lebesgue nullsets, M is Gδ - generated: Theorem 20.2 ([21], Lemma 2.3). Every microscopic set is contained in some microscopic set of type Gδ . Proof. Let A be a microscopic set. Then for each j ∈ N there exists a sequence S {In, j }n∈N of open intervals such that A ⊂ n∈N In, j and λ (In, j ) < ( 21j )n for each T S n ∈ N. Put B = ∞j=1 ∞ n=1 In, j . Obviously, A ⊂ B and B is a microscopic set of type Gδ . t u Simultaneously, the set constructed by A. S. Besicovitch in [10] shows that an approximation of Borel sets by Fσ sets with accuracy to microscopic set is impossible. Besicovitch proved that there exists a Borel set E ⊂ R such that if E = A ∪ N and A is a set of type Fσ , then N is not a set of Hausdorff dimension zero, so also not microscopic, as each microscopic set has Hausdorff dimension zero (compare [3], p. 258-9 or [2], p. 213). A microscopic set can be also described in another way. Theorem 20.3 ([15]). The following conditions are equivalent: 1) A is microscopic, 2) for every η > 0 there exists a sequence {Jn }n∈N of intervals such that ∞

A ⊂ lim sup Jn and n

∑ λ (Jk ) ≤ η n for n ∈ N, k=n

3) for every δ > 0 there exists a sequence {Jn }n∈N of intervals such that A ⊂ lim sup Jn and λ (Jn ) ≤ δ n for n ∈ N. n

Proof. 1) ⇒ 2) Suppose that E is microscopic set and η ∈ (0, 1). Put θ= k

and εk = θ 2 for k ∈ N.

η 1+η

(20.2)

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Let k be a fixed positive integer. As E is microscopic, there exists a sequence of intervals such that

{Ink }n∈N

∞ [

E⊂

Ink and λ (Ink ) < (εk )n .

(20.3)

n=1

Let φ be a function defined in (20.1) and let m ∈ N. There exists a unique pair (k, n) ∈ N × N such that φ (k, n) = m. Put Jm = Ink . Then E ⊂ lim supm Jm . Let p ∈ N and A p = {(k, n) ∈ N × N : φ (k, n) ≥ p}. Using (20.3) and (20.2) we obtain ∞

k

k−1 ·(2n−1)

(εk )n = ∑ θ 2 ·n < ∑ θ 2 m=p (k,n)∈A p (k,n)∈A p (k,n)∈A p (k,n)∈A p  p p ∞ θ θ ≤ = η p. ∑ θ φ (k,n) = ∑ θ m = 1 − θ 1 − θ m=p (k,n)∈A p The other implications are obvious. ∑ λ (Jm ) =



λ (Ink )
0. There exists n0 ∈ N such that n10 < ε. Obviously M ⊂ Moreover 1 1 λ (Kn0 ,i ) = < εi n < (n0 + 1)2 0 n0 i

S2n0

i=1 Kn0 ,i .

for i ∈ {1, ..., 2n0 }. Hence M is a Cantor-type set which is microscopic. As a perfect set M cannot be a strong measure zero set ([9], Corollary 8.1.5) so M ∈ M \ S. A classical result of Marczewski states that every strong measure zero set has universal measure zero ([26], Theorem 5.1), but the σ -ideals UMS and M are incomparable. Indeed, by theorem of Marczewski ([26], Theorem 9.1) a set of reals X belongs to UMS if and only if every set homeomorphic to X has Lebesgue measure zero, so since a microscopic Cantor-type set is homeomorphic to a Cantor set with positive Lebesgue measure, it is not universal measure zero set. On the other hand there exists on the real line a universal measure zero set with Hausdorff dimension one ([32]), it means not microscopic. The next theorem ensures the existence of a microscopic set which is large in a sense of category, i.e. it is residual. Theorem 20.4 ([21], Lemma 2.2). There exists a decomposition of R R = A∪B such that A is of the first category and B is a microscopic set.

20. Properties of the σ - ideal of microscopic sets

331

Proof. Let {rn }n∈N be a sequence of all rational numbers. Let In, j = (rn − 1 , r + 2n·1j+1 ), n ∈ N, j ∈ N. 2n· j+1 n Put ∞ Gj =

[

In, j

n=1

for j ∈ N, B=

∞ \

Gj

j=1

and A = R \ B. For each ε > 0 there exists j ∈ N such that 1/2 j < ε. Obviously, S 1 n n B ⊂ Gj = ∞ n=1 In, j and λ (In, j ) = ( 2 j ) < ε for n ∈ N. Hence B is a microscopic set. Simultaneously, G j is open dense subset of R for each j ∈ N, so B is a residual set and, consequently, A is a set of the first category. t u Consider an equivalent definition of a set of strong measure zero: a set E ⊂ R belongs to S if for each sequence of positive reals {εn }n∈N there exists a sequence of intervals {In }n∈N such that ∞

E ⊂ lim sup In and n

∑ λ (Ik ) < εn for n ∈ N. k=n

Looking at this definition of a strong measure zero set and the definition of a microscopic set from Theorem 20.3 we can notice a similarity to the Borel idea ([27], Lemma 14.1) of describing a Lebesgue measure zero set E by existence of a sequence of intervals {In }n∈N such that ∞

E ⊂ lim sup In and n

∑ λ (In ) < ∞.

n=1

Following E. Borel and M. Frechet ([11], [17]) W. Just and C. Laflamme in [20] and [25] classified measure zero sets according to their open covers and considered some σ -ideals of measure zero sets. One of them is σ -ideal of strong measure zero sets, so it is contained in M. We are going to justify that others are imcomparable with M. Let H denote a collection of sets E ⊂ R with a property that there exists a sequence of positive reals {εn }n∈N converging to zero such that for all nonincreasing sequences {δn }n∈N if δn ≥ εn infinitely often, then there exists a sequence of intervals {In }n∈N such that ∞

E ⊂ lim sup In and n

∑ λ (Ik ) ≤ δn k=n

for n ∈ N.

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Then H is a σ -ideal and any uncountable closed set of measure zero belongs to H ([25]), so it contains both classical Cantor set and Cantor - type microscopic set. The next class L is a collection of sets E ⊂ R with a property that there exists a sequence of positive reals {εn }n∈N such that for all sequences of intervals {In }n∈N if E ⊂ lim supn In , then the condition ∑∞ k=n λ (Ik ) ≥ εn holds for all but finitely many n ∈ N. Since each residual set of measure zero belongs to the L-class ([25]), then by Theorem 20.4 there is a microscopic set in L and if we consider a union of a microscopic comeager set with a nonmicroscopic set of measure zero (for example a classical one-third Cantor set) we get an example of a nonmicroscopic set from L. The next collection U consists of measure zero sets not in L, so E ⊂ R belongs to U if for each sequence of positive reals {εn }n∈N there exists a sequence of intervals {In }n∈N such that ∞

E ⊂ lim sup In and n

∑ λ (Ik ) < εn

for infinitely many n ∈ N.

k=n

Then U is a σ -ideal ([25]) and of course S ⊂ U. Therefore there are microscopic sets in U and comeager microscopic sets do not belong to U. Since H is consistently contained in U ([25]), there are also nonmicroscopic sets in U. Hence U is imcomparable with M. Given a proper σ - ideal I of subsets of the real line, containing all singletons and Gδ - generated, one can consider the family of all sets which can be covered by Fσ - sets from I (see for example [8]). If we denote this family by I ∗ then I ∗ ⊂ I and I ∗ is a σ -ideal. In particular M∗ is a σ - ideal. The next theorem states the results of the comparison of this family with others. Theorem 20.5. 1) C ⊂ M∗ and M∗ \ C 6= 0, / ∗ ∗ 2) M ⊂ M and M \ M 6= 0, / 3) N ∗ \ M 6= 0/ and M \ N ∗ 6= 0, / ∗ ∗ ∗ ∗ 4) M ⊂ N andN \ M 6= 0, / 5) N \ M ∪ N ∗ 6= 0. / Proof. All the above inclusions follow directly from definitions. We only have to show that the listed differences are really nonempty. 1) Any microscopic Cantor-type set belongs to M∗ \ C. 2) Any residual microscopic set belongs to M \ M∗ .

20. Properties of the σ - ideal of microscopic sets

333

3) The classical one-third Cantor set belongs to N ∗ \ M and any residual microscopic set belongs to M \ N ∗ . 4) The family N ∗ \ M∗ is nonempty since it contains N ∗ \ M. 5) The union of the classical one-third Cantor set and any residual null set  belongs to N \ M ∪ N ∗ . t u Since N ∗ ( M ∩ N ,

M∗ ( M ∩ N ,

where M denote a σ -ideal of sets of the first category. Note that the σ -ideal of σ - porous sets satisfies the analogous inclusion, so it is natural to compare it with M∗ . Let us recall the relevant definitions ([31]). For A ⊂ R, x ∈ R and ε > 0 let γ(A, x, ε) := sup{r > 0 : ∃z∈R (z − r, z + r) ⊂ (x − ε, x + ε) \ A}, where we put sup 0/ := 0. The porosity of A at x is defined by p(A, x) := lim sup ε→0+

2γ(A, x, ε) . ε

A is called porous if p(A, a) > 0 for each a ∈ A. It is called σ -porous if it belongs to P – the σ -ideal generated by the porous sets. As was already mentioned P ( M ∩ N and M∗ ( M ∩ N . However Theorem 20.6. P and M∗ are incomparable. Proof. Note that P \ M∗ 6= 0, / since P \ N ∗ 6= 0/ (see [16]). To show M∗ \ P 6= 0/ we use the idea from Lemma 0.2 ([19]). We prove that there exists a sequence of sets {Cn }n∈N , Cn ⊂ (0, 1) for n ∈ N, with the following properties: 1. Cn is closed microscopic nowhere dense set for n ∈ N; 2. Cn ∩Cm = 0/ for n, m ∈ N, n 6= m; 3.

∞ S

Cn is dense in [0, 1].

n=1

Put A1 = (0, 1). By Theorem 2.10 in [21] there exists a closed nowhere dense microscopic set C1 ⊂ A1 . Now we proceed by induction. Let n ∈ N, n ≥ 2. Suppose that we have defined the pairwise disjoint sets C1 ,C2 , ...,Cn−1 closed, nowhere dense and microscopic such that for each p ∈ {1, ..., n − 1}

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 p−1 [ l l +1 , Cp ⊂ \ Ci 2k 2k i=1  where k ∈ N ∪ {0} , l ∈ 0, 1, ..., 2k − 1 and the pair k, l fulfills equality p = 2k + l.  There exists the unique pair k, l of integers such that k ∈ N ∪ {0}, l ∈ 0, 1, ..., 2k − 1 and n = 2k + l. Put 

 An =

 n−1 [ l l +1 , k \ Ci . k 2 2 i=1

Obviously, An is an uncountable set of type Gδ , hence, by Theorem 2.10 in [21], there exists a closed, nowhere dense, microscopic set Cn ⊂ An , so Cn ∩

n−1 [

Ci = 0. /

i=1

It is easy to see that the sequence {Cn }n∈NN fulfills the conditions (1)-(3) S ∗ and ∞ t u i=1 Cn ∈ M \ P.

20.3 Bor4M We consider a σ -field of Borel sets modulo a σ -ideal M: Bor4M := {B4M : B ∈ Bor and M ∈ M}. Clearly Bor4M ⊂ Bor4N = L . We shall prove that Bor4M 6= L. For this purpose we need an auxiliary lemma. Lemma 20.7. The following conditions are equivalent: 1). E ∈ Bor4M 2). there exist two Borel sets A1 , A2 such that A1 ⊂ E ⊂ A2 and A2 \ A1 ∈ M. Proof. 1)⇒2). Let E ∈ Bor4M. Then there exist B ∈ Bor and M ∈ M such that E = B4M. Hence B\M ⊂ E ⊂ B∪M. There exists a microscopic Borel set N (of type Gδ ) such that M ⊂ N. Then B \ N ⊂ E ⊂ B ∪ N, (B ∪ N) \ (B \ N) = N ∈ M and B \ N as well as B ∪ N are Borel sets.

20. Properties of the σ - ideal of microscopic sets

335

2)⇒1). Suppose that there exist two Borel sets A1 , A2 such that A1 ⊂ E ⊂ A2 and A2 \ A1 ∈ M. Then E = A1 ∪ D, where D ⊂ A2 \ A1 , so D ∈ M and consequently E ∈ Bor4M. t u Theorem 20.8. There exists a measurable set E such that E 6∈ Bor4M. Proof. Let C be a classical Cantor set and let B denote the Bernstein set. Put C1 = C ∩ B and C2 = C \ B. Then C = C1 ∪ C2 , so C1 ∈ / M or C2 ∈ / M ( as C∈ / M). Suppose that C1 ∈ / M (the case C2 ∈ / M is analogous). Let A1 , A2 be two arbitrary Borel sets such that A1 ⊂ C1 ⊂ A2 . Suppose that A1 is uncountable. Then from Alexandroff-Hausdorff theorem it contains some Cantor-type set (uncountable and closed). It gives a contradiction with the fact that both B and R \ B meet every uncountable closed subset of the real line. Hence A1 is countable, so C1 \ A1 ∈ / M. Clearly, C1 \ A1 ⊂ A2 \ A1 , and A2 \ A1 ∈ / M. Put E = C1 Using the previous lemma we get a measurable set E such that E∈ / Bor4M. t u Using the notion of a Bernstein set, which is a useful tool to investigate a σ -field of Borel sets modulo a σ -ideal ([7]), we can prove even more. Let A ⊂ R and let A contain a perfect set. Definition 20.9. A set B ⊂ A is called a Bernstein set relatively to A if both B and A \ B meet each perfect subset of A. We will use the following: Proposition 20.10 ([7]). If a σ -ideal I has a Borel base and A ⊂ X is an analytic set such that A ∈ / I then there is no set B in Bor4I which is a Bernstein set relatively to A. Applying the last result with A - the classical one-third Cantor set yields: Corollary 20.11. There exists a Bernstein set B relatively to a perfect nowhere dense Lebesgue null set, such that B ∈ / Bor4M. According to the above corollary, we have: Theorem 20.12. (N ∩ N D) \ (Bor4M) 6= 0, / where N D denotes the family of all nowhere dense sets. To check the countable chain condition for the σ - field Bor4M and M we verify a stronger condition, the "property (D)", which is defined in a more general case. Let X be a perfect Polish space, (X, +) - a metric abelian group, I - an invariant ideal.

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Definition 20.13 ([5]). I has the property (D) if there exists a set B ∈ Bor \ I and a perfect set P ⊂ X such that {B + x : x ∈ P} forms a disjoint family. Proposition 20.14. M has the property (D). Proof. If I and J are ideals such that I ⊂ J and J has the property (D) then I has (D). It was shown in [5] that for every s ∈ (0, 1) the σ -ideal Js of s-dimensional Hausdorff measure zero sets has the property (D). So, since M ⊂ H0 ⊂ Js , we are done. t u Corollary 20.15. (Bor4M, M) does not satisfy ccc.

20.4 Studies on the possibility of replacing Lebesgue nullsets by microscopic sets in the classical theorems of measure theory and theory of real functions. In 2008 A. Karasi´nska and E. Wagner-Bojakowska in [22] studied how "big" can be a set on which a nowhere monotone continuous function can be injective. They proved that a "typical" (in a sense of Baire category) continuous function on [0,1] is nowhere monotone and injective outside a microscopic set. This result is a strenghtening of the result described in [12] (see Ex. 10: 6.6, p. 471). In the paper [22] one can find an example of a continuous nowhere monotone function with a bounded variation on [0,1] (so not a "typical" continuous function), which is injective outside a microscopic set. In [21] A. Karasi´nska, W. Poreda and E. Wagner-Bojakowska proved that the theorem analogous to Sierpi´nski-Erdös Duality Theorem for the family of microscopic sets and sets of the first category on the real line is valid. In 1934, W. Sierpi´nski proved in [29] (assuming CH) that there exists a bijection f : R → R such that f (E) is a nullset if and only if E is of the first category. Sierpi´nski asked whether a stronger theorem is also valid: does there exist a bijection f : R → R that maps each of two classes M and N onto the other. The positive answer to this question was given in 1943 by P. Erdös in [14]. Erdös proved (assuming CH) that there exists a bijection f : R → R such that f = f −1 and f (E) is a nullset if and only if E is a set of the first category. From these properties it follows that f (E) is a set of the first category if and only if E has Lebesgue measure zero. From Erdös result there follows a theorem known as Duality Principle (see [27], Theorem 19.4).

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Observe that if we change the notion of set of Lebesgue measure zero by the notion of a microscopic set, the theorem analogous to Duality Principle will also be true. For this purpose it sufficient to prove that the family M has the following properties: (a) M is a σ -ideal (b) the union of M is equal to R (c) M has a subfamily G such that card(G) ≤ ℵ1 and for each A ∈ M there exists B ∈ G such that A ⊂ B (d) the complement of each set A ∈ M contains a set of cardinality ℵ1 which also belongs to M. The condition (a) was already justified, (b) is obvious, (c) follows from Theorem 20.2 (assuming CH). We concentrate on (d). For any A ∈ M we have λ (A) = 0, so R \ A contains some uncountable closed subset. Using Theorem 2.10 from [21] we obtain that R \ A contains some microscopic set with cardinality ℵ1 . Using Theorem 19.5 in [27] we obtain Theorem 20.16 ([21], Theorem 2.12). (CH). There exists a one-to-one mapping f of the real line onto itself such that f = f −1 and f (E) is a microscopic set if and only if E is a set of the first category. Consequently, for microscopic sets the theorem analogous to Duality Principle holds: Theorem 20.17 (CH). Let P be any proposition involving solely the notions of microscopic set, first category set and notions of pure set theory. Let P∗ be the proposition obtained from P by interchanging the terms "microscopic set" and "set of the first category" whenever they appear. Then each of the proposition P and P∗ implies the other. However, the extended principle, where the notions of measurability and Baire property would be interchanged, is not true. Among many similarities between σ -ideals N and M so called Steinhaus property is worth to be mentioned. Let A and B be two subsets of the real line. By A + B we denote the algebraic sum of A and B, i.e. A+B := {x+y : x ∈ A, y ∈ B}. In 1920 H. Steinhaus proved in [30] that for arbitrary measurable sets A, B of positive measure, so outside σ -ideal N , int(A + B) 6= 0. /

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A category analogue of the theorem of Steinhaus was proved by S. Piccard (see [28]). If A, B ⊂ R are two sets of the second category having the Baire property, then int(A + B) 6= 0. / Observe that for microscopic sets the analogous property is not true: contrary to the σ -ideal N of Lebesgue measure zero sets and to the σ -ideal M of sets of the first category M has not a Steinhaus property. Theorem 20.18. There exists a Borel set A ⊂ R such that A is not microscopic and int(A + A) = 0. / Proof. Let C be a Cantor set and let H be the set of all end-points of component intervals of [0, 1] \C. Put A = C \ H and for each k ∈ N let k

Nk =

3 [



i=0

2i 3k

 .

Then for each k ∈ N we have (A × A) ∩

[

{(x, y) : y = −x + α} = 0. /

α∈Nk

Hence for each k ∈ N (A + A) ∩ Nk = 0, / so (A + A) ∩

[

Nk = 0. /

k∈N

The set

S

k∈N Nk

is dense in [0, 2], so int(A + A) = 0. /

t u

It is well known that the theorem converse to Steinhaus or Piccard results is not true. There exists a nowhere dense set A ⊂ R of measure zero such that int(A + A) 6= 0. / This condition holds for example for Cantor set C because C +C = [0, 2]. Observe, that the analogous property also holds for the σ -ideal M. Theorem 20.19. There exists a microscopic set A ⊂ R such that int(A+A) 6= 0. / Proof. Let A be a microscopic set residual in R (see Theorem 20.4). From theorem of S. Piccard it follows that A + A contains some interval. t u

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20.5 Extension of the notion of a microscopic set in the Euclidean spaces of higher dimensions. In the n-dimensional Euclidean space the notion of microscopic set can be introduced using various differentiation bases (as rectangles with sides parallel to coordinate axes, or cubes for example). Hence we can obtain different notions of microscopic sets. The properties of the sets, their invariance with respect to translation, rotation and other algebraic and set-theoretic operations are investigated in [24]. Definition 20.20. We shall say that A ⊂ R2 is a microscopic set if for each ε > 0 there exists a sequence {In }n∈N of rectangles with sides which are paralS lel to coordinate axes such that A ⊂ n∈N In and λ2 (In ) < ε n for each n ∈ N. Definition 20.21. We shall say that A ⊂ R2 is a strongly microscopic set if for each ε > 0 there exists a sequence {In }n∈N of squares with sides which S are parallel to coordinate axes such that A ⊂ n∈N In and λ2 (In ) < ε n for each n ∈ N. Denote by M2 the family of all microscopic sets in R2 and by M2s the family of all strongly microscopic sets in R2 . Obviously, each strongly microscopic set is microscopic, so M2s ⊂ M2 . In the sequel a rectangle with sides which are parallel to coordinate axes will be called an interval. Analogously as on the real line one can prove the following theorems. Theorem 20.22 ([24], Theorem 3). The families M2 and M2s are the σ -ideals. Theorem 20.23 ([24], Theorem 4). The following conditions are equivalent: (i) A is a microscopic set on the plane. (ii) For each positive number η there exists a sequence {Jn }n∈N of intervals such that n A ⊂ lim supn Jn and ∑∞ k=n λ2 (Jk ) < η for each n ∈ N. (iii) For each positive number δ there exists a sequence {In }n∈N of intervals such that A ⊂ lim supn In and λ2 (In ) < δ n for each n ∈ N. The analogous theorem holds for strongly microscopic sets (the intervals are changed with the squares). Theorem 20.24 ([24], Theorem 5). The plane can be represented as the union of two disjoint sets A and B such that A is a set of the first category and B is a strongly microscopic set.

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Corollary 20.25 ([24], Corollary 6). There exists a strongly microscopic set B ⊂ R2 which is residual. Theorem 20.26 ([24], Theorem 14). If A ∈ M2s and (α, β ) ∈ R2 , then (a) A + (α, β ) = {(x + α, y + β ) : (x, y) ∈ A} ∈ M2s , (b) −A = {(−x, −y) : (x, y) ∈ A} ∈ M2s , (c) (α, β ) · A = {(α · x, β · y) : (x, y) ∈ A} ∈ M2s , (d) if A ∩ {(x, y) : x · y = 0} = 0, / then A−1 = {( 1x , 1y ) : (x, y) ∈ A} ∈ M2s . Clearly the analogous theorem holds for the family M2 . Let us denote by N2 the family of all sets of Lebesgue measure zero on the plane and by C2 - the family of all countable subsets of the plane. Obviously, each countable set is strongly microscopic and if A is microscopic, then A is of Lebesgue measure zero, so we have C2 ⊂ M2s ⊂ M2 ⊂ N2 . Observe that all these inclusions are proper. It is easy to see that the set A = {(x, x) : x ∈ [0, 1]}

(20.4)

is a set of plane measure zero which is not microscopic on the plane, and the set B = [0, 1] × {0} (20.5) is a microscopic set on the plane which is not strongly microscopic. Clearly, each residual strongly microscopic set is uncountable, so C2 ( M2s ( M2 ( N2 . Comparing the sets A and B defined above we see that the family M2 is not invariant under rotation with respect to the origin. For the family M2s the situation is quite different. Theorem 20.27 ([24], Theorem 9). The set A ∈ M2s if and only if for each ε > 0 there exists a sequence {Bn }n∈N of circles on the plane such that A ⊂ S n n∈N Bn and λ2 (Bn ) < ε for each n ∈ N. Consequently, the family M2s is invariant under rotation and if A ∈ M2s , then the projection of A onto any line is a microscopic set. If E ⊂ X × Y and x ∈ X, the set Ex = {y ∈ Y : (x, y) ∈ E} is called the xsection of E.

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Fubini Theorem underlines a close connection between the measure of any plane measurable set and the linear measure of its sections perpendicular to an axis. In [27], Theorem 14.2 one can find an elementary proof of the fact that if E is a plane set of measure zero, then Ex is a linear nullset for all x outside a set of linear measure zero. Fubini Theorem has a category analogue. Kuratowski and Ulam in 1932 proved (compare [27], Theorem 15.1) that if E is a plane set of the first category, then Ex is a linear set of first category for all x except those belonging to a certain set of the first category. Note that for strongly microscopic sets the result analogous to Fubini theorem also holds. It is not difficult to observe it because there is a close connection between the area of the square and the length of its side.The result analogous to Fubini theorem for microscopic sets on the plane is also valid. Theorem 20.28 ([24], Theorem 17). Let E ⊂ R2 be a microscopic set on the plane. Then Ex is a microscopic set on the real line for each x ∈ R outside some microscopic set on the real line, i.e. the set {x ∈ R : Ex is not a microscopic set on the real line} is microscopic on R. Using Theorem 20.28 we proved Theorem 20.29 ([24], Theorem 18). A product set A × B is microscopic on the plane if and only if at least one of the sets A or B is microscopic on the real line.

20.6 Additional remarks In 2003 G. Horbaczewska and E. Wagner-Bojakowska introduced a definition of convergence of a sequence of functions with respect to the σ -ideal of microscopic sets. The idea comes from the Riesz theorem which states that a sequence of measurable functions { fn }n∈N is convergent in measure to the function f if and only if for every increasing sequence {nm }m∈N there exists a subsequence {nm p } p∈N such that the sequence { fnm p } p∈N is convergent to f almost everywhere (i.e. outside the set of measure zero). Therefore convergence in measure (in a finite measure space) can be defined using only the notion of a nullset (compare Chapter 6). This enables us to define convergence of the sequence of functions for different σ -ideals. In [18] two kinds of such

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a convergence, for the σ -ideal of microscopic sets and for the σ -ideal of sets of first Baire category were compared with the convergence in measure and with the convergence introduced by G. Beer using the Hausdorff metric. It was shown that even for continuous functions we have different types of convergence.

References [1] J. Appell, Insiemi ed operatori "piccoli" in analisi funzionale, Rend. Ist. Mat. Univ. Trieste 33 (2001), 127–199. [2] J. Appell, A short story on microscopic sets, Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 52 (2004), 229-233. [3] J. Appell, E. D’Aniello, M. Väth, Some remarks on small sets, Ric. Mat. 50 (2001), 255–274. [4] M. Balcerzak, A generalisation of the theorem of Mauldin, Comment. Math. Univ. Carolin. 26, (1985), 209-220. [5] M. Balcerzak, Can ideals without ccc be interesting?, Topology Appl. 55 (1994), 251–260. [6] M. Balcerzak, Classification of σ -ideals, Math. Slovaca 37, no 1 (1987), 63–70. [7] M. Balcerzak, On Borel sets modulo a σ -ideal, Demonstratio Math. 29 (2), (1996), 309–316. [8] M. Balcerzak, J. E. Baumgartner, J. Hejduk, On certain σ -ideals of sets, Real Anal. Exchange 14, (1988–89), 447–453. [9] T. Bartoszy´nski, H. Judah, Set theory: On the Structure of the Real Line, A. K. Peters, Ltd., Wellesley, MA, 1995. [10] A. S. Besicovitch, An approximation in measure to Borel sets by Fσ -sets, Journal London Math. Soc. 29 (1954), 382–383. [11] E. Borel, Les Eléments de la théorie des ensembles, Albin Michel, Paris, 1949. [12] A. M. Bruckner, J. B. Bruckner, B. S. Thomson, Real Analysis, Prentice-Hall, Upper Saddle River, New Jersey 07458, 1997. [13] J. Cicho´n, A. Kharazishvili, A. We¸glorz, Subsets of the Real Line, Wydawnictwo Uniwersytetu Łódzkiego, 1995. [14] P. Erdös, Some remarks on set theory, Ann. Math. 44 (2) (1943), 643–646. [15] M. Filipczak, E. Wagner-Bojakowska, Remarks on small sets on the real line, Tatra Mt. Math. Publ. 42 (2009), 73–80. [16] J. Foran, P. D. Humke, Some set theoretic properties of σ -porous sets, Real Anal. Exchange 6 (1980-81), 114–119. [17] M. Frechet, Les probabilites nulles et la rarefaction, Ann. scient. Ec. Norm. Sup., 30 serie, 80 (1963), 139–172. [18] G. Horbaczewska, E. Wagner-Bojakowska, Some kinds of convergence with respect to small sets, Reports on Real Analysis, Conference at Rowy (2003), 88–97. [19] G. Ivanova, E. Wagner-Bojakowska, On some modification of Darboux property, Math. Slovaca (to appear). [20] W. Just, C. Laflamme, Classifying measure zero sets with respect to their open covers, Trans. Amer. Math. Soc. 321, (1990), 621–645.

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[21] A. Karasi´nska, W. Poreda, E. Wagner-Bojakowska, Duality Principle for microscopic sets in monograph Real Functions, Density Topology and Related Topics, Łód´z University Press, 2011, 83–87. [22] A. Karasi´nska, E. Wagner-Bojakowska, Nowhere monotone functions and microscopic sets Acta Math. Hungar. 120 (3) (2008), 235–248. [23] A. Karasi´nska, E. Wagner-Bojakowska, Homeomorphisms of linear and planar sets of the first category into microscopic sets, Topology Appl. 159 (7) (2012), 1894– 1898. [24] A. Karasi´nska, E. Wagner-Bojakowska, Microscopic and strongly microscopic sets on the plane. Fubini Theorem and Fubini property, Demonstratio Math. (to appear). [25] C. Laflamme, A few σ -ideals of measure zero sets related to their covers, Real Anal. Exchange 17 (1991–92), 362–370. [26] A. W. Miller, Special subsets of the real line In: Kunen K., Vaughan J.E. (Eds.), Handbook of Set-theoretic Topology, Elsevier, North Holland, Amsterdam, 1984, 201–233. [27] J. C. Oxtoby, Measure and Category, Springer - Verlag New York Heidelberg Berlin, 1980. [28] S. Piccard, Sur les ensembles de distance, Mémoires Neuchatel Université, 1938–39. [29] W. Sierpi´nski, Sur la dualité entre la première catégorie et la mesure nulle, Fund. Math. 22 (1934), 276–280. [30] H. Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fund. Math. 1 (1920), 93–104. [31] L. Zajíˇcek, Porosity and σ -porosity, Real Anal. Exchange 13 (2) (1987-88), 314– 350. [32] O. Zindulka, Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps, Fund. Math. 218 (2012), 95–119.

˙ G RA ZYNA H ORBACZEWSKA Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

´ A LEKSANDRA K ARASI NSKA Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

˙ E L ZBIETA WAGNER -B OJAKOWSKA Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 21

Topological and algebraic aspects of subsums of series

ARTUR BARTOSZEWICZ, MAŁGORZATA FILIPCZAK, FRANCISZEK ´ PRUS-WISNIOWSKI

2010 Mathematics Subject Classification: 40A05, 11B05, 28A75. Key words and phrases: subsums of series, achievment set of sequence, M-Cantorvals.

The investigation of topological properties of sets of subsums for absolutely convergent series has been initiated almost one hundred years ago by Soichi Kakeya [11], [12]. A major step in the research took place in 1988 when J.A. Guthrie and J.E. Nymann published the full topological classification of the sets of subsums [8] – wider than Kakeya thought. However, their theorem is ineffective in the sense that it lists all four possible (up to homeomorphisms) types of sets of subsums, but provides no tool for recognition of the type for a given series. Finding a complete analytic characterization of the Guthrie-Nymann classification remains a challenging problem and we present the current state of research in this direction. Starting with a new exposition of the Guthire-Nymann Classification Theorem (based upon [21]), we survey all known examples of series leading to M-Cantorvals together with very recently discovered sufficient conditions for such series. The topological classification of the sets of subsums induces a natural division of the classic Banach space l1 into four disjoint sets. Interesting algebraic and topological properties of the division are also discussed in the survey.

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21.1 Sets of subsums of series Let (an )n∈N be a sequence of real numbers tending to zero. For a series Σn∈N an and a given a given set B ⊂ N, we will say that the series: ∑n∈B an is a subseries of ∑n∈N an . If B is finite, we will say that ∑n∈B an is a finite subseries of ∑n∈N an . We agree to write ∑n∈0/ an = 0. We are going to investigate the set of subsums of a series, that is, the set ) ( x∈R: ∃B⊂N

E = E ( an ) :=

∑ an = x

.

n∈B

We will also write E =



∑ εn an :

εn ∈ {0, 1} ,

assuming tacitly, that we consider such choises of (εn )n∈N only that lead to convergent subseries. The restricted definition allows for a very nice, transparent and natural classification of series. We start with a classification of series from the point of view of their behaviour under rearrangements. We will say that a series ∑ an is strongly divergent if ∑ aπ(n) diverges for every permutation π of its terms. We will say that a series ∑ an is absolutely convergent if ∑ aπ(n) converges for every permutation π of its terms. We know from the elementary theory of series that ∑ an is absolutely convergent if and only if ∑ |an | converges ((an ) ∈ l1 ). Other series which are neither strongly divergent nor absolutely convergent will be called potentially non-absolutely convergent. These are extactly series ∑ an for which there are permutations π1 and π2 of N such that ∑ aπ1 (n) converges and ∑ aπ2 (n) diverges. Thus the potentially non-absolutely convergent series are exactly the series that are non-absolutely convergent or can be rearranged into a non-absolutely convergent series. All three classes of series defined above have very transparent characterizations in terms of subseries of all positive terms and of all negative terms. With the classic definitons a+ n := max{an , 0 }

and

a− n := max{−an , 0 },

we get the following well known characterizations. Theorem 21.1. A series ∑ an is absolutely convergent if and only if both series − ∑ a+ n and ∑ an converge. A series ∑ an is potentially non-absolutely convergent if and only if both − series ∑ a+ n and ∑ an diverge.

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A series ∑ an is strongly divergent if and only if exactly one of the series − ∑ a+ n and ∑ an converges. Another characterization of our classification of series can be given in terms of their sets of subsums. We need an auxilliary fact (which would be false if not the initial agreement that general terms of a series must tend to 0) (cf. [3]). Lemma 21.2. If ∑ an is a divergent series of positive terms then every positive number is the sum of an infinite subseries of ∑ an and hence E(an ) = [0, +∞). We are ready for a theorem that tells us how to recognize the type of a series by looking at its sets of subsums E = E(an ). Theorem 21.3. A series ∑ an is: (i) strongly divergent if and only if the set E is a half-line. (ii) potentially non-absolutely convergent if and only if E = R. (iii) absolutely convergent if and only if E is bounded. Proof. Since our classification of series forms a division of the set all series, it suffices to prove implications from the left to the right in all three cases. − First, consider the case A := ∑ a+ n < +∞ and ∑ an = −∞. Applying Lemma 21.2 to the series ∑(−a− n ), we conclude that its sets of subsums is the half-line (−∞, 0]. It follows that E(∑ an ) = (−∞, A]. The proof in the case ∑ a+ n = +∞ − and B := − ∑ an > −∞ is analogous and leads to the conclusion that E := [B, ∞) which completes the proof of left-to-right implication in (i). Next, if the series ∑ an : is potentially non-absolutely convergent then using − the Lemma 21.2 to both series ∑ a+ n and ∑(−an ) and taking into account the the sum of an empty subseries is 0, we obtain E = R. − Finally, if the values A = ∑ a+ n and B = − ∑ an both are finite, we get E ⊂ [B, A]. t u If an absolutely convergent series ∑ an has only finitely many non-zero terms, then E = E(an ) is a finite subset of R and therefore presents no topological mysteries whatsoever. On the other hand, the removal of all zero terms from any series does not change their sets of subsums. Therefore we may and from now on we will assume that all terms of the investigated sequences (an ) are non-zero. Even more, we are now going to show that in order to describe all topological properties of sets of subsums of absolutely convergent series it suffices to consider only series of positive terms. Indeed, let α be the sum of all positive terms of a series ∑ an and let β be the sum of all negative terms, that is,

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α :=

and

an



β :=

an >0



an .

an 0 }

A− := {n ∈ A : an < 0 }.

and

Now, if the series ∑ an is absolutely convergent, we may use associativity and commutativity of infinite addition freely and hence

∑ an = ∑

n∈A

an +

n∈A+

=



an =

n∈A−





|an | +



|an | +

an 0 ∃N ∈ N ∀ n ≥ N In ⊂ (x − ε, x + ε), that is, if the closures of the intervals tend to the singleton : {x} : in the Vietoris topology. Observe three simple properties of the classic Cantor set C. Fact 21.15. The classic Cantor set C enjoys the following properties: (c1) Given any two C-gaps, there are an even order C-gap and an odd order C-gap between them. (c2) Every point of C is the limit of a sequence of C-gaps of even order and of a sequence of C-gaps of odd order. (c3) The set of endpoints of all odd order C-gaps is dense in C. The Guthrie-Nymann set is defined to be GN := C ∪

∞ [ n=1

G2n−1 = [0, 1] \

∞ [

G2n ,

n=1

where Gk denotes the union of all C-gaps of order k. Clearly, GN is a nonempty bounded perfect set, since no two GN-gaps have common endpoints. The GNgaps are exactly C-gaps of even order. The GN-intervals are exactly closures of C-gaps of odd order. GN has infinitely many component intervals and therefore is not homeomorphic to the classic Cantor set. How can we characterize all perfect subsets of R that are homeomorphic to GN? We start with a list of topological properties of the set GN. Fact 21.16. The Guthrie-Nymann set enjoys the following properties: (GN1) GN-gaps and GN-intervals have no common endpoints. (GN2) Endpoints of all GN-gaps are limits of sequences of GN-intervals and limits of sequences of GN-gaps. (GN3) Given any two GN-intervals (or any two GN-gaps, or a GN-gap and a GN-interval), there are a GN-interval and a GN-gap between them. (GN4) The union of all GN-intervals is dense in GN.

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Theorem 21.17. A nonempty bounded perfect set P ⊂ R is homeomorphic to the Guthrie-Nymann set if and only if (i) P-gaps and P-intervals have no common endpoints and (ii) the union of all P-intervals is dense in P. Proof. The necessity of both conditions (i) and (ii) follows easily from properties (GN1) and (GN4). Assume now that P ⊂ R is a nonempty bounded perfect set with properties (i) and (ii) and denote a := inf P, b := sup P. We are going to construct a homeomorphism h : [0, 1] → [a, b] (an increasing continuous surjection) such that h(GN) = P. Let (IiGN )i∈N be a joint enumeration of all GN-intervals and all closures of interior GN-gaps. Analogously, let (IiP )i∈N denotes a sequence of all P-intervals and closures of all interior P-gaps. It follows from the property (i) that the last sequence is infinite, indeed. S S We are ready for inductional construction of a function f : i IiGN → i IiP . Take the interval I1GN . If it is a GN-interval, then we map it in the linear increasing manner onto the first P-interval in the sequence (IiP )i∈N . If I1GN is the closure of a bounded GN-gap, then we define f I GN to be the increasing linear 1

map of the interval onto the first closure of a P-gap in the sequence (IiP )i∈N . Suppose now that n is a positive integer such that there is an increasing conS S tinuous injection f of ni=1 IiGN into i IiP such that f takes GN-intervals onto P-intervals and takes closures of GN-gaps onto closures of P-gaps. Consider GN . Exactly one of the following cases holds: the interval In+1 GN lies between the intervals I GN and I GN for some i, j ≤ n. (a) In+1 i j GN lies to the right of all I GN for i = 1, . . . , n. (b) In+1 i GN lies to the left of all I GN for i = 1, . . . , n. (c) In+1 i In the case (a), if IiGN is a GN-interval (the closure of a GN-gap), then we map it in the linear and increasing manner onto the P-interval (the closure of a P-gap) with the smallest index in the sequence (IiP )i∈N among indices of all P-intervals (of all closures of a P-gaps) lying between f (IiGN ) and f (I GN j ). In GN the case (b), if Ii is a GN-interval (the closure of a GN-gap), then we map it in the linear and increasing manner onto the P-interval (the closure of a P-gap) with the smallest index in the sequence (IiP )i∈N among indices of all P-intervals (of all closures of a P-gaps) lying to the right of all f (IiGN ) for i = 1, . . . , n. The case (c) is fully analogous to the case (b). S This construction yields an increasing continuous surjection f of i IiGN S P onto i Ii such that the image under f of the union of all GN-intervals is the union of all P-intervals.

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Let us recall that a bounded and increasing continuous function g : A → R defined on a set A dense in a closed interval [α, β ] can be extended to a continuous function g : [α, β ] → R if and only if lim g(x) = lim− g(x)

x→x0+

x→x0

for every x0 ∈ (α, β ) \ A. Then g is an increasing function from [α, β ] onto [limt→α + g(t), limt→β − g(t)]. The constructed by us function f is defined on [0, 1] except for the loose points of GN which are not endpoints of interior GN-gaps. Let x0 be such an exceptional point. The function f is increasing and bounded and hence there exist finite limits limx→x− f (x) ≤ limx→x+ f (x). Suppose that the two limits 0 0   are distinct. Then the open interval limx→ x− f (x), limx→x+ f (x) has no com0 0  S S mon points with f i IiGN = i IiP which contradicts the fact that the last set is dense in [a, b]. Hence, it must be limx→x− f (x) = limx→x+ f (x) which proves 0 0 that f can be uniquely extended to a homeomorphism h : [0, 1] → [a, b]. It remains to show that h(GN) = P. Since h is a homeomorphism, we get h(A) = h(A) for any A ⊂ [0, 1]. In particular, choosing A to be the union of all GN-intervals, we get h(GN) = P by the property (ii) and by our construction of h. t u A set homeomorphic to the GN set will be called an M-Cantorval. Another characterization of M-Cantorvals was given by Mendes and Oliveira in [16]. Theorem 21.18. A nonempty bounded perfect set P ⊂ R is an M-Cantorval if and only if all endpoints of P-gaps are limits of sequences of P-intervals and limits of sequences of P-gaps. Proof. A short outline of a direct constructional proof of the Mendes-Oliveira characterization of M-Cantorvals can be found in the Appendix of [16]. We are going to present here another proof based on Thm. 21.17. Suppose that P is an M-Cantorval. Then it has properties (i) and (ii) of Thm. 21.17. Observe that if a, b (with a < b) are points of P such that (∗) a, b are not endpoints of the same P-gap and (∗∗) a and b do not belong to the same P-interval, then the open interval (a, b) contains a P-interval. Indeed, the interval (a, b) must contain a point of the complement of P by (∗∗). Hence, since a, b ∈ P, the interval must contain a P-gap. Now, at least one of the endpoints of the P-gap

21. Topological and algebraic aspects of subsums of series

357

must lie in (a, b) by (∗). This endpoint cannot be an endpoint of a P-interval because of (i). Hence, by (ii), there is at least one P-interval contained in (a, b). Let x ∈ P be an endpoint of a P-gap. The point x does not belong to any P-interval by (i). On the other hand, since P is perfect, there is a sequence (xn ) of points of P monotonically convergent to x. Passing, if necessary, to a subsequence, we may assume that any two consecutive terms of the sequence are neither endpoints of the same P-gap nor belong to the same P-interval. Thus, by our earlier observation, there is a P-interval Pn between xn and xn+1 . The sequence (Pn ) converges to x and Pi ∩ Pj = 0/ for i 6= j. If Gn denotes any P-gap lying between Pn and Pn+1 , then the sequence (Gn ) converges to x as well. Hence all endpoints of P-gaps are limits of sequences of P-intervals and limits of sequences of P-gaps. Now, let P ⊂ R be a nonempty bounded perfect set such that all endpoints of P-gaps are limits of sequences of P-intervals and limits of sequences of Pgaps. This property implies instantly that a P-gap and a P-interval cannot have a common endpoint, that is, P satisfies the property (i) of Thm. 21.17. Let x be a point of P not belonging to any of P-intervals. Take any sequence (xn ) of points of P monotonically convergent to x. Passing, if necessary, to a subsequence, we may assume that no two consequtive terms of the sequence belong to the same P-interval or are endpoints of the same P-gap. Therefore, given any positive integer n, there is a P-gap between xn and xn+1 such that at least one of the endpoints of the gap belongs to the open interval with endpoints xn and xn+1 . According to our assumption about P, the endpoint of the gap is a limit of a sequence of P-intervals. Hence the open interval with endpoints xn and xn+1 contains infinitely many P-intervals. Choosing one of them and denoting it by Pn , we obtain a sequence (Pn ) of P-intervals convergent to x. Hence x belongs to the closure of the union of all P-intervals. Since x ∈ P was arbitrary, we conclude that the set P has the property (ii) of Thm. 21.17 as well. Then P is an M-Cantorval by the Thm. 21.17. t u

21.3 Sets of subsums of series and Cantorvals The first essential appearance of an M-Cantorval popped up in the paper [23] and it was given as a counterexample to a hypothesis on sets of subsums of an absolutely convergent series. M-Cantorvals turned out to be one of four possible topological types of sets of subsums of an absolutely convergent series [8]. However Guthrie and Nymann did not use the name; they wrote about sets homeomorphic to the set T of subsums ∑ βn where β2n−1 = 3/4n and

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β2n = 2/4n (n = 1, 2, . . . ). The Guthrie-Nymann set was given as a transparent example of a set homeomorphic to the set T in [8]. Finally, when Mendes and Oliveira characterized topological types of algebraic sums of homogeneous Cantor sets in [16], they defined various types of Cantorvals, including the M-Cantorvals, and used the name explicitly. We need first a theorem that tells us that the set of subsums is locally identical near endpoints of its gaps and it will be the crucial tool in proving the topological classification of sets of subsums of absolutely convergent series. It was proved in [18] and a number of versions of it were developed in more general settings (Lemma 3.3, [19] and Proposition 2.1, [1]). Theorem 21.19 (Nymann-Sáenz Theorem). If (a, b) is an interior E-gap, then the following equalities hold  b + [0, ε] ∩ E = [b, b + ε] ∩ E and  [1 − ε, 1] ∩ E − (1 − a) = [a − ε, a] ∩ E for all sufficiently small ε > 0. Proof. We start with an Observation 1: E ∩ [0, ε] = Ek ∩ [0, ε]

for ε < ak .

The inclusion ⊃ above is obvious. On the other hand, if x ∈ E and x < ak , then x is the sum of some terms less than ak , that is, some terms with indices greater than k. Hence, x ∈ Ek . Observation 2: Let b be the right endpoint of an interior E-gap. Let k and (k) f j = b be as in the Second Gap Lemma (Fact 21.11). If j = t(k) (see page 350), then b = ∑kn=1 an . Hence if x ∈ E and x > b, then x ∈ b + Ek by the Second Gap Lemma. The inclusion b + Ek ⊂ E is obvious. Hence (b, +∞) ∩ E = b + Ek and thus [b, b + ε] ∩ E = [b, b + ε] ∩ (b + Ek )

for every ε > 0. (k)

(k)

If j < t(k), then taking x ∈ [b, b + ε] ∩ E, where ε < f j+1 − f j , and its representation x = ∑n∈A an , we look at the trivial equality x =

∑ an

n∈A n≤k

+

∑ an .

n∈A n>k

21. Topological and algebraic aspects of subsums of series (k)

359 (k)

Clearly, x˜ := ∑n∈A an ∈ Fk . If x˜ < b, then x˜ ≤ f j−1 , and x ≤ f j−1 + rk . Thus, n≤k

by the Second Gap Lemma, x ≤ a < b, a contradiction. If x˜ > b, then x˜ ≥ (k) (k) f j+1 > f j + ε = b + ε, a contradiction. Therefore, it must be x˜ = b and hence x = b + ∑n∈A an ∈ b + Ek . We have proved that n>k

[b, b + ε] ∩ E = [b, b + ε] ∩ (b + Ek ) (k)

(k)

(k)

for 0 < ε < f j+1 − f j .

(k)

Finally, given ε < min{ak , f j+1 − f j }, we get b + [0, ε] ∩ E

 Obs. 1   = b + [0, ε] ∩ Ek = [b, b + ε] ∩ (b + Ek Obs. 2

= [b, b + ε] ∩ E.

The proof of the second equality in the thesis of the Thm. 21.19 is analogous. t u We are now ready for the main Guthrie-Nymann Classification Theorem (Thm. 1, [8]). Theorem 21.20. The set E of all subsums of an absolutely convergent series always is of one of the following four types: (i) a finite set; (ii) a union of a finite family of bounded closed intervals; (iii) a Cantor set; (iv) an M-Cantorval. Proof. Clearly, E is a finite set if and only if almost all terms of the series are zeros. It remains to look at the case when ∑ an is a convergent series of positive terms and of sum 1. Assume that E is then neither a union of a finite family of closed intervals nor a Cantor set. The first assumption tells us that an > rn for infinitely many n by the Cor. 21.14. The second assumption tells us that E contains at least one closed interval by the Thm. 21.7. Then 0 is the limit of a sequence of E-gaps by the First Gap Lemma. Since E is symmetric with respect to the point 21 , 1 is the limit of a sequence of E-gaps as well. A union of a finite family of nowhere dense sets is nowhere dense. Hence, since E contains a component interval, it follows from the Fact 21.5 that sets Ek contain at least one component interval Pk . Since En = [0, rn ]∩E = [0, an )∩E for all n ∈ A := {i : ai > ri }, it follows that the intervals Pk are intervals of E

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for those n as well. The sequence of intervals (Pn )n∈A converges to 0, because rn → 0. By symmetry again, the point 1 ∈ E is the limit point of a sequence of E-intervals as well. Now, by the Nymann-Sáenz Thm., we conclude that every endpoint of every E-gap is the limit of a sequence of E-gaps and of a sequence of E-intervals. Finally, an application of the Mendes-Oliveira Thm. 21.18 shows that E is an M-Cantorval. t u The latter theorem states that the space l1 can be decomposed into four sets c00 , C, I and MC, where I consists of sequences (xn ) with E(xn ) equal to a finite union of intervals, C consists of sequences (xn ) with E(xn ) homeomorphic to the Cantor set, and MC consists of sequences (xn ) with E(xn ) being Cantorvals. Let us recall some examples of absolutely summable sequences belonging to MC. We use the original notations proposed by the authors. The notation will be unified later in the chapter. A. D. Weinstein and B. E. Shapiro in [23] gave an example of a sequence (an ) defined by the formulas: a5n+1 = 0, 24·10−n , a5n+2 = 0, 21·10−n , a5n+3 = 0, 18 · 10−n , a5n+4 = 0, 15 · 10−n , a5n+5 = 0, 12 · 10−n . So,   3·8 3·7 3·6 3·5 3·4 3·8 , , , , , ,... . (an ) = 10 10 10 10 10 100 However, they did not justify why the interior of E(an ) is non-empty. Independently, C. Ferens ([7]) constructed a sequence (bn ) putting b5l−m = l−1 (m + 3) 233l for m = 0, 1, 2, 3, 4 and l = 1, 2, . . . . Therefore   1 1 1 1 2 1 (bn ) = 7 · , 6 · , 5 · , 4 · , 3 · , 7 · 2 , . . . . 27 27 27 27 27 27 Finally, in Jones’ paper [10] there is presented a sequence   3 2 2 2 3 19 2 19 2 19 2 19 3 19 2 (dn ) = , , , , · , · , · , · , ·( ) ,... . 5 5 5 5 5 109 5 109 5 109 5 109 5 109 In fact, R. Jones shows a continuum of sequences generating Cantorvals, indexed by a parameter q, by proving that, for any positive number q with ∞ 2 1 6 ∑ qn < 5 n=1 9

(i.e.

1 6

6q
k2 > · · · > km be positive integers and K = ∑m i=1 ki . Assume that there exist positive integers n0 and n such that each of numbers n0 , n0 + 1, . . . , n0 + n can be obtained by summing up the numbers k1 , k2 , . . . , km (i.e. n0 + j = ∑m i=1 εi ki with εi ∈ {0, 1}, j = 1, . . . , n). If 1 km 6q< n+1 K + km then E(k1 , . . . , km ; q) is a Cantorval. Now we can easily check that sequences (an ) , (bn ) and (dn ) generate Cantorvals, because they belong to appropriate one-parameter families, indexed by q. Example 21.22. The Weinstein-Shapiro sequence ([23]). It is clear that if E(xn ) is a Cantorval, α 6= 0 and (αxn ) = (αx1 , αx2 , . . . ), then E(αxn ) is a Cantorval too. To simplify a notation we multiply the sequence (an ) by 10 3 and consider the family of sequences aq = (8, 7, 6, 5, 4; q) for q ∈ (0, 12 ). Summing up 8, 7, 6, 5 and 4, we can get any natural number between n0 = 4 and n + n0 = 26. Therefore, by Theorem 21.21, for any q satisfying inequalities 4 1 6q< , 23 34 1 the sequence aq generates a Cantorval. Obviously, the number 10 used in [23] 1 4 4 belongs to [ 23 , 34 ). It is not difficult to check that aq ∈ I for q > 34 .

Example 21.23. The Ferens sequence ([7]). For the family of sequences bq = (7, 6, 5, 4, 3; q)

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1 3 K is equal to 25, n0 = 3 and n = 19. Hence, for any q ∈ [ 20 , 28 ), bq generates a 2 Cantorval. In particular, the sequence (7, 6, 5, 4, 3; 27 ), obtained from the Ferens sequence by multiplication by a constant, generates a Cantorval. Note that 3 bq ∈ I, for q > 28 .

Example 21.24. The Jones-Velleman sequence ([10]). Applying Theorem 21.21 to the sequence dq = (3, 2, 2, 2; q) 2 ), E(dq ) is a Cantorval we obtain K = 9, n0 = 2 and n = 5, so for any q ∈ [ 16 , 11 ∞ 1 2 2 n set. Clearly, ∑n=1 q ∈ [ 5 , 9 ), for such q. Moreover dq ∈ I for q > 11 . We can also consider analogous sequences for more than three 2’s. In fact, any sequence xq = (3, 2, . . . , 2; q) | {z } k−times

1 2 [ 2k , 2k+5 ),

with q ∈ generates a Cantorval set. Note that for k = 1 and k = 2 the argument of Theorem 21.21 breaks down, 1 2 because 2k > 2k+5 . However, we can apply Theorem 21.21 to "shortly defined" sequences. Indeed, for the sequence (4, 3, 2; q), numbers K, n0 and n are the same as for dq . km 1 It is not difficult to check that, to keep the interval [ n+1 , K+k ) non-empty, m m should be greater than 2. There is a natural question if Theorem 21.21 precisely describes the set of q with (k1 , . . . , km ; q) ∈ MC. The upper bounds, for all mentioned examples are km . However, this is not true for exact, because (k1 , . . . , km ; q) ∈ I, for q > K+k m all sequences satisfying the assumptions of Theorem 21.21. Example 21.25. For the sequence hq = (10, 9, 8, 7, 6, 5, 2; q), we have K = 47, km 1 1 2 n0 = 5 and n = 37. Therefore the interval [ n+1 , K+k ) = [ 38 , 49 ) is non-empty. m 2 However, for h = (10, 9, 8, 7, 6, 5, 2; 49 ) and any n ∈ N, we have ∑i>7n−1 h(i) = 2

·47

2 n−1 2 n−1 49 ( 49 ) (2 + 1− < h(7n − 1). It means that h ∈ / I. Note that in 2 ) = 4( 49 ) 49

the second part of the proof of 21.21(compare [5]) only the inequality q ≥ 2 1 is used. Since 49 > 38 , we have h ∈ / C and so h ∈ MC. 3 Again, it is not difficult to check that hq ∈ / I if and only if q < 50 .

1 n+1

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21.4 Topological and algebraic properties of C, I and MC Let us observe that all the sets c00 , C, I and MC are dense in `1 . Moreover, c00 is an Fσ -set of the first category. We are interested in studying the topological size and Borel classification of considered sets. To do it, let us consider the hyperspace H(R), that is the space of all non-empty compact subsets of reals, equipped with the Vietoris topology (see [13], 4F, pp. 24-28). Recall, that the Vietoris topology is generated by the subbase of sets of the form {K ∈ H(R) : K ⊂ U} and {K ∈ H(R) : K ∩U 6= 0} / for all open sets U in R. This topology is metrizable by the Hausdorff metric dH given by the formula dH (A, B) = max{max d(t, B), max d(s, A)} t∈A

s∈B

where d is the natural metric in R. It is known that the set N of all nowhere dense compact sets is a Gδ -set in H(R) and the set F of all compact sets with finite number of connected components is an Fσ -set. To see this, it is enough to observe that • K is nowhere dense if and only if for any set Un from a fixed countable base of natural topology in R there exists a set Um from this base, such that cl(Um ) ⊂ Un and K ⊂ (cl(Um ))c ; • K has more then k components if and only if there exist pairwise disjoint open intervals J1 , J2 , . . . , Jk+1 , such that K ⊂ J1 ∪J2 ∪· · ·∪Jk+1 and K ∩Ji 6= 0/ for i = 1, 2, . . . , k + 1. Now, let us observe that if we assign the set E(x) to the sequence x ∈ `1 , we actually define the function E : `1 → H(R). It is not difficult to check (compare Lemma 3.1, [2]) that the function E is Lipschitz with Lipschitz constant L = 1, and consequently it is continuous. Now we can prove that Theorem 21.26 ([2]). The set C is a dense Gδ -set (and hence residual), I is a true Fσ -set (i.e. it is Fσ but not Gδ ) of the first category, and MC is in the class (Fσ δ ∩ Gδ σ ) \ Gδ . Proof. Let us observe that C ∪ c00 = E −1 [N] and I ∪ c00 = E −1 [F] where N, F, E are defined as before. Hence C ∪ c00 is Gδ -set and I ∪ c00 is Fσ -set. Thus C is Gδ -set (because c00 is Fσ -set) and I ∪ MC is Fσ . Moreover, I = (I ∪ c00 ) ∩ (I ∪ MC) is Fσ -set, too. By the density of C, C is residual. Since I is dense of the first category, it cannot be Gδ -set. For the same reason, MC also cannot be Gδ -set. Since MC is a difference of two Fσ -sets, it is in the class Fσ δ ∩ Gδ σ . t u

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Jones in a very nice paper [10] gives the following example. Let (xn ) = (1/2n ) and (yn ) = (1/3n ). Then clearly (xn ) ∈ I and (yn ) ∈ C. Moreover, (xn + yn ) ∈ C and (xn − yn ) ∈ I. Since, for any n ∈ N, xn = (xn + yn ) − yn and yn = −(xn − yn ) + xn , then neither I nor C is closed under pointwise addition. However, the sets C, I and MC contain large (c-generated) algebraic structures. Assume that V is a linear space (linear algebra). A subset E ⊂ V is called lineable (algebrable) whenever E ∪ {0} contains an infinite-dimensional linear space (infinitely generated linear algebra, respectively). For a cardinal κ > ω, let us observe that the set E is κ-algebrable (i.e. it contains κ-generated linear algebra), if and only if it contains an algebra which is a κ-dimensional linear space . Moreover, we say that a subset E of a commutative linear algebra V is strongly κ-algebrable, if there exists a κ-generated free algebra A contained in E ∪ {0}. The subset M of a Banach space X is spaceable if M ∪ {0} contains infinitely dimensional closed subspace Y of X. (More information of such structures and a rich bibliography is presented in chapter 14.) In [2] it is proved that Theorem 21.27. C and I are strongly c-algebrable. MC is c-lineable. Theorem 21.28. Let I1 be a subset of I which consists of those x ∈ `1 for which E(x) is an interval. Then I1 is spaceable. Moreover, for any infinitedimensional closed subspace Y of `1 , there is (yn ) ∈ Y such that E(yn ) contains an interval. Note that from the last assertion it follows that the set C - the biggest in the topological sense - is not spaceable.

References [1] R. Anisca, Ch. Chlebovec, On the structure of arithmetic sums of Cantor sets with constant rations of dissection, Nonlinearity 22 (2009), 2127–2140. [2] T. Banakh, A. Bartoszewicz, S. Głab, ˛ E. Szymonik, Algebraic and topological properties of some sets in l1 , Colloq. Math. 129 (2012), 75–85. [3] C. R. Banerjee, B. K. Lahiri, On subseries of divergent series, Amer. Math. Monthly 71 (1964), 767–768 [4] E. Barone, Sul condominio di misure e di masse finite, Rend. Mat. Appl. 3 (1983), 229–238. [5] A. Bartoszewicz, M. Filipczak, E. Szymonik, Muligeometric sequences and Cantorvals, to appear in CEJM. [6] C. A. Cabrelli, K. E. Hare, U. M. Molter, Sums of Cantor sets, Ergodic Theory Dynamical systems 17 (1997), 1299–1313.

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[7] C. Ferens, On the range of purely atomic measures, Studia Math. 77 (1984), 261–263. [8] J. A. Guthrie, J. E. Nymann, The topological structure of the set of subsums of an infinite series, Colloq. Math. 55 (1988), 323–327. [9] H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monasth. Math. Phys. 49 (1941), 316–320. [10] R. Jones, Achievement sets of sequences, Amer. Math. Monthly 118, no. 6 (2011), 508–521. [11] S. Kakeya, On the partial sums of an infinite series, Tôhoku Sci. Rep. 3, no. 4 (1914), 159–164. [12] S. Kakeya, On the set of partial sums of an infinite series, Proc. Tokyo Math.-Phys. Soc. 2nd ser. 7 (1914), 250–251. [13] A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Math. 156, Springer, New York, 1995. [14] P. Kesava Menon, On a class of perfect sets, Bull. Amer. Math. Soc. 54 (1948), 706–711. [15] S. Koshi, H. Lai, The ranges of set functions, Hokkaido Math. J. 10 (special issue) (1981), 348–360. [16] P. Mendes, F. Oliveira, On the topological structure of arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), 329–343. [17] M. Morán, Fractal series, Mathematica 36 (1989), 334–348. [18] J. E. Nymann, R. A. Sáenz, On the paper of Guthrie and Nymann on subsums of an infinite series, Colloq. Math. 83 (2000), 1–4. [19] J. E. Nymann, R. A. Sáenz, The topoplogical structure of the set of P-sums of a sequence, II, Publ. Math. Debrecen 56 (2000), 77–85. [20] G. Pólya und G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Bd. I, Berlin, 1925. [21] F. Prus-Wi´sniowski, Beyond the sets of subsums, preprint, Łód´z University, 2013. [22] R. Wituła, Continuity and the Darboux property of nonatomic finitely additive measures, in: Generalized Functions and Convergence, Memorial Volume for Professor Jan Mikusi´nski (eds. P. Antosik and A. Kami´nski), World Scientific 1990. [23] A. D. Weinstein, B. E. Shapiro, On the structure of the set of α-representable numbers, Izv. Vyssh. Uchebn. Zaved. Mat. 24 (1980), 8–11.

A RTUR BARTOSZEWICZ Institute of Mathematics, Łód´z University of Technology ul. Wólcza´nska 215, 90-924 Łód´z, Poland E-mail: [email protected]

M AŁGORZATA F ILIPCZAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

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´ F RANCISZEK P RUS -W I SNIOWSKI Institute of Mathematics, University of Szczecin ul. Wielkopolska 15, PL-70-453 Szczecin, Poland E-mail: [email protected]

Chapter 22

On ψ-density topologies on the real line and on the plane

MAŁGORZATA FILIPCZAK, MAŁGORZATA TEREPETA

2010 Mathematics Subject Classification: 28A05, 54A10. Key words and phrases: density topology, ψ-density topology, comparison of topologies.

In this chapter we discuss topologies called ψ-density topologies. The definition of them is based on Taylor’s strengthening the Lebesgue Density Theorem. All of ψ-density topologies are essentially weaker than the density topology Td but still essentially stronger than Tnat . The notion of ψ-density topology was involved in the research work of many mathematicians. They concentrated mostly on the differences between density topology and ψ-density topologies on the real line. We would like to present the main results of that research but we will focus on ordinary and strong ψ-density topologies on the plane.

22.1 The density topology on the real line The classic Lebesgue Density Theorem [19] claims that for any Lebesgue measurable set A ⊂ R the equality λ (A ∩ [x − h, x + h]) =1 h→0+ 2h lim

(22.1)

holds for all points x ∈ A except for the set of Lebesgue measure zero. Denoting

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 λ (A ∩ [x − h, x + h]) Φd (A) = x ∈ R : lim =1 h→0+ 2h we can equivalently say that λ (A∆ Φd (A)) = 0 for any A ∈ L. The operator Φd is a lower density operator i.e. for any A, B ∈ L it has the following properties: (1) (2) (3) (4)

Φd (0) / = 0, / Φd (R) = R; Φd (A ∩ B) = Φd (A) ∩ Φd (B); λ (A∆ B) = 0 =⇒ Φd (A) = Φd (B); λ (A∆ Φd (A)) = 0.

It is well known that a family Td = {A ∈ L : A ⊂ Φd (A)} forms a topology called the density topology and denoted by Td . Let us recall its several properties. Theorem 22.1. The density topology has the properties: If λ (N) = 0 then N is Td −closed; Tnat Td ; (R, Td ) is neither first countable, nor Lindelöf, nor separable; A is Td −compact ⇐⇒ A is finite; λ (N) = 0 ⇐⇒ N is Td −nowhere dense ⇐⇒ N is Td −meager; (R, Td ) is a Baire space; intTd (A) = A ∩ Φd (B), where B is a measurable kernel of A; (R, Td ) is completely regular but not normal; A is connected in (R, Td ) ⇐⇒ A is connected in (R, Tnat ); Td is invariant under translations and multiplications by nonzero numbers. The proofs of these properties will be presented in the next chapter. Notice that: - properties (a)-(d) follow from properties (1)-(3) of the operator Φd ; - properties (e)-(g) are true by the Lebesgue Density Theorem (compare [19]); - a proof of completely regularity is much more complicated and connected with the Lusin-Menchoff Theorem (compare [11]); (R,√Td ) is not normal, because there is no possibility to separate Q from Q + 2 by Td -open sets (compare [11]); - (i) was proved by Goffman and Waterman via properties of approximately continuous functions (see [12]); - (j) again follows straightforward from the properties of Lebesgue measure. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

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22.2 Taylor’s strengthening the Lebesgue Density Theorem In The Scottish Book one can find a problem formulated by Stanisław Ulam (1936) as follows: "It is known that in sets of positive measure there exist points of density 1. Can one determine the speed of convergence of this ratio for almost all points of the set?" (Problem 146, [20]). In other words it is the question about possibility of strengthening the Lebesgue Density Theorem. The answer to Ulam’s question was given by S. J. Taylor in 1959, [21]. Taylor modified the condition (22.1) by introducing in denominator of the fraction a new factor ψ, which is a nondecreasing continuous function from (0, ∞) to (0, ∞) such that limx→0+ ψ(x) = 0 (the family of such functions will be denoted b He proved two important theorems ([21], Th.3 and 4). by C). The First Taylor’s Theorem For any Lebesgue measurable set A ⊂ R there exists a function ψ ∈ Cb such that λ (A0 ∩ I) =0 d(I)→0 λ (I)ψ(λ (I)) lim

for almost all x ∈ A, where I is any interval containing x. The Second Taylor’s Theorem For any function ψ ∈ Cb and a real number α, 0 < α < 1, there exists a perfect set E ⊂ [0, 1] such that λ (E) = α and lim sup d(I)→0

λ (E 0 ∩ I) =∞ λ (I)ψ(λ (I))

for all x ∈ E. In [22] S.J. Taylor formulated an alternative form of Egoroff’s theorem and, by a consequence, he obtained the following theorem. The Third Taylor’s Theorem Given any Lebesgue measurable set E in m-dimensional Euclidean space, there exist ψ ∈ Cb and S ⊂ E such that λm (E \ S) = 0 and for x ∈ S   1 λm (I ∩ E) −1 = 0 (22.2) lim λm (I) d(I)→0 ψ (d (I)) where I is any rectangle containing x with slides parallel to the coordinate axes of Rm and d (I) stands for a diameter of I.

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22.3 ψ-density topology on the real line Following Taylor, in [23] there was introduced a notion of ψ-density point b which involved only intervals I with center at x. Fix ψ ∈ C. Definition 22.2 ([23]). We say that x ∈ R is a ψ-density point of a set A ∈ L if λ (A0 ∩ [x − h, x + h]) = 0. h→0+ 2hψ(2h) lim

In particular, if ψ = id we obtain superdensity introduced by Zajicek in [15]. For any set A ∈ L we denote Φψ (A) = {x ∈ R : x is a ψ-density point of A}. The operator Φψ is not a lower density operator, in fact the properties (1)-(3) are fulfilled but (4) fails by The Second Taylor’s Theorem. However, for any measurable set A, Φψ (A) is measurable (in fact it is a Fσ δ set) and a family Tψ = {A ∈ L : A ⊂ Φψ (A)} b Tnat ⊂ constitutes a topology called ψ-density topology. Clearly, for any ψ ∈ C, Tψ ⊂ Td and null sets are Tψ −closed. Therefore, the space (R, Tψ ) is neither first countable, nor second countable, nor Lindelöf, nor separable; each set compact in Tψ is finite; a set is measurable if it is Borel in Tψ ([7]). Moreover, using the condition (i) from Theorem 22.1, we can easy conclude that the family of connected sets in topology Tψ coincides with the family of connected sets in Tnat (and Td ). The set R \ Q ∈ Tψ \ Tnat . Let E be the set constructed in The Second Taylor’s Theorem. It is easy to see that Φd (E) ∈ Td \ Tψ . Therefore, Tnat



Td .

Now we will look at the properties which distinguish topologies Tψ and Td . The set constructed in The Second Taylor’s Theorem has positive measure and is Tψ −nowhere dense. In density topology nowhere dense sets must have measure zero. E. Wagner-Bojakowska proved: Theorem 22.3 (compare [24], Th.8). There exist Tψ −closed and Tψ −nowhere S dense sets En ⊂ R such that R = ∞ n=1 En .  Hence R, Tψ is not a Baire space while (R, Td ) is. To describe the interior operation in ψ-density topology let Φψ1 (A) = Φψ (A). If α is an ordinal number, 1 < α < Ω and α = β +1, where 1 ≤ β < Ω ,

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β

then Φψα (A) = Φψ (Φψ (A)). If α is a limit number, α < Ω , then Φψα (A) = β 1≤β 0 Theorem 22.4. (a) For each ψ ∈ C, intψ (A) ⊂ A ∩ Φψα (A). (b) For each ψ ∈ Cb and A ∈ L there exists an ordinal β , 1 ≤ β < Ω such that β

intψ (A) = A ∩ Φψ (A). (c) For each ψ ∈ Cb and each countable ordinal α > 0 there exists A ∈ L such that intψ (A) 6= A ∩ Φψα (A).  The spaces (R, Td ) and R, Tψ satisfy different separate axioms. In [4] it is proved that ψ−density topologies does not satisfy the Lusin-Menchoff   Theorem and R, Tψ is not regular. Since Tnat ⊂ Tψ , R, Tψ is a Hausdorff space. b any translation of a set belonging to Tψ belongs to Tψ Clearly, for ψ ∈ C, either. It seems interesting that the invariance under multiplication depends on ψ. In fact, it is strictly connected with the condition which we call (∆2 ), by analogy with well known condition used in Orlicz spaces. We will say that ψ ∈ Cb fulfills (∆2 ) condition (ψ ∈ ∆2 ) if lim sup x→0+

ψ(2x) < ∞. ψ(x)

In [9] it is proved that: Theorem 22.5. The topology Tψ is invariant under multiplication if and only if ψ ∈ ∆2 . ( 1 x x for x ∈ (0, 1) does not satisfies (∆2 ), but functions Note that ψ(x) = 1 for x > 1 ψ(x) = xα satisfy this condition for any α > 0. It is obvious that [ b ⊂ Td . {Tψ : ψ ∈ C} In [27] it was proved that the inclusion is proper and topology generated by this union is equal to density topology. Moreover, (compare [23]),

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\

b = {U \ N : U ∈ Tnat ∧ λ (N) = 0}. {Tψ : ψ ∈ C}

It is evident that the same ψ-density topologies can be obtained via differψ1 (x) ψ1 (x) b If lim sup ent functions from C. x→0+ ψ2 (x) < ∞ and lim infx→0+ ψ2 (x) > 0 then Tψ = Tψ ([23]). However, there exist functions ψ1 , ψ2 ∈ Cb such that they 1

2

ψ1 (x) 1 (x) fulfill the conditions 0 < lim supx→0+ ψ ψ2 (x) < ∞ and lim infx→0+ ψ2 (x) = 0 and Tψ1 = Tψ2 . The necessary and sufficient condition were given by E. WagnerBojakowska and W. Wilczy´nski in [26]. For ψ1 , ψ2 ∈ Cb and k ∈ N they put:   1 + Ak = x ∈ R+ ; ψ1 (2x) < ψ2 (2x) , k   1 + Bk = x ∈ R+ ; ψ2 (2x) < ψ1 (2x) , k + Ak =A+ k ∪ (−Ak ),

+ Bk = B+ k ∪ (−Bk ).

and proved: b Theorem 22.6 ([26], Th. 8 ). Let ψ1 , ψ2 ∈ C, εk = lim sup x→0+

m(Ak ∩ [−x, x]) m(Bk ∩ [−x, x]) , ηk = lim sup 2xψ1 (2x) 2xψ2 (2x) x→0+

for k ∈ N. The topologies Tψ1 , Tψ2 are equal if and only if limk→∞ εk = limk→∞ ηk = 0. A brief survey of ψ-density topologies one can find in [2] and [3]. Some other properties of ψ-density topologies were also examined. G. Horbaczewska considered resolvability of ψ-density topologies and proved that for any function ψ ∈ Cb such topologies are maximally resolvable and, assuming Martin’s Axiom, extraresolvable ([10]). A. Go´zdziewicz-Smejda and E. Łazarow introduced the notion of ψ-sparse sets and ψ-sparse topologies ([13], [14]). E. Łazarow and A. Vizváry examined the category analogue of ψ-density topologies ([16]). In [17] E. Łazarow and K. Rychert introduced the notion of ψporosity and ψ-superporosity and they compared them with the classical notions of porosity and superporosity.

22.4 ψ-density topologies on the plane. Defining density points and ψ−density points on the real line there are used the closed intervals with the common center. On Rm there are two standard dif-

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ferentiation basis - we can use cubes or rectangles. To simplify considerations we will present the definitions and results obtained for R2 , but all of them can be applied in the same manner for m > 2. 2 - the natural topology on R2 . For any x = (x , x ) ∈ We will denote by Tnat 1 2 2 R and h, k > 0, let Sq(x, h) denote a square [x1 − h, x1 + h] × [x2 − h, x2 + h] and R(x, h, k) denote a rectangle [x1 − h, x1 + h] × [x2 − k, x2 + k]. Recall that x ∈ R2 is an ordinary density point of a set A ∈ L2 if λ2 (A0 ∩ Sq(x, h)) = 0. h→0+ 4h2 lim

Let Φdo (A) = {x ∈ R2 : x is an ordinary density point of a set A} for A ∈ L2 and Tdo denotes the family of all sets A ∈ L2 such that A ⊂ Φdo (A). The family Tdo is a topology called the ordinary density topology on the plane (see [28], section 4). Analogously, x ∈ R2 is a strong density point of a set A ∈ L2 if λ2 (A0 ∩ R (x, h, k)) = 0. h→0+ 4hk lim

k→0+

In the same way for A ∈ L2 we define the set Φds (A) = {x ∈ R2 : x is a strong density point of a set A}. The family Tds = {A ∈ L2 : A ⊂ Φds (A)} is a topology called the strong density topology on the plane. Both operators Φdo and Φds are lower density operators, so topologies Tdo and Tds satisfy properties analogous to (a)-(g) from Theorem 22.1. We will study properties of ordinary and strong ψ−density on the plane. Because the results have never been published in English we will present proofs of some theorems. More detailed information about this can be found in [5]. b We say that x ∈ R2 is an ordinary ψ-density point of a Suppose that ψ ∈ C. set A ∈ L2 if λ2 (A0 ∩ Sq(x, h)) = 0. lim h→0+ 4h2 ψ(4h2 ) Analogously, we say that x ∈ R2 is a strong ψ−density point of A if λ2 (A0 ∩ R (x, h, k)) = 0. h→0+ 4hkψ (4hk) lim

k→0+

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We say that x ∈ R2 is an ordinary (strong) ψ-dispersion point of a set A if x is an ordinary (strong) ψ-density point of A0 . We denote by Φψo (A) (Φψs (A)) the set of all ordinary (strong) ψ-density points of a set A. Using The Third Taylor’s Theorem we can easy obtain on the plane a result analogous to The First Taylor’s Theorem for ordinary ψ−density. Theorem 22.7. For any A ∈ L2 there exists a function ψ ∈ Cb such that  λ2 A \ Φψo (A) = 0. Proof. By The Third Taylor’s Theorem, there exists ψ ∗ ∈ Cb such that   λ2 (Sq (x, h) ∩ A) 1   −1 = 0 lim √ h→0+ ∗ 4h2 ψ 2h q  t for almost all x ∈ A. Let ψ (t) = ψ ∗ for t > 0. Then ψ ∈ Cb and , for any 2 h>0   λ2 (A0 ∩ Sq (x, h)) 1 λ2 (Sq (x, h)) − λ2 (Sq (x, h) ∩ A) = = ψ (4h2 ) 4h2 ψ (4h2 ) 4h2   1 λ2 (Sq (x, h) ∩ A)   = 1− . √ 4h2 ψ∗ 2h t u We will prove that the analogues result for strong ψ−density is not valid. Observe, that The Third Taylor’s Theorem refers to rectangles, but in the denominator of the formula (22.2) we have a diameter of I. We will show that - roughly speaking - we can not put λ2 (I) instead of d (I). Lemma 22.8. If B ∈ L1 satisfies the property λ (B ∩ [−h, h]) > 0 for any h > 0 then (0, 0) is not a strong ψ−density point of a set A = B0 × R b for any ψ ∈ C. b We will show that for any ε > 0, δ > 0 and h ∈ (0, δ ) there Proof. Let ψ ∈ C. is a number k ∈ (0, δ ) such that λ2 (A0 ∩ R (x, h, k)) > ε. 4hk · ψ (4hk)

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Fix ε > 0, δ > 0 and h ∈ (0, δ ). Since λ (B ∩ [−h, h]) > 0, there exists a positive number λ (B ∩ [−h, h]) . α= 2h A number h is fixed and limt→0+ ψ (t) = 0, so there is k ∈ (0, δ ) such that ψ (4hk) < αε . Therefore, λ2 (A0 ∩ R (x, h, k)) λ (B ∩ [−h, h]) · 2k α = = > ε. 4hk · ψ (4hk) 2h · 2k · ψ (4hk) ψ (4hk) t u Moreover, if λ (B ∩ [x1 − h, x1 + h]) > 0 for any h > 0, then (x1 , x2 ) is not a b strong ψ−density point of a set A = B0 × R for any x2 ∈ R and any ψ ∈ C. Observe that this property of a strong ψ−density is rather unusual. It is easy to check that Proposition 22.9. If x1 is a density point of B ∈ L1 then for any x2 ∈ R, (x1 , x2 ) is an ordinary and strong density point of B × R. Proposition 22.10. If x1 is a ψ−density point of B ∈ L1 , for some ψ ∈ Cb then for any x2 ∈ R, the point (x1 , x2 ) is a ψ ∗ −ordinary density point of B × R,  ∗ 2 where ψ (t) = ψ t for t > 0. On the other hand, if for some ψ ∈ Cb and b B ∈ L1 , (x1 , x2 ) is a √ ψ−ordinary point of B × R, then x1 is a ψ−density point  b (t) = ψ t . of B, for ψ For strong ψ−density we obtain a "strong strengthening" of the Second Taylor’s Theorem. Theorem 22.11. For any α ∈ (0, 1) there is a set E ⊂ [0, 1] × [0, 1] such that λ2 (E) = α and for any ψ ∈ Cb no point of E is a strong ψ−density point of a set E. Indeed, we can take a nowhere dense set C ⊂ [0, 1] of measure α and put A = C × [0, 1] and use Lemma 22.8. On the other hand, using Proposition 22.10, we can prove a theorem analogous to The Second Taylor’s Theorem for ordinary ψ−density. Theorem 22.12. For each function ψ ∈ Cb and number α ∈ (0, 1) there is a set E ⊂ [0, 1] × [0, 1] such that λ2 (E) = α and no point of E is its ordinary ψ−density point. Straightforward from the definitions of Φψo (A) and Φψs (A) we obtain (compare [23], Th.1.3):

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Theorem 22.13. For any A, B ∈ L2 (1) (2) (3) (4) (5)

if A ⊂ B then Φψo (A) ⊂ Φψo (B); if A ∼ B then Φψo (A) = Φψo (B); Φψo (0) / = 0/ and Φψo (R2 ) = R2 ; o Φψ (A ∩ B) = Φψo (A) ∩ Φψo (B); Φψo (A) ⊂ Φdo (A).

The same properties are satisfied for the operator Φψs . Put Tψo = {A ∈ L2 : A ⊂ Φψo (A)} and Tψs = {A ∈ L2 : A ⊂ Φψs (A)}. b The families T o and T s form topologies on the Theorem 22.14. Let ψ ∈ C. ψ ψ 2 and weaker then the ordinary plane, stronger than the Euclidean topology Tnat density topology Tdo . Moreover Tψs Tψo ∩ Tds . 2 ⊂ T s ⊂ T o ⊂ T o follow immediately from the definiProof. Inclusions Tnat ψ ψ d tions. To prove that Tψo (Tψs ) is a topology, it is enough to observe that the union of an arbitrary subfamily of Tψo (Tψs ) belongs to Tψo (Tψs ). The only difficulty is to show that it is a measurable set. It is true because Tψs ⊂ Tψo ⊂ Tdo and Tdo is closed under arbitrary unions, and Tdo ⊂ L2 . ∞ √ S b (t) = ψ t and A = Let ψ (an , bn ), 0 < bn+1 < an < bn for n = 1, 2, ... n=1

b be an interval set such that 0 is a right ψ−density point of A (for example 1 1 1 1 b an = 2n+1 + 4n+1 · ψ( 2n ) and bn = 2n ) . Clearly, the set B = −A∪{0}∪A belongs to Tψb and Td . Therefore, the set B × R, by Proposition 22.9, belongs to Tds , and - by Proposition 22.10 - belongs to Tψo . However, from Lemma 22.8 it follows that (0, 0) is not a strong ψ−density of B × R, so B × R ∈ / Tψs . 2 . Finally, we will define a set D such that The set R2 \ (Q × Q) ∈ Tψs \ Tnat 0 o o D ∈ Td \ Tψ . There is n0 ∈ N such that ψ( 21n0 ) 6 1. Let D=

∞ [ n=n0



 1 1 1 1 − ψ( ), × R. 2n 2n 4n−1 2n

 1 1 , 2n For simplicity we write Sq(h) instead of Sq ((0, 0), h). For each h ∈ 2n+1 2    2 1 λ2 (D ∩ Sq(h)) λ2 D ∩ Sq( 21n ) 1 2n ψ 4n−1 2n 6 6 = 4ψ . 1 1 n−1 4h2 4 4 (2n+1 n 2 4 )

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Since n → ∞ when h → 0+, the point (0, 0) is an ordinary density point of D0 . On the other hand, for any n > n0 , λ2 (D ∩ Sq( 21n ))

>

4( 21n )2 ψ(4( 21n )2 )

1 λ2 ([ 21n − 21n ψ( 4n−1 ), 21n ] × [− 21n . 21n ]) 1 ) 4 41n ψ( 4n−1

=

1 2

2 , and (0, 0) is not an ordinary ψ-density point of D0 . The set D0 \ {(0, 0)} ∈ Tnat 0 o o t u so D ∈ Td \ Tψ .

For any function ψ ∈ Cb the spaces (R, Tψo ) and (R, Tψs ) are Hausdorff and not separable. Any set of two-dimensional measure zero is closed. Each compact subspace of (R, Tψo ) or (R, Tψs ) is finite. Moreover b Then Theorem 22.15. Let ψ ∈ C. \

b = {Tψs : ψ ∈ C}

\

b = {U \ P : U ∈ T 2 ∧ λ2 (P) = 0}. {Tψo : ψ ∈ C} nat

Proof. It is not difficult to check  that a measurable set A belongs to the family 2 U \ P : U ∈ Tnat ∧ λ2 (P) = 0 if and only if   ∀ (x ∈ A) ∃ (δx > 0) ∀ h, k ∈ (0, δx ) λ2 A0 ∩ R(x, h, k = 0 .  2 ∧ λ (P) = 0 , then A ∈ T s , for any ψ ∈ C. b Therefore, if A ∈ U \ P : U ∈ Tnat 2 ψ  T 2 ∧ λ (P) = 0 ⊂ {T s : ψ ∈ C}. b Hence U \ P : U ∈ Tnat ψ  2 2 ∧ λ (P) = 0 . Therefore, there is a Suppose, that A ∈ / U \ P : U ∈ Tnat 2 point x ∈ A such that    1 0 >0 λ2 A ∩ Sq x, n   is decreasing and tends to for any n ∈ N. The sequence λ2 A0 ∩ Sq x, 1n b 0. There exists a function ψ ∈ C, such that      4 1 0 ψ = λ2 A ∩ Sq x, . n2 n Since

λ2 A0 ∩ Sq x, 1n  4 ψ n42 n2

 =

n2 , 4

A∈ / Tψo ⊃ Tψs . Therefore, \

b ⊂ {Tψs : ψ ∈ C}

\

 b ⊂ U \ P : U ∈ T 2 ∧ λ2 (P) = 0 . {Tψo : ψ ∈ C} nat t u

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 o and Φ s are lower density operators, so R2 , T o Recall that operators Φ d d d  and R2 , Tds are Baire spaces.   Theorem 22.16. The plane is a first category set in R2 , Tψs and in R2 , Tψo b for any ψ ∈ C. Proof. Let (Cn )n∈N be a sequence of Cantor-type sets on the real line of S positive one-dimensional Lebesgue measure, such that λ (R \ ∞ n=1 Cn ) = 0. b All sets Cn × R are T s −closed sets and, by Lemma Fix a function ψ ∈ C. ψ 22.8, Φψs (Cn × R) = 0. / Therefore, there are Tψs −nowhere dense sets. The set S s (R \ ∞ n=1 Cn ) × R is Tψ −nowhere dense because it is a set of measure zero. Note that, by The First Taylor’s Theorem, there are functions ψ ∈ Cb such that sets Cn × R have a nonempty Tψo −interior. In the next part of the proof we will divide the plane differently for  different ψ. Let ψ ∈ Cb and ψ ∗ (t) = ψ t 2 . There exist Tψ ∗ −closed and Tψ ∗ −nowhere S dense sets En ⊂ R such that R = ∞ n=1 En (Theorem 22.3). Therefore, / t u / The sets En × R are Tψo −closed and intTψo (En × R) = 0. intTψ ∗ (En ) = 0.  b Theorem 22.17. The space R2 , Tψs is not regular for any ψ ∈ C. Proof. As in the first part of the proof of Theorem 22.16 we can show an "universal" closed set and a point which can not be separated by Tψs −open sets b Fix a point (x0 , y0 ) ∈ R2 . In [11], Th. 5 it is proved that the for any ψ ∈ C. set L = {(x0 , y) : y ∈ R} \ {(x0 , y0 )} can not be separated from this point by Tds −open sets. b the set L is T s −closed, as a null set. Suppose that there are For every ψ ∈ C, ψ  disjoint sets U and V , open in R2 , Tψs such that (x0 , y0 ) ∈ U and L ⊂ V . Then U and V belong to Tds and separate (x0 , y0 ) from L, which gives a contradiction. t u A proof of the analogous property for ordinary ψ−density is more complicated.  b Theorem 22.18. The space R2 , Tψo is not regular for any ψ ∈ C.  Proof. Let ψ ∈ Cb and ψ ∗ (t) = ψ t 2 . There exists an increasing sequence (En )n∈N of closed, Tψ ∗ −nowhere dense subsets of [0, 1] such that ! λ

[0, 1] \

∞ [

n=1

En

=0

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(compare [4]). Let E = ∞ n=1 En and A = E × R. Observe that, for any n ∈ N, o En × R is Tψ −closed and Tψo −boundary set, and A ∩ ((0, 1) × R) ∈ Tψo .  Suppose that R2 , Tψo is regular. Fix z = (z1 , z2 ) ∈ A ∩ ((0, 1) × R). Then there exists a set U ∈ Tψo such that z ∈ U and ClTψo (U) ⊂ A ∩ ((0, 1) × R).  We will find a sequence x(n) n∈N of elements of the set U such that x = limn→∞ x(n) ∈ / A and S

lim sup n→∞

λ2 (U ∩ Sq (x, h)) > 0. 4h2

/ the set U \ (E1 × R) has positive Since intTψo (E1 × R) = 0/ and intTψo (U) 6= 0,   (1) (1) measure, so we can choose a point x(1) = x1 , x2 ∈ U \ (E1 × R). The se(1)

quence (En )n∈N is increasing and x1 ∈ E \ E1 , so there is k1 > 1 such that (1) (1) x1 ∈ Ek1 \ Ek1 −1 . Therefore, x1 belongs to some component J of R \ Ek1 and J is open. Denote it by (a1 , b1 ). Let   (1) (1) ε1 = min x1 − a1 , b1 − x1 . The set U belongs to Tψo and x(1) ∈ U. Therefore, x(1) ∈ Φψo (U) ⊂ Φdo (U) and there exists a number r1 ∈ (0, ε1 ) such that  λ2 U ∩ Sq x(1) , r1 3 ≥ . 2 4 4r1  It is not difficult to check, that for any y ∈ Sq x(1) , 14 r1  λ2 U ∩ Sq y, 34 r1 1 ≥ .  2 4 4 3 r1 4

Moreover, for any y ∈ Sq x(1) , 41 r1 the distance between y and Ek1 −1 × R (and between y and Ei × R for i < k1 − 1) is greater then 34 r1 . The set     1 1 1 1 (1) (1) (1) (1) U1 = U ∩ x1 − r1 , x1 + r1 × x2 − r1 , x2 + r1 4 4 4 4   (2) (2) belongs to Tψo . Since intTψo (Ek1 × R) = 0, / there is a point x(2) = x1 , x2 ∈ 

(2)

U \ (E1 × R). Therefore, there is k2 > k1 such that x1 ∈ Ek2 \ Ek2 −1 . Denote (2) by (a2 , b2 ) the component of R \ Ek2 such that x1 ∈ (a2 , b2 ) and by ε2 the

380

Małgorzata Filipczak, Małgorzata Terepeta (2)

(2)

minimum of numbers x1 − a1 , b1 − x1 and 41 r1 . Let r2 ∈ (0, ε2 ) be such a number that  λ2 U ∩ Sq x(2) , r2 3 ≥ . 2 4 4r2  We now proceed by induction and find a sequence x(n) n∈N of elements of U, a sequence (kn )n∈N of natural numbers and a decreasing sequence (rn )n∈N tending to zero such that   (n) (n−1) 1 (22.3) x ∈ U ∩ Ekn ∩ Sq x , rn , 4  λ2 U ∩ Sq x(n) , rn 3 ≥ 2 4rn 4 for any n > 1 and   3 dist x(n) , Ei × R ≥ rn 4

(22.4)

for any i < kn .  From (22.3) we know that the sequence x(n) n∈N is convergent. By (22.4),  x = limn→∞ x(n) ∈ / A. Finally, for any n ∈ N, x ∈ Sq x(n) , 41 rn and consequently  λ2 U ∩ Sq x, 34 rn 1 ≥ . 2 3 4 4 rn 4

It follows that lim sup n→∞

λ2 (U ∩ Sq (x, h)) 1 ≥ . 4h2 4

and x ∈ ClTdo (U). Since ClTdo (U) ⊂ ClTψo (U) , the set ClTψo (U) is not a subset  of A ∩ ((0, 1) × R). This contradiction proves that the space R2 , Tψo is not regular. t u From the definitions of operators and topologies Tψo and Tψs it follows that, b if A ∈ T o and x ∈ R then x + A = {x + a : a ∈ A} ∈ T o for any function ψ ∈ C, ψ ψ s (if A ∈ Tψ , then x + A ∈ Tψs ). As it would be expected, invariance under multiplications is connected with (∆2 ) condition. Observe first that: ψ(α0 t) b If there is α0 > 1 such that lim sup Lemma 22.19. Let ψ ∈ C. t→0+ ψ(t) = ∞

then lim supt→0+ ψ(αt) ψ(t) = ∞ for any α > 1. b Theorem 22.20. Let ψ be a function from the family C.

22. On ψ-density topologies on the real line and on the plane

381

ψ(t) o 1. If A ∈ Tψo , α > 0 and lim supt→0+ ψ(α 2 t) < ∞ then αA = {αa : a ∈ A} ∈ Tψ . ψ(t) 2. If α > 0 and lim supt→0+ ψ(α / 2 t) = ∞ then there is such a set B that αB ∈ o Tψ .

Proof. Suppose that A ∈ Tψo . To prove the first condition of the theorem it is enough to show that for any x ∈ A, αx is an ordinary ψ-density point of αA. Let x ∈ A. The point (0, 0) is an ordinary ψ-density point of a set A − x = {a − x : a ∈ A}. For any h > 0     h 0 0 0 . (α(A − x)) ∩ Sq(h) = α(A − x) ∩ Sq(h) = α (A − x) ∩ Sq α Therefore, lim sup h→0+

6

λ2 ((α(A − x))0 ∩ Sq(h)) 6 4h2 ψ(4h2 )

λ2 ((A − x)0 ∩ Sq( αh )) 1 ψ(t) · lim sup lim =0 h h 2 2 2 α h→0+ 4( α ) ψ(4( α ) ) t→0+ ψ(α t)

what means that (0, 0) is an ordinary ψ-density point of a set αA − αx. In the second part of the proof we will use a construction from the real line. Let ψ ∗ (t) = ψ(t 2 ) for t > 0. Hence lim sup t→0+

ψ(t) ψ ∗ (t) = lim sup = ∞. ∗ 2 ψ (αt) t→0+ ψ(α t)

Repeating the proof of Theorem 2.8 from [23] we can construct an interval set S A= ∞ n=1 [an , bn ] with 0 < bn+1 < an < bn for n ∈ N and lim bn = 0 such that 0 n→∞

is a ψ ∗ -dispersion point of A and is not a ψ ∗ -dispersion point of a set αA. It is easy to check that A0 ∈ Tψ ∗ and (αA)0 6∈ Tψ ∗ . Therefore, the set A0 × R is open in topology Tψo but the set (αA)0 × R = α · (A0 × R) is not. t u Corollary 22.21. An ordinary ψ−density topology is invariant under multiplication by positive numbers if and only if ψ ∈ ∆2 . One of the clearest differences between Tdo and Tds is connected with rotations. It is well known that the ordinary density topology on the plane is invariant under rotations and the strong density topology is not. It can be surprising, that invariance Tψo under rotations again depends on the (∆2 ) condition. b If ψ ∈ ∆2 then, for any set A ∈ T o , the Theorem 22.22. Suppose that ψ ∈ C. ψ set B received from A by turning around a fixed point, belongs to topology Tψo .

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Małgorzata Filipczak, Małgorzata Terepeta

Proof. We will show that if the point (0,0) is an ordinary ψ-dispersion point of a set A then (0,0) is an ordinary ψ-dispersion point of the set Aθ received from A by turning around (0,0) of an angle θ ∈ (0, 2π). Let Sqθ (h) denotes a square Sq(h) turned around (0,0) of θ . Because √ λ2 (Aθ ∩ Sq(h)) = λ2 (A ∩ Sq−θ (h)) 6 λ2 (A ∩ Sq(h 2), we obtain √ √ λ2 (A ∩ Sq(h 2) λ2 (Aθ ∩ Sq(h)) λ2 (A ∩ Sq(h 2) ψ(2 · 4h2 ) √ √ 6 = . · 1 2 2 4h2 ψ(4h2 ) 4h2 ψ(4h2 ) ψ(4h2 ) 2 4(h 2) ψ(4(h 2) ) 2

) From the assumption we know that lim supt→0+ ψ(2·4h < ∞. Therefore, ψ(4h2 )

lim sup h→0+

λ2 (Aθ ∩ Sq(h)) = 0. 4h2 ψ(4h2 )

Fix a point s = (s1 , s2 ) ∈ R2 and an angle θ ∈ (0, 2π). We will show that if x is an ordinary ψ-dispersion point of a set A then the point y, received from x by rotating x around the point s of θ , is an ordinary ψ-dispersion point of the set B received from A by the same rotate. Suppose that λ2 (A0 ∩ Sq(x, h)) = 0. h→0+ 4h2 ψ(4h2 ) lim

Denote by Sq∗ (y, h) a square received from Sq(y, h) by rotating of an angle −θ 0 0 ∗ around s. Thus, √ like in previous case, λ2 (B ∩ Sq(y, h)) = λ2 (A ∩ Sq (y, h)) ≤ 0 λ2 (A ∩ Sq(x, 2)) and λ2 (B0 ∩ Sq(y, h)) = 0. h→0+ 4h2 ψ(4h2 ) lim

t u b If ψ ∈ Theorem 22.23. Suppose that ψ ∈ C. / ∆2 then, for any angle θ ∈ (0, π4 ], there exists a set A ∈ L2 such that (0, 0) is not an ordinary ψ-dispersion point of a set A, but is an ordinary ψ-dispersion point of a set A rotated of an angle −θ around the point (0, 0). √

Proof. Fix an angle θ ∈ (0, π4 ]. Since cos( π4 −θ ) > 22 , a number 2 cos2 ( π4 −θ ) is greater then one. Let α ∈ (1, 2 cos2 ( π4 −θ )). We know that lim supt→0+ ψ(αt) ψ(t) = ∞. Therefore, there is a sequence (tn )n∈N & 0 such that lim

n→∞

ψ(αtn ) = ∞. ψ(tn )

(22.5)

22. On ψ-density topologies on the real line and on the plane

We will construct a sequence (dn )n∈N & 0 such that ψ(4d1 )
n · ψ(4dn2 ), q √ √ π ψ(4dn2 ) < 2 · cos( − θ ) − α, 4 q dn+1 6 dn · ψ(4dn2 ) for any n ∈ N. Let bn =

√ tn 2 .

(22.6) (22.7) (22.8)

From (22.5) it follows that ψ(α · 4b2n ) = ∞. n→∞ ψ(4b2 n) lim

Thus, we can choose a subsequence (cn )n∈N of a sequence (bn )n∈N , such that (cn )n∈N satisfies (22.6). Since limn→∞ cn = 0, almost all cn satisfy (22.7). We can choose a subsequence (dn )n∈N such that (22.8) is true and ψ(4d1 ) < 41 . Let q hi = di · ψ(4di2 ) (22.9) and denote by Ai a triangle with vertices (di , di ), (di , di − hi ) and (di − hi , di ). We define ∞ A=

[

Ai .

i=1

Observe that from (22.8) and (22.9) it follows λ2 (

∞ [

Ak ) = λ2 (Ai ) + λ2 (

k=i

∞ [

Ak ) 6 λ2 (Ai ) + λ2 ([0, di+1 ]2 ) =

(22.10)

k=i+1 2 = λ2 (Ai ) + di+1 6 λ2 (Ai ) + h2i = 3λ2 (Ai ),

for any i ∈ N. It is obvious that (0, 0) is not an ordinary ψ-dispersion point of a set A because, for each i, 1 2 λ2 (A ∩ Sq(di )) 1 2 hi > = . 2 2 2 2 4di ψ(4di ) 4di ψ(4di ) 8

We will prove that (0, 0) is an ordinary ψ-dispersion point of a set √ B, received from A by turning by −θ around (0, 0). Notice, that λ2 (B∩Sq( 2di )) = λ2 (A∩Sq(di )) for any i ∈ N and θ ∈ (0, π4 ]. Let t be an arbitrary point of (0, d1 ]. √ √ There is i ∈ N such that t ∈ ( 2di+1 , 2di ]. We will consider two cases: √ √ 10 . If t ∈ [ αdi , 2di ] then from (22.10) and (22.9) it follows

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Małgorzata Filipczak, Małgorzata Terepeta

√ λ2 (B ∩ Sq(t)) λ2 (B ∩ Sq( 2di )) λ2 (A ∩ Sq(di )) 3λ2 (Ai ) 6 = 6 6 2 2 2 2 2 2 4t ψ(4t ) 4αdi ψ(4αdi ) 4αdi ψ(4αdi ) 4αdi2 ψ(4αdi2 ) 3 · 12 di2 ψ(4di2 ) 3 = ; 6 8αi 4αdi2 iψ(4di2 ) √ √ 20 . Let t ∈ ( 2di+1 , αdi ). Note that n√ π  √ π o {(di , di )}θ = 2di cos + θ , 2di cos −θ . 4 4 By (22.7) we have    q π √ √ √ π 2 − θ − ψ(4di ) > di α > di , 2di ·cos( −θ )−hi = di 2 cos 4 4 √ so Aθi ∩ ([0, di ] × [0, di ]) = 0. / Therefore, B ∩ Sq(t) = B ∩ Sq( 2di+1 ). Reminding that α < 2, we obtain from (22.9) and (22.6) √ λ2 (A ∩ Sq(di+1 )) λ2 (B ∩ Sq(t)) λ2 (B ∩ Sq( 2di+1 )) 6 = 6 2 2 2 ψ(4 · 2d 2 ) 2 2 4t ψ(4t ) 4 · 2di+1 ψ(4 · 2di+1 ) 8 · di+1 i+1 3λ2 (Ai+1 ) 3 6 6 . 2 2 8 · di+1 ψ(α · 4di+1 ) 16(i + 1) Since i → ∞ when t → 0+, it follows that (0, 0) is an ordinary ψ-dispersion point of a set B. t u As we can expect, a strong ψ-density topology is not invariant under rotation, either. Firstly, we construct a set C ∈ L2 such that (0, 0) is a strong ψ-dispersion point of this set and λ2 (C ∩ Sq (r)) > 0 for any r > 0. Example 22.24. Suppose that ψ ∈ Cb , (bn )n∈N is a decreasing sequence tending to 0 and (an )n∈N is a sequence of positive numbers with 1 an+1 ≤ √ an 2 for n ∈ N. Let, for any n ∈ N, cn = an+1 · bn ·

q  ψ 4a2n+1 ,

Cn = [an − cn , an ] × [an − cn , an ]

22. On ψ-density topologies on the real line and on the plane

and C=

∞ [

385

Cn .

n=1

Obviously, λ2 (C ∩ Sq (r)) > 0 for any r > 0 and λ2 (Cn+1 ) ≤ 12 λ2 (Cn ) for any √  n ∈ N. Moreover, there is n0 ∈ N such that ψ 4a2n+1 < 1 and bn < 2 − 1 for n > n0 . Therefore, for n > n0 , q √  √ an − cn ≥ 2an+1 − an+1 bn ψ 4a2n+1 > 2an+1 − an+1 bn > an+1 , so squares Cn and Cn+1 are disjoint. We will check that (0, 0) is a strong ψ-dispersion point of C. Fix arbitrary k, h ∈ (0, an0 ]. There is n ≥ n0 such that min (k, h) ∈ (an+1 , an ]. From the construction of the set C it follows that C ∩ R ((0, 0), h, k) ⊂ C ∩ ([0, an ] × [0, an ]) . Hence

λ2 (C ∩ R ((0, 0), h, k)) b2n c2n  = ≤ 2 . 4hk · ψ (4hk) 2 4an+1 · ψ 4a2n+1

Now we will use a lemma which is, in fact, the strengthening of Lemma 22.8. Lemma 22.25. Suppose that A ∈ L2 . If there exists a sequence (an )n∈N & 0 such that all points (an , 0) belong to an interior of A (in the natural topology b on the plane) then (0, 0) is not a strong ψ-dispersion point of A for any ψ ∈ C. Proof. Fix a function ψ ∈ Cb and n ∈ N. There exists a positive number δn < min {an , 1}, such that Sq ((an , 0) , δn ) ⊂ A. Hence, for any k ∈ (0, δn ), λ2 (A ∩ R ((0, 0) , an , k)) ≥ 2kδn . Since lim ψ (t) = 0, there is εn ∈ (0, δn ) such that ψ (4an k) < t→0+

δn an

for any k ∈

(0, εn ]. Therefore, λ2 (A ∩ R ((0, 0) , an , k)) 2kδn 1 ≥ > . 4an k · ψ (4an k) 4an k · ψ (4an k) 2  1 Let k1 = ε1 and kn = min εn , 2 kn−1 for n ≥ 2. Then lim sup n→∞

λ2 (A ∩ R ((0, 0) , an , kn )) 1 ≥ 4an kn · ψ (4an kn ) 2

and, consequently, (0, 0) is not a strong ψ-dispersion point of A.

t u

386

Małgorzata Filipczak, Małgorzata Terepeta

From the latter example and lemma it follows that (0, 0) it is not a strong ψ-dispersion point of the set C rotated of π4 . Therefore, R2 \ C ∈ Tψs and π R2 \C 4 6∈ Tψs . Modifying a bit the construction in Example 22.24 we can construct, for any angle θ ∈ 0, π4 , the set D ∈ L2 such that (0, 0) it is a strong ψ-dispersion point of D and (0, 0) it is not a strong ψ-dispersion point of the set D rotated of −θ (compare [5], Example 2.12). Corollary 22.26. For any ψ ∈ Cb the strong ψ-density topology on the plane is not invariant under rotation.

References [1] V. Aversa, W. Wilczy´nski, ψ-density topology for discontinuous regular functions, Atti Sem. Mat. Fis. Univ. Modena 48(2) (2000), 473–480. [2] A. K. Banerjee, P. Das, A note on ψ-density topology, Kochi J. Math. 3 (2008), 211–216. [3] P. Das, B.K. Lahiri, Ψ -density topologies, East-West J. Math. 5 (2003), 1–18. [4] M. Filipczak, σ -ideals, topologies and multiplication, Bull. Soc. des Sci. LII, Łód´z 2002, 11–16. [5] M. Filipczak, Topologie ψ-ge¸sto´sci na płaszczy´znie, Wydawnictwo Uniwersytetu Łódzkiego, Łód´z 2004. [6] M. Filipczak, ψ-density topology is not regular, Bull. Soc. des Sci. LIV, Łód´z 2004, 21–25. [7] M. Filipczak, M. Terepeta, Consequences of Theorem of S. J. Taylor, Report on Real Analysis, Conference at Rowy (2003), 67–75. [8] M. Filipczak, M. Terepeta, ψ-continuous functions, Rend. Circ. Mat. Palermo 58 (2009), 245–255. [9] M. Filipczak, M. Terepeta, On ∆2 -condition in density-type topologies, Demonstratio Mathematica XLIV(2) (2011), 423–432. [10] G. Horbaczewska, Resolvability of ψ-density topology, Bull. Soc. Sci. Lett. LII, Łód´z 2002, 5–9. [11] C. Goffman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J. Volume 28(4) (1961), 497–505. [12] C. Goffman, D. Waterman, Approximately continuous transformations, Proceedings of the American Mathematical Society Vol. 12(1) (1961), 116–121. [13] A. Go´zdziewicz-Smejda, E. Łazarow, Comparison of ψ-sparse topologies, Jan Długosz University in Cz˛estochowa Scientific Issues, Mathematics XIV, Cz˛estochowa 2009, 21–36. [14] A. Go´zdziewicz-Smejda, E. Łazarow, Topologies generated by the ψ-sparse sets, Real Anal. Exchange Volume 36(2) (2010), 257–268. [15] J. Lukes, J. Maly, L. Zajicek, Fine Topology Methods in Real Analysis and Potential Theory, Springer-Verlag 1189, 1986.

22. On ψ-density topologies on the real line and on the plane

387

[16] E. Łazarow, A. Vizváry, ψI -density topology, Jan Długosz University in Cz˛estochowa Scientific Issues, Mathematics XV, Cz˛estochowa (2010), 67—80. [17] E. Łazarow, K. Rychert, ψI -superporosity, Real Functions, Density Topology and Related Topics, Łód´z Univ. Press 2011, 69-–76. [18] K. Ostaszewski, Continuity in the density topology, Real Anal. Exchange 7(2) (1981–82), 259–270. [19] S. Saks, Theory of the Integral, Monografie Matematyczne, Warszawa 1937. [20] The Scottish Book, http://kielich.amu.edu.pl/Stefan_Banach/ e-scottish-book.html. [21] S. J. Taylor, On strengthening of the Lebesgue Density Theorem, Fund. Math. 46(3) (1959), 305–315 . [22] S. J. Taylor, An alternative form of Egoroff’s theorem, Fund. Math. 48(2) (1959), 169–174. [23] M. Terepeta, E. Wagner-Bojakowska, Ψ -density topology, Rend. Circ. Mat. Palermo 48, Ser. II (1999), 451–476. [24] E. Wagner-Bojakowska, Remarks on ψ-density topology, Atti Sem. Mat. Fis. Univ. Modena IL (2001), 79–87. [25] E. Wagner-Bojakowska, W. Wilczy´nski, The interior operation in a ψ-density topology, Circ. Mat. Palermo 48 (1999), 5–26. [26] E. Wagner-Bojakowska, W. Wilczy´nski, Comparison of ψ-density topologies, Real Anal. Exchange 25(2) (1999/2000), 661–672 . [27] E. Wagner-Bojakowska, W. Wilczy´nski, The union of ψ-density topologies, Atti Sem. Mat. Fis. Univ. Modena 50(2) (2002), 313–326. [28] W. Wilczy´nski, Density Topologies, Chapter 15 in Hadbook of Measure Theory, edited by E. Pap, Elsevier 2002.

M AŁGORZATA F ILIPCZAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

M AŁGORZATA T EREPETA Center of Mathematics and Physics, Łód´z University of Technology al. Politechniki 11, 90-924 Łód´z, Poland E-mail: [email protected]

Chapter 23

Density type topologies generated by functions. f -density as a generalization of hsi-density and ψ-density

TOMASZ FILIPCZAK

2010 Mathematics Subject Classification: 54A10, 28A05, 26A15, 54C30 . Key words and phrases: lower density operator, density topology, f -density, comparison of topologies.

The notions of a density point and an approximately continuous function have been defined at the beginning of XX century. The density topology was defined by Haupt and Pauc in the fifties ([17]) and re-invented by Goffman and Waterman in 1961 ([16]). Let us recall the basic notions. We write A ∼ B instead of λ (A 4 B) = 0. A point x ∈ R is called a density point of a set A ∈ L if λ (A ∩ [x − h, x + h]) = 1. h→0+ 2h lim

The set of all density points of a set A ∈ L we denote by Φd (A). The operator Φd is a lower density operator and a family Td := {A ∈ L : A ⊂ Φd (A)} is a topology called the density topology. Over the last thirty years several density-type topologies has been studied by many mathematicians. All such topologies are generated by operators called lower density operators or slight different operators (studied by Hejduk in [19] and called almost density operators).

390

Tomasz Filipczak

We start by describing the properties of density-type topologies that arise from the definition of a lower density operator and an almost lower density operator in the sense of Hejduk. We recall definitions of an hsi-density topology and a ψ-density topology (described in the previous chapter). Then we define an f -density topology which is a generalization of several density-type topologies. In particular, hsi-density topologies and ψ-density topologies are f -density topologies. At first sight these topologies have quite different properties. However, they can be studied as special cases of the more general concept. For any A ⊂ R we denote by KA a measurable kernel of A, and we set −A := {−a : a ∈ A} and A + x := {a + x : a ∈ A}, x ∈ R.

23.1 Density-type topologies consisting of measurable sets We will study operators Φ : L → P (R). Consider the following properties: (D0) (D1) (D2) (D3) (D4)

if A ∈ Tnat then A ⊂ Φ (A), Φ (0) / = 0/ and Φ (R) = R, Φ (A ∩ B) = Φ (A) ∩ Φ (B), if A ∼ B then Φ (A) = Φ (B), Φ (A) ∼ A

and (D4’) λ (Φ (A) \ A) = 0 for measurable A and B. If Φ fulfils (D1)-(D4) then it is called a lower density operator. An operator fulfilling (D1)-(D3) and (D4’) has been named by Hejduk an almost lower density operator. Remark 23.1. If Φ fulfils (D2) then Φ is monotonic i.e. (D2’) if A ⊂ B then Φ (A) ⊂ Φ (B). Remark 23.2. From (D2’) and (D3) it follows that for any A, B ∈ L such that λ (A \ B) = 0 we have Φ (A) ⊂ Φ (B). We define TΦ := {A ∈ L : A ⊂ Φ (A)}. Remark 23.3. If Φ1 (A) ⊂ Φ2 (A) for A ∈ L then TΦ1 ⊂ TΦ2 . Consequently, if Φ1 = Φ2 then TΦ1 = TΦ2 .

23. f -density as a generalization of hsi-density and ψ-density

391

We will prove that if an operator Φ satisfies conditions (D0)-(D3) and (D4’) then TΦ forms a topology. We will examine properties of such operators and topologies generated by them. We will also consider operators satisfying the condition (D4), stronger than (D4’). To make the text self-contained we shortly repeat some proofs known in the literature. A lot of proofs are based on considerations for density topology (compare [23] and [29]). Some properties can be proved under the weaker assumptions (see [22] and [19]). Recall that N stands for the σ -ideal of null sets on the real line. Theorem 23.4. If Φ : L → P (R) fulfils (D0)-(D3) and (D4’) then (1) TΦ forms a topology on R, (2) intTΦ E ⊂ E ∩ Φ (KE ) for E ⊂ R, (3) if A ∈ N then A is TΦ -closed, TΦ -nowhere dense and TΦ -discrete, (4) A ∈ N if and only if A is TΦ -closed and TΦ -discrete, (5) A is a TΦ -compact set if and only if A is finite, (6) the space (R, TΦ ) is neither first countable, nor Lindelöf, nor separable, (7) cardTΦ = 2c , (8) Tnat $ TΦ , (9) A ∈ L if and only if A is a TΦ -Borel set, (10) (R, TΦ ) is a Hausdorff space, (11) (R, TΦ ) is not a normal space. Proof. (1) Of course 0/ ∈ TΦ . If A, B ∈ TΦ then by (D2), A ∩ B ⊂ Φ (A) ∩ Φ (B) = Φ (A ∩ B), which implies A ∩ B ∈ TΦ . Assume that At ∈ TΦ for S t ∈ T and A := t∈T At . Since At \ KA ∈ N , Remark 23.2 shows that Φ (At ) ⊂ Φ (KA ), and consequently KA ⊂ A =

[ t∈T

At ⊂

[

Φ (At ) ⊂ Φ (KA ) .

t∈T

From (D4’) it follows that A ∼ KA , so A ∈ L. Using (D2’) again, we conclude that A ⊂ Φ (KA ) ⊂ Φ (A), which gives A ∈ TΦ . (2) Since intTΦ E is a measurable subset of E, intTΦ E \ KE ∈ N . From Remark 23.2 we obtain intTΦ E ⊂ Φ (intTΦ E) ⊂ Φ (KE ). (3) Let A ∈ N . By (D3) and (D1), R \ A ⊂ R = Φ (R) = Φ (R \ A), hence A is TΦ -closed. The set A is nowhere dense because from (2), (D1) and (D3) we have intTΦ A ⊂ Φ (A) = 0. / If x ∈ A then (R \ A) ∪ {x} ∈ TΦ , and consequently {x} is TΦ -open in A. Thus A is TΦ -discrete. (4) Suppose that A is TΦ -closed and TΦ -discrete. For any x ∈ A there exists a TΦ -open set Ux such that A ∩Ux = {x}. Thus we have

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x ∈ Ux ⊂ Φ (Ux ) = Φ (Ux \ {x}) ⊂ Φ (R \ A) , and so A ⊂ Φ (R \ A). Therefore A ⊂ Φ (R \ A) \ (R \ A) ∈ N , by (D4’). (5) Suppose that A is infinite. Let B ⊂ A be a countable infinite subset of A. Then {(R \ B) ∪ {x}}x∈B is a TΦ -open cover of A without finite subcover. (6) If An , n ∈ N, are TΦ -open neighbourhoods of x and xn ∈ An \ {x} then the set A1 \ {xn : n ∈ N} is a TΦ -open neighbourhood of x which does not include any An . Thus (R, TΦ ) is not first countable. Let C be the Cantor ternary set. Then {(R \C) ∪ {x}}x∈C is a TΦ -open cover of A without countable subcover, and consequently (R, TΦ ) is not Lindelöf. Since any countable set is TΦ -closed, (R, TΦ ) is not separable. (7) Any subset of the Cantor set C is TΦ -closed. Hence cardTΦ ≥ cardP (C) = 2c . (8) By (D0), Tnat ⊂ TΦ . Moreover (R \ Q) ∈ TΦ \ Tnat . (9) The inclusion BorTΦ ⊂ L follows from TΦ ⊂ L. From (3) and (8) we conclude that N ⊂ BorTΦ and BorTnat ⊂ BorTΦ . Thus we have L ⊂ BorTΦ , because every measurable set is a sum of a Borel set and a null set. (10) It follows from (8). (11) (compare [22, Prop. 7.17]) Suppose, contrary to our claim, that (R, TΦ ) is a normal space. Write F := {C ∩ Φ (A) : A ∈ L}, where C is the Cantor set. For any measurable set A there is a Borel set A0 such that A ∼ A0 , so card {Φ (A) : A ∈ L} ≤ c. Therefore, one can find a set H ⊂ C such that H ∈ / F. Since H and F := C \ H are TΦ -closed, there are disjoint TΦ -open sets AH , AF containing H and F. Thus Φ (AH ) ∩ Φ (AF ) = Φ (AH ∩ AF ) = 0/ and H ⊂ C ∩ AH ⊂ C ∩ Φ (AH ) ⊂ C \ Φ (AF ) ⊂ C \ AF ⊂ C \ F = H. Hence H = C ∩ Φ (AH ), which gives a contradiction, because C ∩ Φ (AH ) ∈ F. t u Remark 23.5. The assumption (D0) has not been used in the proof of conditions (1)-(7) and (11). If Φ fulfils additionally the condition (D4), we obtain stronger results. Theorem 23.6. If Φ : L → P (R) fulfils (D0)-(D4) then (1) Φ (A) ∈ L for A ∈ L, (2) Φ (Φ (A)) = Φ (A) for A ∈ L, (3) TΦ = {Φ (A) \ N : A ∈ L, N ∈ N }, (4) intTΦ E = E ∩ Φ (KE ) for E ⊂ R, (5) intTΦ A ∼ A ∼ clTΦ A for A ∈ L,

23. f -density as a generalization of hsi-density and ψ-density

393

(6) Φ (A) = intTΦ (clTΦ A) for A ∈ L, (7) A ∈ N ⇔ A is TΦ -nowhere dense ⇔ A is TΦ -meager, (8) A ∈ L ⇔ A has TΦ -Baire property ⇔ A is a sum of a TΦ -open set and a TΦ -closed set, (9) (R, TΦ ) is a Baire space. Proof. (1)-(2) From (D4) we have Φ (A) ∼ A. Hence Φ (A) ∈ L and Φ (Φ (A)) = Φ (A), by (D3). (3) If A ∈ TΦ then A = Φ (A) \ (Φ (A) \ A), and by (D4), Φ (A) \ A ∈ N . Let A ∈ L and N ∈ N . From (D3) and (2) it follows that Φ (Φ (A) \ N) = Φ (Φ (A)) = Φ (A) ⊃ Φ (A) \ N, which gives Φ (A) \ N ∈ TΦ . (4) The inclusion intTΦ E ⊂ E ∩ Φ (KE ) follows from Theorem 23.4. Let x ∈ E ∩ Φ (KE ) and A := KE ∪ {x}. The set A ∩ Φ (A) is a TΦ -open neighbourhood of x, because Φ (A ∩ Φ (A)) = Φ (A) ∩ Φ (Φ (A)) = Φ (A) ⊃ A ∩ Φ (A) . Since A ∩ Φ (A) ⊂ A ⊂ E, we have x ∈ intTΦ E. (5) From (D4) and (4) we conclude that intTΦ A = A ∩ Φ (A) ∼ A and clTΦ A = R \ intTΦ (R \ A) ∼ R \ (R \ A) = A. (6) For any measurable set A we have A ∼ clTΦ A and clTΦ A = R\intTΦ (R \ A) ⊃ R \ Φ (R \ A) ⊃ Φ (A). Hence intTΦ (clTΦ A) = clTΦ A ∩ Φ (clTΦ A) = clTΦ A ∩ Φ (A) = Φ (A) . (7) According to Theorem 23.4, it is sufficient to prove that each TΦ -nowhere dense set is a null set. Let A be a TΦ -nowhere dense set. Then clTΦ A is also TΦ -nowhere dense. Using (6) we get Φ (A) = intTΦ (clTΦ A) = 0, / which yields A ∼ 0. / (8) If A ∈ L then A = (A ∩ Φ (A)) ∪ (A \ Φ (A)), A ∩ Φ (A) is TΦ -open and A \ Φ (A) is TΦ -closed (because it is a null set). Of course, a sum of a TΦ -open set and a TΦ -closed set has TΦ -Baire property. Finally, if A = U 4 N where U is TΦ -open and N is TΦ -meager, then U ∈ L and N ∈ N , by (7). Consequently, A is measurable. (9) If A is TΦ -open and TΦ -meager, then A ⊂ Φ (A) and A ∈ N , so A = Φ (A) = 0. / t u Remark 23.7. From (6) it follows that A is TΦ -regular open if and only if Φ (A) = A. In the following chapter there are constructed an operator Φ, satisfying (D0)-(D3) and (D4’), and closed sets of positive measure F0 , F1 such that

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Φ (F0 ) = 0/ and Φ (F1 ) is a singleton. In fact, it is proved that for any function f with lim infx→0+ f (x) x = 0 there are closed sets of positive measure such that Φ f (F0 ) = 0/ and Φ f (F1 ) = {0}, where Φ f is an f -density operator defined in the section 5 (see Theorem 24.6 and Theorem 24.7 in chapter 24). Using this result one can easy check that properties (2)-(9) from Theorem 23.6 can be false if we replace (D4) by (D4’). Note that in the following theorem items start from (2) to stress the similarity to properties of Theorem 23.6. Theorem 23.8. If Φ : L → P (R) fulfils (D0)-(D3) and (D4’) and F0 , F1 are closed sets of positive measure such that Φ(F0 ) = 0, / Φ(F1 ) = {0} then (2) Φ (Φ (F1 )) = 0/ 6= Φ (F1 ), (3) Φ (F1 ) ∈ / TΦ , (4) intTΦ F1 = 0/ 6= F1 ∩ Φ (F1 ), (5) λ (F0 \ intTΦ F0 ) = λ (F0 ) > 0 and λ (clTΦ (R \ F0 ) \ (R \ F0 )) = λ (F0 ) > 0, (6) Φ (F1 ) = {0} 6= intTΦ clTΦ F1 , (7) F0 is TΦ -nowhere dense, but is not a null set; R is TΦ -meager, but is not TΦ -nowhere dense, (8) each nonmeasurable set is TΦ -meager, and consequently has TΦ -Baire property, (9) (R, TΦ ) is not a Baire space. Proof. Conditions (2)-(6) are clear. Obviously, F0 is TΦ -nowhere dense. By S Smital’s Lemma, the set E := q∈Q (F0 + q) has a full measure (compare [20], p. 65). Thus R \ E is TΦ -nowhere dense. Since every F0 + q is TΦ -nowhere dense too, R is TΦ -meager. This implies (8)-(9). t u Now we construct an easy example of an operator Φ : L → P (R), satisfying (D0)-(D3) and (D4’), such that the set Φ (A) need not be measurable, for measurable A. Example 23.9. Suppose that Φ : L → P (R) fulfils (D0)-(D3) and (D4’), F0 is a closed set of positive measure such that Φ(F0 ) = 0/ and D is a nonmeasurable subset of F0 . Write b (A) := (Φ (A) \ D) ∪ (Φd (A) ∩ D) Φ b fulfils (D0)-(D3) and (D4’). But Φ b (F0 ) = for A ∈ L. It is easy to check that Φ Φd (F0 ) ∩ D ∈ / L.

23. f -density as a generalization of hsi-density and ψ-density

395

23.2 hsi-density Reminding the concept of ordinary density point it is worth observing that x ∈ Φ (A) if and only if   λ A ∩ x − n1 , x + 1n = 1. lim 2 n→∞

n

 1

Replacing the sequence n n∈N by a fixed sequence (sn )n∈N decreasingly tending to zero, we obtain the notion of a density generated by the sequence (sn )n∈N . Denote by Se the family of all nonincreasing and tending to zero sequences of positive numbers and fix hsi = (sn )n∈N ∈ Se and A ∈ L. If λ (A ∩ [x − sn , x + sn ]) =1 n→∞ 2sn lim

then x is called an hsi-density point of a set A. Analogously, if   λ (A ∩ [x, x + sn ]) λ (A ∩ [x − sn , x]) lim = 1 lim =1 n→∞ n→∞ sn sn then we say that x is a right-hand (left-hand) hsi-density point of A. The set of all hsi-density points (right-hand, left-hand hsi-density points) of a set A − + we denote by Φhsi (A) (Φhsi (A), Φhsi (A), respectively). Clearly, Φhsi (A) = − + Φhsi (A) ∩ Φhsi (A). We will write Thsi instead of TΦhsi , i.e.  Thsi = A ∈ L : A ⊂ Φhsi (A) . e Obviously, Td = Th 1 i and Td ⊂ Thsi for any hsi ∈ S. n The notion of an hsi-density point has been defined in [12]. Properties of an operator Φhsi and a topology Thsi have been studied also in [11], [13] and [21]. Note that the authors used nondecreasing sequences tending to ∞ instead of nonincreasing sequences tending to zero, and considered s1n instead of sn . Let us recall the basic properties of Φhsi and Thsi . Theorem 23.10 ([12]). For any sequence hsi ∈ Se (1) Φhsi satisfies (D0)-(D4), (2) Thsi is a topology, (3) Φhsi and Thsi satisfy all conditions of theorems 23.4 and 23.6.

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e Theorem 23.11 ([12]). Let hsi ∈ S. (1) If lim infn→∞ sn+1 sn > 0 then Thsi = Td . (2) If lim infn→∞ sn+1 sn = 0 then Thsi % Td . The density topology is invariant under translation and under multiplication by nonzero numbers. However, hsi-density topologies bigger than Td are not invariant under multiplication. Theorem 23.12 ([12]). Let hsi ∈ Se and m ∈ R. (1) The topology Thsi is invariant under translation. (2) If |m| ≥ 1 then Thsi is invariant under multiplication by m. (3) If Thsi % Td and |m| < 1 then Thsi is not invariant under multiplication by m. All hsi-density topologies fulfil the same separating axioms as Td .  Theorem 23.13 ([21]). For any hsi ∈ Se the space R, Thsi is completely regular but not normal.  e A set A is connected in R, Thsi if and only Theorem 23.14 ([13]). Let hsi ∈ S. if A is connected in (R, Tnat ). Despite the fact that hsi-density topologies have very similar properties, there are a lot of nonhomeomorphic hsi-density topologies. Theorem 23.15 ([13]). Suppose that Thsi 6= Td 6= Thti .  (1) The spaces (R, Td ) and R, Thsi are not homeomorphic.   (2) If Thsi * Thti and Thti * Thsi then the spaces R, Thsi and R, Thti are not homeomorphic.    (3) For any m > 1, Th 1 si $ Thsi $ Thmsi and the spaces R, Th 1 si , R, Thsi , m m  R, Thmsi are homeomorphic.

23.3 ψ-density In [25] Terepeta and Wagner-Bojakowska, based on the results of Taylor ([24]), introduced the notion of a ψ-density point. They defined a ψ-density operator and a ψ-density topology, and studied the basic properties of these notions. The other interesting results can be found in [27], [28], [1], [26] and [14]. Some of them are presented in the previous chapter of this book.

23. f -density as a generalization of hsi-density and ψ-density

397

Recall that Cb denotes the family of all continuous and nondecreasing functions ψ : (0, ∞) → (0, ∞) with limx→0+ ψ (x) = 0. We say that x is a ψ-density point of a measurable set A if λ ([x − h, x + h] \ A) = 0. h→0+ 2hψ (2h) lim

The set of all ψ-density points of A is denoted by Φψ (A). We write Tψ instead of TΦψ i.e.  Tψ = A ∈ L : A ⊂ Φψ (A) . The Second Taylor’s Theorem (see section 22.2 in the previous chapter) implies:  Theorem 23.16. For each ψ ∈ Cb there exists a set E such that λ E \ Φψ (E) is positive. Therefore, the Lebesgue Density Theorem does not hold for ψ-density, and no Φψ is a lower density operator. However, Φψ (A) ⊂ Φd (A), so  λ Φψ (A) \ A = 0 for A ∈ L. We also have: Proposition 23.17 ([25]). For any ψ ∈ Cb and A ∈ L, Φψ (A) is the set of type Fσ δ . Theorem 23.18 ([25]). For any ψ ∈ Cb (1) Φψ satisfies (D0)-(D3) and (D4’), (2) Tψ is a topology, (3) Φψ and Tψ satisfy all conditions of Theorem 23.4. b Tnat $ Tψ $ Td and the family of Tψ -connected Moreover, for any ψ ∈ C, sets is the same as in (R, Tnat ) (compare [25]). Since Φψ does not satisfy (D4), there are several differences between the density topology and ψ-density topologies. For example, the space R, Tψ is not regular ([2]) and it is not a Baire space ([26]). More properties of ψ-density topologies are described in section 22.3 in the previous chapter.

23.4 The definition and basic properties of f -density By A we denote the family of all functions f : (0, ∞) → (0, ∞) such that

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(A1) limx→0+ f (x) = 0, (A2) lim infx→0+ f (x) x < ∞, (A3) f is nondecreasing. Let f ∈ A, A ∈ L and x ∈ R. We say that x is a right-hand (left-hand) f density point of A if   λ ([x − h, x] \ A) λ ([x, x + h] \ A) =0 lim =0 . lim h→0+ h→0+ f (h) f (h) − By Φ + f (A) (Φ f (A)) we denote the set of all right-hand (left-hand) f -density − points of A. If x ∈ Φ f (A) := Φ + f (A)∩Φ f (A) then we say that x is an f -density point of A.

Remark 23.19. Condition (A2) is essential because limh→0+ λ ([x,x+h]\A) h ≤ f (h) → 0. f (h)

f (h) h

= ∞ implies

h→0+

The definition of f -density was introduced in [3]. In this paper continuity of functions from the family A was assumed. In subsequent papers this condition was omitted (compare Theorem 23.34). Considering functions from the family A, we often define f (x) only for x ∈ (0, δ ), for some δ > 0. To distinguish the notion of f -density from ψ-density (considered in the previous section) we will use Latin letters ( f , g) defining density generated by function and Greek letters (ψ) in the second case. Straightforward from the properties of Lebesgue measure it follows: Proposition 23.20. Φ f (A + x) = Φ f (A) + x for f ∈ A, A ∈ L and x ∈ R. Since functions from the family A are monotonic, we can describe an f density point in equivalent way. Proposition 23.21 ([3]). Let f ∈ A, A ∈ L and x ∈ R. Then x is an f -density point of A if and only if lim

h→0,k→0 h≥0,k≥0,h+k>0

λ ([x − h, x + k] \ A) = 0. f (h + k)

Proof. Sufficiency is evident. Since λ ([x − h, x + k] \ A) λ ([x − h, x] \ A) λ ([x, x + k] \ A) ≤ + f (h + k) f (h) f (k) for h, k > 0, we obtain necessity.

t u

23. f -density as a generalization of hsi-density and ψ-density

399

− Theorem 23.22 ([3]). If f ∈ A and A ∈ L then Φ f (A), Φ + f (A) and Φ f (A) are the sets of type Fσ δ .

Proof. For any h > 0 the function Fh (x) = orem follows from the equality Φ+ f (A)

=

\ [

λ ([x−h,x]\A) f (h)

\

n∈N δ ∈Q+ h∈(0,δ )

Fh−1

is continuous. Thus the-

  1 0, . n t u

The following theorem states that an f -density point of a set A can not be a dispersion point of A. Thus an operator Φ f satisfies the condition (D4’). Theorem 23.23 ([3]). Let f ∈ A, A ∈ L and x ∈ R. λ ([x,x+h]\A) = 0. (1) If x ∈ Φ + f (A) then lim infh→0+ h (2) Φ f (A) ⊂ R \ Φd (R \ A).

> 0. From (A2) it follows that Proof. Suppose that lim infh→0+ λ ([x,x+h]\A) h lim sup h→0+

λ ([x, x + h] \ A) λ ([x, x + h] \ A) h ≥ lim inf · lim sup >0 h→0+ f (h) h h→0+ f (h)

which gives x ∈ / Φ+ f (A). The second condition is a consequence of the first. t u Theorem 23.24 ([3]). Let f ∈ A. The operator Φ f fulfils (D0)-(D3) and (D4’) i.e. for any A, B ∈ L we have: (0) if A ∈ Tnat then A ⊂ Φ f (A), (1) Φ f (0) / = 0/ and Φ f (R) = R, (2) Φ f (A ∩ B) = Φ f (A) ∩ Φ f (B), (3) if A ∼ B then Φ f (A) = Φ f (B), (4’) λ (Φ f (A) \ A) = 0. Proof. Conditions (0), (3), Φ f (R) = R and Φ f (A ∩ B) ⊂ Φ f (A) ∩ Φ f (B) are obvious. The equality Φ f (0) / = 0/ follows from (A2). Since λ (I \ (A ∩ B)) ≤ λ (I \ A) + λ (I \ B) for every interval I, we have Φ f (A) ∩ Φ f (B) ⊂ Φ f (A ∩ B). From Theorem 23.23 we conclude that Φ f (A) \ A ⊂ (R \ A) \ Φd (R \ A) .

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Thus the Lebesgue Density Theorem implies (4’).

t u

The family TΦ f will be denoted by T f , i.e.  T f = A ∈ L : A ⊂ Φ f (A) . According to Theorem 23.24, Φ f and T f fulfil all conditions of Theorem 23.4: Theorem 23.25. For any f ∈ A (1) T f forms a topology on R, (2) intT f E ⊂ E ∩ Φ f (KE ) for E ⊂ R, (3) if A ∈ N then A is T f -closed, T f -nowhere dense and T f -discrete, (4) A ∈ N if and only if A is T f -closed and T f -discrete, (5) A is a T f -compact set if and only if A is finite, (6) the space (R, T f ) is neither first countable, nor Lindelöf, nor separable, (7) cardT f = 2c , (8) Tnat $ T f , (9) A ∈ L if and only if A is a T f -Borel set, (10) (R, T f ) is a Hausdorff space, (11) (R, T f ) is not a normal space. In general, there are two possibilities to define density points. We can do it "in a symmetric way", examining the set on intervals centered at x, or "in onesided way" - on intervals [x − h, x] and [x, x + h]. In this sense, the definition of f -density is "one-sided". However, "symmetric" definition leads us to the same notion (using, if necessary, another function). Let f ∈ A, A ∈ L and x ∈ R. We say that x is a symmetric f -density point of A if λ ([x − h, x + h] \ A) = 0. lim h→0+ f (2h) By Φ sf (A) we denote the set of all symmetric f -density points of A. For any function f defined on (0, ∞) we set f ∗ (x) := f (2x). An easy verification shows that: Proposition 23.26 ([15]). (1) f ∈ A if and only if f ∗ ∈ A, (2) Φ f ∗ = Φ sf for f ∈ A, o  n (3) Φ f : f ∈ A = Φ sf : f ∈ A . Of course, the equality Φ f = Φ sf does not have to happen.

23. f -density as a generalization of hsi-density and ψ-density

Example 23.27. Let f (x) :=

1 n!

for x ∈



1 1 (n+1)! , n!



 1 1 , . (n + 1)! 2 · n!

∞ [

A := (−∞, 0] ∪

n=2

i

401

and

 i 1 1 1 Clearly, f ∈ A. It is easy to check that λ ([−h,h]\A) ≤ for h ∈ , n f (2h) (n+1)! n! , and 1 λ ([0, ]\A) consequently, 0 ∈ Φ sf (A). But f n!1 > 12 , which gives 0 ∈ / Φ f (A). ( n! ) It should be mentioned that the notion of f -density could be defined even more generally. In [18] Hejduk considered f -density points (and symmetric f -density points) assuming only condition (A2). Theorem 23.28 ([18]). Let f : (0, ∞) → (0, ∞) be a function satisfying lim inf x→0+

f (x) < ∞. x

Then (1) Φ f and Φ sf satisfy conditions (D0)-(D3) and (D4’), (2) T f and T fs form topologies such that Tnat ⊂ T fs ⊂ T f .

23.5 The comparison of f -density topologies Let f , g ∈ A. We ask for conditions under which the inclusion T f ⊂ Tg holds. The necessary and sufficient condition is presented in the following chapter (Theorem 24.2). Now, we show an easy sufficient condition. We also formulate a necessary and sufficient condition to compare T f with Td . Using it, we divide the family of all f -density topologies into two parts. The first consists of topologies similar to hsi-density topologies and the second - similar to ψdensity topologies. Obviously, different functions can generate the same operator (for example Φ f = Φ2 f ). Fortunately, different operators generate different topologies. Moreover, to prove the inclusion T f ⊂ Tg , it is enough to show that the condi+ tion 0 ∈ Φ + f (A) implies 0 ∈ Φg (A). Theorem 23.29 ([5], [9]). For each f , g ∈ A the following conditions are equivalent   + (A) , (1) ∀A∈L 0 ∈ Φ + (A) ⇒ 0 ∈ Φ g f

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(2) ∀A∈L (0 ∈ Φ f (A) ⇒ 0 ∈ Φg (A)), (3) ∀A∈L Φ f (A) ⊂ Φg (A), (4) T f ⊂ Tg . + Proof. Implication (1)⇒(2) follows from 0 ∈ Φ − f (A) ⇔ 0 ∈ Φ f (−A). By x ∈ Φ f (A) ⇔ 0 ∈ Φ f (A − x), we obtain (2)⇒(3). Implication (3)⇒(1) is clear. Assume now that T f ⊂ Tg , 0 ∈ Φ + / Φg+ (A). We can find ε > 0 and f (A) but 0 ∈ a decreasing sequence (xn ) tending to zero such that

λ ([0, xn ] \ A) ≥ε g (xn ) for every n. Defining an = xn − λ ((xn+1 , xn ) \ A) and B := (−∞, 0] ∪

∞ [

(xn+1 , an )

n=1

we obtain that 0 ∈ / Φg+ (B). On the other hand 0 ∈ Φ + f (B), because λ ([0, x] \ B) ≤ λ ([0, x] \ A) for any x > 0. Consequently, B ∈ T f \ Tg , which gives a contradiction.

t u

According to the latter theorem, we will usually formulate conditions for topologies and operators, and will prove only the condition 23.29 from Theorem 23.29. f (x) Let f , g ∈ A. We say that f precedes g if lim supx→0+ g(x) < ∞. We denote f (x) = ∞ we write f ⊀ g (compare [25]). it by f ≺ g. If lim supx→0+ g(x)

Theorem 23.30 ([3]). Let f , g ∈ A. If f ≺ g then T f ⊂ Tg . Proof. Suppose that 0 ∈ Φ + f (A). Then lim sup h→0+

λ ([0, h] \ A) λ ([0, h] \ A) f (h) f (h) ≤ lim sup ·lim sup = 0·lim sup = 0, g (h) f (h) h→0+ h→0+ g (h) h→0+ g (h)

which means that 0 ∈ Φg+ (A).

t u

Example 23.31 ([5]). f , g ∈ A such that f ⊀ g andiT f ⊂ Tg . Let h There exist  1 1 1 1 ; n!1 . Then f ≥ g f (x) := n! for x ∈ (n+1)! ; n! and g(x) := n!1 for x ∈ (n+1)! f( 1 ) and f ⊀ g because g n!1 = n. Suppose that 0 ∈ Φ + f (A). Since f (x) = g (x) for ( n! )

23. f -density as a generalization of hsi-density and ψ-density

x 6=

1 n! ,

403

to prove 0 ∈ Φg+ (A) it is enough to show that limn→∞

λ ([0, n!1 ]\A)

For n ≥ 2 we have 0≤

λ

g( n!1 )

= 0.

    1   λ 0, 1  \ A + 1 1 λ 0, 1+n! \A 0, n! \ A 1+n! 1 (n!)2    ≤ = + −→ 0, 1 1 1 n! n→∞ g n! f 1+n! f 1+n!

which completes the proof. Example 23.31 shows that Theorem 23.30 can not be reversed. Fortunately, if one of considered topologies is Td , the opposite implication also holds. Theorem 23.32 ([5], [10]). For any f ∈ A we have (1) T f ⊂ Td if and only if lim supx→0+ f (x) x < ∞. f (x) (2) Td ⊂ T f if and only if lim infx→0+ x > 0. Proof. By Theorem 23.30, inequalities under consideration are sufficient conditions for inclusions of topologies. Suppose that lim supx→0+ f (x) x = ∞. Let S A := ∞ (2x , x ), where (x ) is a decreasing sequence of positive numn+1 n n n=1 λ (A∩[0,2xn ]) f (xn ) + 1 < 2, 0 ∈ / Φd (A). But bers such that 2xn+1 < xn < n . Since 2xn 0 ∈ Φ+ f (A), because for h ∈ (xn+1 , xn ] we have pose now that lim infx→0+ f (x) x xn+1 < f (xn ) < n1 xn . Then

= 0. Let A :=

λ ([0,h]\A) n+1 ≤ f2x f (h) (xn+1 ) S∞ n=1 (xn+1 , xn − f


> 1− , xn xn n and we easily obtain 0 ∈ Φd+ (A). But 0 ∈ / Φ+ f (A) since

λ ([0,xn ]\A) f (xn )

≥ 1.

t u

Corollary 23.33. Let f ∈ A. (1) T f (2) T f (3) Td (4) T f

f (x) = Td if and only if 0 < lim infx→0+ f (x) x ≤ lim supx→0+ x < ∞. f (x) $ Td if and only if 0 = lim infx→0+ f (x) x ≤ lim supx→0+ x < ∞. f (x) $ T f if and only if 0 < lim infx→0+ f (x) x < lim supx→0+ x = ∞. f (x) * Td and Td * T f if and only if 0 = lim inf f (x) x < lim supx→0+ x = ∞.

Thus the family A splits into four subfamilies. However, it turns out that properties of the topology T f depend mainly on whether id ≺ f . Therefore we define two subfamilies of A as follows:

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Tomasz Filipczak



 f (x) A := { f ∈ A : id ≺ f } = f ∈ A : lim inf >0 , x→0+ x   f (x) =0 . A0 := { f ∈ A : id ⊀ f } = f ∈ A : lim inf x→0+ x 1

23.6 f -density and hsi-density In this section we will describe a relation between hsi-density and f -density for f ∈ A1 . First of all, we will show that any operator Φ f generated by f ∈ A is generated by some continuous function f1 ∈ A and by some function f2 ∈ A constant on intervals. By Ac we denote the family of continuous functions from A, and by As the family of functions from A for which there exist decreasing and tending to 0 sequences (xn ) and (an ) such that f (x) = an for x ∈ (xn+1 , xn ].    Theorem 23.34 ([5], [6]). Φ f : f ∈ A = Φ f : f ∈ Ac = Φ f : f ∈ As .  a , 2an Proof. Let f ∈ A and a := f (1). As f is nondecreasing, the set f −1 2n+1 is either empty or is an interval (may be degenerated). Let (kn ) be an increasing sequence of all numbers n for which In := f −1 2kna+1 , 2akn is a nondegenerated interval, and let xn be the right endpoint of In . Evidently, xn+1 is the left endpoint of In . Let us denote g(x) :=

a for x ∈ (xn+1 , xn ] . 2kn

Obviously, g ∈ As . It is not difficult to prove that Φ f = Φg (see [5], Th. 1). Assume now that f ∈ As , i.e. f (x) = an for x ∈ (xn+1 , xn ], where (xn ) and (an ) are decreasing sequences tending to 0 and such that lim inf axnn < ∞. Put n→∞  δn := min xn −x2 n+1 , an+1 and n  an for x ∈ [xn+1 + δn , xn ] , g(x) := linear on [xn+1 , xn+1 + δn ] . Of course, g ∈ Ac . One can check that Φ f = Φg (see [5, Th. 1]).

t u

Now we are in the position to prove that hsi-density operators and hsidensity topologies are specific cases of f -density operators and f -density topologies. Let  s for x ∈ (sn+1 , sn ] , fhsi (x) := n s1 for x > s1

23. f -density as a generalization of hsi-density and ψ-density

405

e Obviously, fhsi ∈ A and fhsi (x) ≥ x for every x. for hsi ∈ S. Theorem 23.35 ([5]). (1) If hsi ∈ Se then fhsi ∈ As and Φhsi = Φ fhsi . n o  (2) Φhsi : hsi ∈ Se ⊂ Φ f : f ∈ A1 . Proof. The condition fhsi ∈ As is obvious. To prove that Φhsi = Φ fhsi it is sufficient to show that for any measurable A + 0 ∈ Φhsi (A) ⇔ 0 ∈ Φ + fhsi (A) .

The first implication follows from the inequality λ ([0,x]\A) ≤ λ ([0,ssnn ]\A) for f (x) x ∈ (sn+1 , sn ], and the second from the equality f (sn ) = sn . The inclusion (2) follows immediately from the definition of fhsi . t u Proposition 23.36 ([8]). For any function f ∈ A, there is a sequence hsi ∈ Se such that T f ⊂ Thsi . Proof. By (A2), there exist a sequence hsi ∈ Se and a positive number M such that f (ssnn ) < M for every n. It is easily seen that T f ⊂ Thsi . t u Using this inclusion we can characterize the family of T f -connected sets. Theorem 23.37 ([8]). For any f ∈ A, the family of T f -connected sets is equal to the family of sets connected in the natural topology. e Proof. The latter proposition shows that Tnat ⊂ T f ⊂ Thsi for some hsi ∈ S. Thus our claim follows from Theorem 23.14. t u Recall that any Φhsi is a lower density operator. The same is true for Φ f with f ∈ A1 . Theorem 23.38 ([4]). If f ∈ A1 then Φ f satisfies (D4). Proof. By Theorem 23.32, Td ⊂ T f . Hence Φd (A) ⊂ Φ f (A) for A ∈ L, and consequently, A \ Φ f (A) ⊂ A \ Φd (A), which gives λ (A \ Φ f (A)) = 0. This completes the proof, because by Theorem 23.24, Φ f fulfils (D4’). t u By Theorem 23.22, Φ f (A) ∈ L for each f ∈ A and A ∈ L. From Theorem 23.6 we obtain:

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Tomasz Filipczak

Theorem 23.39. If f ∈ A1 then (1) Φ f (Φf (A)) = Φ f (A) for A ∈ L, (2) T f = Φ f (A) \ N : A ∈ L, N ∈ N , (3) intT f E = E ∩ Φ f (KE ) for E ⊂ R, (4) intT f A ∼ A ∼ clT f A for A ∈ L, (5) Φ f (A) = intT f clT f A for A ∈ L, (6) A ∈ N ⇔ A is T f -nowhere dense ⇔ A is T f -meager, (7) A ∈ L ⇔ A has T f -Baire property ⇔ A is a sum of a T f -open set and a T f -closed set, (8) (R, T f ) is a Baire space. Theorem 23.40 ([3]). If f ∈ A and α ≥ 1 then T f is invariant under multiplication by α. Proof. From

λ ([0,h]\αA) f (h)

=

αλ ([0, αh ]\A) f (α αh )

≤α

λ ([0, αh ]\A) f ( αh )

it follows that 0 ∈ Φ f (A)

implies 0 ∈ Φ f (αA). Let A ∈ T f and x ∈ αA. Then  x x ⊂ Φ f (αA − x) 0 ∈ A− ⊂ Φf A− α α and consequently, 0 ∈ intT f (αA − x) = intT f (αA) − x, which ends the proof. t u Theorems formulated up to now show that topologies T f generated by f ∈ A1 have properties similar to the properties of topologies generated by sequences. Any hsi-density topology is completely regular. In the next chapter it will be proved that f -density topologies are completely regular for f ∈ A1 (Theorem 24.15). There is a natural question if there exists an f -density topology, generated by a function from A1 , which is not generated by a sequence. The positive answer was obtained in [5]. However, the example presented in this paper is quite complicated and the proof of its correctness is rather laborious. The much simpler example is presented in [10]. It is an example of a topology T f which is bigger than the density topology and invariant under multiplication by nonzero numbers. By Theorem 23.12, such topology can not be generated by a sequence. The proof is based on the (∆2 ) condition which is presented in the next chapter (compare Example 24.24). Now we present the example from the paper [5], omitting technical details. Using this example, it can be shown that the family of topologies generated by functions from A1 is "much bigger" than the family of topologies generated by sequences.

23. f -density as a generalization of hsi-density and ψ-density

407

Example 23.41 ([5], Theorem 6). Let us define sequences hwi := (2, 2, 3, 3, 3, 4, 4, 4, 4, . . .) , hri := (1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) , an−1 an−1 . a0 := 1, an := 2 and bn := an wn = wn wn Of course, limn→∞ abn−1 = limn→∞ abnn = ∞. The function n  fb(x) :=

an−1 for x ∈ (bn , an−1 ] , bn rn for x ∈ (an , bn ]

e belongs to A1 , and T fb 6= Thsi for each hsi ∈ S. Slightly modifying the function fb one can n o obtain continuum different topolo 1 e gies from T f : f ∈ A \ Thsi : hsi ∈ S . Example 23.42 ([7], Cor. 7 and Th. 8). Let fb be a functionndefined in example o  / Thsi : hsi ∈ Se and 23.41 and fbα (x) := fb αx for α > 1. Then fbα ∈ A1 , T fbα ∈ T fb & T fbα for β > α > 1. β

The result from Example 23.41 can be also strengthen in a different way. e If Thbi & Thai then there exists Theorem 23.43 ([7, Th. 11]). Let hai , hbi ∈ S. e f ∈ A1 such that Thbi ⊂ T f ⊂ Thai and T f 6= Thsi for hsi ∈ S.

23.7 f -density and ψ-density By Theorem 23.34, it is clear that a ψ-density topology is a specific case b we have Tψ = T s , where of an f -density topology. Namely, for ψ ∈ C, fψ fψ (x) := xψ (x). Therefore: Proposition 23.44.  n o (1)  (2)  f (x) b = 0 ⊂ T f : f ∈ A0 . Tψ : ψ ∈ C ⊂ T f : f ∈ A ∧ lim x→0+ x Note that both above inclusions are proper. From Theorem 23.32, the equal0 ity limx→0+ f (x) x = 0 implies T f $ Td . On the other hand, there is f ∈ A such that topologies T f and Td are not comparable. Thus the inclusion (2) is proper. A question concerning (1) is more interesting and more difficult. In fact, it is

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the question whether one can replace monotonicity of ψ by monotonicity of xψ (x), in the definition of ψ-density point. The negative answer was obtained in [15]. The proof is based on the condition which is presented in the next chapter (Theorem 24.2). Theorem 23.45 ([15], Theorem 1.3).  n o  f (x) b =0 . Tψ : ψ ∈ C $ T f : f ∈ Ac ∧ lim x→0+ x  The family T f : f ∈ A0 is much bigger than the family of all ψ-density topologies and it contains topologies incomparable with the density topology. Nevertheless, the properties of f -density topologies generated by functions from A0 are very similar to the properties of ψ-density topologies. These topics are more precisely desribed in the next chapter.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

V. Aversa, W. Wilczy´nski, Ψ -Density Topology for Discontinuous Regulator Functions, Atti Sem. Mat. Fis. Univ. Modena 48 (2000), 473-479. M. Filipczak, ψ-density topology is not regular, Bull. Soc. Sci. Lett. Łód´z Sr. Rech. Déform. 43 (2004), 21-25. M. Filipczak, T. Filipczak, A generalization of the density topology, Tatra Mt. Math. Publ. 34 (2006), 37-47. M. Filipczak, T. Filipczak, Remarks on f -density and ψ-density, Tatra Mt. Math. Publ., 34 (2006), 141-149. M. Filipczak, T. Filipczak, Density topologies generated by functions and by sequences, Tatra Mt. Math. Publ. 40 (2008), 103-115. M. Filipczak, T. Filipczak, On f -density topologies, Topology Appl. 155 (2008), 1980-1989. M. Filipczak, T. Filipczak, On the comparison of the density type topologies generated by sequences and by functions, Comment. Math. Vol. 49, No. 2 (2009), 161-170. M. Filipczak, T. Filipczak, On homeomorphisms of density type topologies generated by functions, Tatra Mt. Math. Publ. 46 (2010), 7-13. M. Filipczak, T. Filipczak, On the comparison of density type topologies genarated by functions, Real Anal. Exchange 36 (2011), 341-352. M. Filipczak, T. Filipczak, On ∆2 condition for density-type topologies generated by functions, Topology Appl. 159 (2012), 1838-1846. M. Filipczak, T. Filipczak, J. Hejduk, On the comparison of the density type topologies, Atti Sem. Mat. Fis. Univ. Modena 52 (2004), 37-46. M. Filipczak, J. Hejduk, On topologies associated with the Lebesgue measure, Tatra Mt. Math. Publ. 28 (2004), 187-197. M. Filipczak, J. Hejduk, W. Wilczy´nski, On homeomorphisms of the density type topologies, Annales Societatis Mathematicae Polonae, XLV (2) (2005), 151-159.

23. f -density as a generalization of hsi-density and ψ-density

409

[14] M. Filipczak, M. Terepeta, ψ-continuous functions and functions preserving ψdensity points, Tatra Mt. Math. Publ. 42 (2009), 175-186. [15] T. Filipczak, The comparison of f -density and ψ-density, in: Real Functions, Density Topology and Related Topics, Łód´z University Press, 2011, 23-28. [16] C. Goffman, D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116-121. [17] O. Haupt, C. Pauc, La topologie approximative de Denjoy envisagée comme vraie topologie, C. R. Acad. Sci. Paris 234 (1952), 390–392. [18] J. Hejduk, On the density topologies generated by functions, Tatra Mt. Math. Publ. 40 (2008), 133-141. [19] J. Hejduk, On the abstract density topologies, Selected Papers of the 2010 International Conference on Topology and its Applications, 2012, 79-85. [20] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN, 1985. [21] A. Loranty, Separation axioms of the density type topologies, Reports on Real Analysis Conference - Rowy 2003, Łód´z, 2004, 119–128. [22] J. Lukeš, J. Malý, L. Zajíˇcek, Fine topology methods in Real Analysis and Potential Theory, Springer-Verlag, 1986. [23] J. C. Oxtoby, Measure and Category, Springer-Verlag, 1980. [24] S. J. Taylor, On strengthening the Lebesgue Density Theorem, Fund. Math., 46 (1959), 305-315. [25] M. Terepeta, E. Wagner-Bojakowska, ψ-density topology, Rend. Circ. Mat. Palermo, 48 (1999), 451-476. [26] E. Wagner-Bojakowska, Remarks on ψ-density topology, Atti Sem. Mat. Fis. Univ. Modena, IL, (2001), 79-87. [27] E. Wagner-Bojakowska, W. Wilczy´nski, Comparison of ψ-density topologies, Real Analysis Exchange 25 (1999-2000), 661-672. [28] E. Wagner-Bojakowska, W. Wilczy´nski, The interior operation in a ψ-density topology, Rend. Circ. Mat. Palermo, 49 (2000), 5-26. [29] W. Wilczy´nski, Density topologies, in: Handbook of Measure Theory, Elsevier Science B. V., 2002, 675-702.

T OMASZ F ILIPCZAK Institute of Mathematics, Łód´z Technical University ul. Wólcza´nska 215, 93-005 Łód´z, Poland E-mail: [email protected]



Chapter 24

Density type topologies generated by functions. Properties of f -density

MAŁGORZATA FILIPCZAK, TOMASZ FILIPCZAK

2010 Mathematics Subject Classification: 54A10, 28A05, 26A15, 54C30 . Key words and phrases: lower density operator, density topology, f -density, comparison of topologies.

Notions of f -density operators, f -density topologies and their basic properties were described in the previous chapter. Recall that by A we denote the family of all nondecreasing functions f : (0, ∞) → (0, ∞) with limx→0+ f (x) = 0 and lim infx→0+ f (x) x < ∞. We say that x ∈ R is a right-hand f -density point of a measurable set A for a fixed f ∈ A if lim

h→0+

λ ([x, x + h] \ A) = 0. f (h)

By Φ + f (A) we denote the set of all right-hand f -density points of A, and in analogous way we define a left-hand f -density point and the set Φ − f (A). Fi− + nally, if x ∈ Φ f (A) := Φ f (A) ∩ Φ f (A) then we say that x is an f -density point  of A. The family T f = A ∈ L : A ⊂ Φ f (A) forms a topology called f -density topology. In chapter 23 f -density is treated mainly as a generalization of hsi-density and ψ-density. Now we will focus our attention on the more advanced properties, which are generally more difficult to prove. All presented results are known but proofs contained in this chapter are considerably shortened and simplified.

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Małgorzata Filipczak, Tomasz Filipczak

24.1 Comparison of f -density topologies A simple sufficient condition for the inclusion T f1 ⊂ T f2 is presented in Theorem 23.30. There is also formulated a necessary and sufficient condition to distinguish T f from Td . Theorem 23.32 says that Td ⊂ T f (T f ⊂ Td ) if and f (x) only if lim infx→0+ f (x) x > 0 (lim supx→0+ x < ∞). Consequently, we divide the family A into two subfamilies:     f (x) f (x) A1 := f ∈ A : lim inf > 0 and A0 := f ∈ A : lim inf =0 . x→0+ x→0+ x x Topologies T f generated by functions from the family A1 are bigger then the density topology, and any hsi-density topology is an f -density topology generated by some f ∈ A1 (compare Theorem 23.34). Topologies generated by f ∈ A0 are smaller then Td or incomparable with Td . Any ψ-density topology is an f -density topology for some f ∈ A0 (compare Proposition 23.44 and [5]). Now we will formulate the necessary and sufficient condition for the inclusion T f1 ⊂ T f2 . The analogous condition for ψ-density topology was formulated in [16]. However, the proof for f -density is much shorter and simpler than that for ψ-density. In further considerations we will use the observation that to prove T f1 ⊂ T f2 it suffices to show that, for any measurable set A, 0 ∈ Φ + f1 (A) implies 0 ∈ + Φ f2 (A) (see Theorem 23.29). We will also need the following Lemma (compare [6]). Lemma 24.1. Let f ∈ A, t, h ∈ (0, ∞) and A be a measurable set satisfying > t. There is an interval [a, b] ⊂ (0, h) such that lim supx→0+ λ ([0,x]∩A) f (x) λ ([a, x] ∩ A) λ ([a, b] ∩ A) ≥ t and ≤ t for x ∈ (a, b] . f (b) f (x) Proof. Since lim supx→0+ λ ([0,x]∩A) > t, there is a number y ∈ (0, h) such f (x) λ ([0,y]∩A) > t. From the continuity f (y) λ ([a,y]∩A) = t for some a ∈ (0, y). Let f (y)

that

of Lebesgue measure, it follows that

  λ ([a, x] ∩ A) b := inf x ∈ [a, y] : ≥t . f (x)

24. Density type topologies generated by functions. Properties of f -density

413

λ ([a,b]∩A) ≥ t. f (b) λ ([a,b0 ]∩A) < t, and f (b)

Obviously, a < b ≤ y. To finish the proof it remains to check that Suppose that consequently

λ ([a,b]∩A) f (b)

< t. Thus there is b0 > b such that

λ ([a, x] ∩ A) λ ([a, b0 ] ∩ A) ≤ 0. We look for a measurable set E + such that 0 ∈ Φ + f2 (E) \ Φ f1 (E). There is a positive number t such that lim supx→0+ λ (Afn1∩[0,x]) > t for sufficiently large n. We can assume that this (x) inequality holds for every n.Applying Lemma 24.1, we can define intervals [an , bn ] such that bn+1 < min an , 1n f2 (an ) , λ ([an , bn ] ∩ An ) λ ([an , x] ∩ An ) ≥ t and ≤ t for x ∈ (an , bn ] . f1 (bn ) f1 (x) Set E := R \

∞ [

(An ∩ [an , bn ]) .

n=1 n ]\E) n) Since λ ([0,b ≥ λ ([afn1,b(bnn]∩A ≥ t > 0, 0 ∈ / Φ f1 (E). f1 (bn ) ) Let us consider x ∈ (0, b1 ]. We first assume that x ∈ (an , bn ] for some n. If λ ([an , x] ∩ An ) = 0 then

λ ([0, x] \ E) bn+1 1 ≤ < . f2 (x) f2 (an ) n If λ ([an , x] ∩ An ) > 0 then one can find y, z ∈ [an , x] such that z < y ≤ x, λ (An ∩ [y, x]) = 0, z ∈ An and y − z < f1 (an ). Thus λ ([0, x] \ E) λ ([an , y] ∩ An ) + bn+1 λ ([an , z] ∩ An ) y−z bn+1 ≤ ≤ + + < f2 (x) f2 (z) f2 (z) n f1 (z) f2 (an ) t 1 1 t +2 . < + + = n n n n Assume now that x ∈ (bn+1 , an ]. From the previous case we obtain t +2 λ ([0, x] \ E) λ ([0, bn+1 ] \ E) ≤ < . f2 (x) f2 (bn+1 ) n+1 This gives 0 ∈ Φ f2 (E), which completes the proof.

t u

As a straightforward consequence we obtain: Theorem 24.3 ([6]). Let f1 , f2 ∈ A. If limx→0+

f1 (x) f2 (x)

= 0 then T f1 $ T f2 .

Proof. There are δ > 0 and n0 ∈ N such that An f1 f2 ⊂ (0, δ ) and An f2 f1 ∩ (0, δ ) = 0/ for n ≥ n0 . Clearly, limn→∞ εn f2 f1 = 0, so T f1 ⊂ T f2 . Since An f1 f2 ∩ (0, x) = (0, x) for x ∈ (0, δ ) and n ≥ n0 , we have εn f1 f2 = lim supx→0+ f1x(x) . By the definition of the family A, we have lim supx→0+ f1x(x) > 0, and consequently limn→∞ εn f1 f2 > 0. Therefore T f1 6= T f2 . t u

24. Density type topologies generated by functions. Properties of f -density

Theorem 23.32 shows that the condition limx→0+ for T f1 $ T f2 .

f1 (x) f2 (x)

415

= 0 is not necessary

24.2 Properties of f -density for f ∈ A0 In Theorem 23.38 it is proved that λ (Φ f (A) 4 A) = 0 for f ∈ A1 and A ∈ L. Thus for any f from A1 an operator Φ f and a topology T f have properties similar to the properties of "classical" density operator Φd and the density topology Td (compare Theorem 23.39). Now we will study properties of Φ f and T f for f ∈ A0 . The essential role in these considerations is played by the result analogous to The Second Taylor’s Theorem (compare [14] and chapter 22) for f -density. We begin by defining a Cantor-type set generating by two sequences. In nth step of the construction of the Cantor ternary set, two subintervals of the length 31n are chosen from any component. In our construction, we will choose kn subintervals of the length rn each. Let (rn )n=0,1,... be a sequence of positive numbers and (kn )n=0,1,... be a sequence of positive integers such that k0 = r0 = 1, kn ≥ 2 and kn rn < rn−1 for n ≥ 1. We define inductively a decreasing sequence (Fn )n=0,1... of closed sets consisting of pn := k0 · . . . · kn pairwise disjoint closed intervals Iin of the length rn . For n = 0 we put F0 := I10 := [0, 1]. Suppose that we have defined disjoint closed intervals I1n , . . . , Ipnn and the set Fn := I1n ∪ . . . ∪ Ipnn for some n ≥ 0. For any i ∈ {1, . . . , pn } we define kn+1 pairwise disjoint closed subintervals n+1 I(i−1)k , . . . , Iikn+1 of the interval Iin , of the length rn+1 each. We choose them n+1 n+1 +1 n+1 in such a way that the left endpoint of I(i−1)k is the left enpoint of Iin , the n+1 +1

right endpoint of Iikn+1 is the right endpoint of Iin and distances between subinn+1 tervals are the same. Let Fn+1 := I1n+1 ∪ . . . ∪ Ipn+1 . Thus we have defined the n+1 sequence (Fn )n=0,1... . Put F := F ((rn ) , (kn )) :=

∞ \

Fn .

n=0

Remark 24.4. From now on we will assume that F0 = I10 := [0, 1] and will define sequences (rn ) and (kn ) for n ≥ 1. Lemma 24.5. For any ε ∈ (0, 1) and any tending to zero sequence (xn ) of positive numbers there exists a subsequence (rn ) of the sequence (xn ) and a se-

416

Małgorzata Filipczak, Tomasz Filipczak

quence (kn ) of positive integers such that the set F := F ((rn ) , (kn )) satisfies λ (F) > 1 − ε and ε ε < λ (Iin \ Fn+1 ) < for n = 0, 1, . . . , i = 1, . . . , pn . pn 2n+2 pn 2n+1

(24.2)

Proof. Fix a natural number m and suppose that we have defined r j = xt j and k j for j = 1, . . . , m. As rm+1 we choose any element xt from the sequence (xn ) such that t > tm and xt < pm 2εm+2 . Now we put 

ε km+1 := max k ∈ N : rm − krm+1 > pm 2m+2

 .

From the definition of km+1 it follows that ε ε ε < rm − km+1 rm+1 ≤ + rm+1 < . pm 2m+2 pm 2m+2 pm 2m+1 Since λ (Iim \ Fm+1 ) = rm − km+1 rm+1 , we obtain (24.2). Moreover ∞

λ ([0, 1] \ F) =



∑ λ (Fn \ Fn+1 ) =

n=0



pn

∑ λ (Iin \ Fn+1 )
> > > . n n+2 f (r ) f (r ) f (r ) 2 4 f λ Iin n n n Hence limn→∞ tion 23.21).

λ (Iinn \F ) f (λ (Iinn ))

= ∞, and consequently x ∈ / Φ f (F) (compare Proposit u

24. Density type topologies generated by functions. Properties of f -density

417

Using the set F we will construct a closed set with exactly one f -density point. Theorem 24.7 ([9]). For any function f ∈ A0 there exists a closed set F1 ⊂ [0, 1] such that λ (F1 ) > 0 and Φ f (F1 ) = {0}. Proof. Since f ∈ A0 ,there exists a sequence  (xn ) such that nxn+1 < f (xn ) < xn 1 for n ≥ 2. Let Jn := xn+1 + n f (xn+1 ) , xn . According to Theorem 24.6 there is a closed set En ⊂ Jn with Φ f (En ) = 0/ and λ (Jn \ En ) < n1 f (xn+1 ). Put S E := ∞ / It remains to check that 0 ∈ Φ + n=2 En . Of course, Φ f (E) = 0. f (E). For x ∈ [xn+1 , xn ] we have   1 [0, x] \ E ⊂ 0, xn+2 + f (xn+2 ) ∪ (Jn+1 \ En+1 ) ∪ n+1   1 ∪ xn+1 , xn+1 + f (xn+1 ) ∪ (Jn \ En ) . n Therefore 1 1 1 1 λ ([0, x] \ E) xn+2 + n+1 f (xn+2 ) + n+1 f (xn+2 ) + n f (xn+1 ) + n f (xn+1 ) 5 < < , f (x) f (xn+1 ) n

and consequently 0 ∈ Φ + f (E). Thus F1 := E ∪ (−E) ∪ {0} is the desired set. t u The sets constructed in Theorem 24.6 and Theorem 24.7 show that properties of f -density operators for f ∈ A0 differ considerably from the properties of the classical density operator Φd . Theorems 23.6 and 23.8 contain the list of differences. In particular we have. Proposition 24.8. If f ∈ A0 then the space (R, T f ) is of the first category. In chapter 23 it is proved that intT f (A) = A ∩ Φ f (A) for f ∈ A1 and A ∈ L. This equality need not be true for f ∈ A0 . If F1 is the set constructed in Theorem 24.7 then F1 ∩ Φ f (F1 ) = {0} but intT f (F1 ) = 0. / Now we will describe an interior operation in an f -density topology (in the general case). Let f ∈ A and A ∈ L. By induction we define Φ αf (A) for 1 ≤ α < ω1 :  Φ 1f (A) := Φ f (A) , Φ α+1 (A) := Φ f Φ αf (A) , f Φ αf (A) :=

\

β

Φ f (A) if α is a limit number.

1≤β 0, and the set A1 := (A \ F1 ) ∩ Fi1 has positive measure for some i1 > 1. Let a1 ∈ A1 ∩ Φd (A1 ). There is a natural number n1 such that [a1 − tn1 , a1 + tn1 ] ∩ F1 = 0/ and

λ (A1 ∩ [a1 − tn1 , a1 + tn1 ]) 1 > . 2tn1 2

Let us denote 1 and y1:= a1 + tn1 .  x1 := a1 − tnS 1 Note that λ A ∩ (x1 , y1 ) \ ij=1 Fj > 0 and the set A2 :=

A ∩ (x1 , y1 ) \

i1 [ j=1

! Fj ∩ Fi2

422

Małgorzata Filipczak, Tomasz Filipczak

is of a positive measure for some i2 > i1 . Fix a2 ∈ A2 ∩ Φd (A2 ). There is a natural number n2 > n1 such that the points x2 := a2 − tn2 and y2 := a2 + tn2 satisfy [x2 , y2 ] ⊂ (x1 , y1 ) , [x2 , y2 ] ∩

i1 [

Fj = 0/ and

j=1

λ (A2 ∩ [x2 , y2 ]) 1 > . y2 − x2 2

Proceeding by induction we define increasing sequences (ik ), (nk ) of natural numbers and a decreasing sequence of closed intervals ([xk , yk ]) such that ik−1

yk − xk = 2tnk , [xk , yk ] ∩

[

Fj = 0/ and

j=1

Let {x} =

∞ \

λ (A ∩ [xk , yk ]) 1 > . yk − xk 2

[xk , yk ] .

k=1

Then x ∈ /

S∞

j=1 Fj

= B, but for every k we have

λ (A ∩ [xk , yk ]) λ (A ∩ [xk , yk ]) 2tnk 1 = · > , f (yk − xk ) 2tnk f (2tnk ) 2 which implies that x is not a T f -interior point of R \ A, and consequently x ∈ clT f (A). This establishes (24.4) and completes the proof. t u

24.4 Homeomorphisms of f -density topologies Theorem 24.17 ([4]). If f1 , f2 ∈ A and h : (R, T f1 ) → (R, T f2 ) is a homeomorphism, then (1) h and h−1 are continuous (in a usual sense), strictly monotonic and satisfy Lusin’s condition (N), (2) the sets  A := x : there exists derivative h0 (x) , n o 0 B := x : there exists derivative h−1 (h (x)) have full measure, (3) if h0 (x) = 1 for every x ∈ A ∩ B, then T f1 = T f2 .

24. Density type topologies generated by functions. Properties of f -density

423

Proof. Let I be an open interval. By Theorem 23.37,   E : E is T f1 -connected = E : E is T f2 -connected = {E : E is connected} , so h−1 (I) is an interval, too. Since no end of an interval can be its f1 -density point, the interval h−1 (I) has to be open. Thus h is continuous. Since h is also an injection, it is strictly monotonic. Let P be a null set. Then P and all subsets of P are closed in T f1 . Consequently, h (P) and all its subsets are closed in T f2 , and so they are measurable. Hence h (P) is of measure zero, which finishes the proof of (1). Any monotonic function is almost so A has full n everywhere differentiable, 0 o −1 −1 measure. Observe that B = h y; there exists h (y) . Using Lusin’s condition (N) for h−1 , we conclude that B has full measure too. Suppose that h0 (x) = 1 for x ∈ A ∩ B. By (1) and Banach-Zarecki theorem, we deduce that h (x) is absolutely continuous on any interval [a, b] (see [11]). Since h0 (x) = 1 almost everywhere, h (x) = x, which gives T f1 = T f2 . t u Theorem 24.18 ([4]). Let f1 , f2 ∈ A1 . If topological spaces (R, T f1 ) and (R, T f2 ) are homeomorphic, then topologies T f1 and T f2 are comparable i.e. T f1 ⊂ T f2 or T f2 ⊂ T f1 . Proof. Suppose, contrary to our claim, that T f1 and T f2 are not comparable. Let h be a homeomorphism from (R, T f1 ) onto (R, T f2 ). By Theorem 24.17, and for some x0 there exist derivatives h0 (x0 ), 0 h is strictly monotonic −1 0 h (h (x0 )) with h (x0 ) = c 6= 1. Since f -density topologies are invariant with respect to translations and symmetries, we can assume that h is increasing and h (x0 ) = x0 = 0. We can also assume that 0 < c < 1 (we replace h by h−1 , if necessary). Since T f1 \ T f2 6= 0, / Theorem 23.29 shows that there is a measurable set A such that 0 ∈ Φ + (A) \ Φ+ f1 f2 (A). Thus there exist a positive number η and a n ]\A) > η for every n. decreasing and tending to 0 sequence (hn ) such that λ ([0,h f2 (hn ) It is not difficult to define sequences (bn ) and (cn ) satisfying bn+1 < cn < bn ≤ hn and λ ([cn , bn ] \ A) > η. f2 (bn )

Let us define an := bn − λ ([cn , bn ] \ A) and B :=

∞ [ n=1

(bn+1 , an ) ∪ {0} ∪

∞ [ n=1

(−an , −bn+1 ) .

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Of course, cn ≤ an ≤ bn . An easy computation shows that λ ([0, x] \ B) ≤ λ ([0, x] \ A) for x ≤ b1 . Hence 0 ∈ Φ + f1 (B) and B ∈ T f1 . The proof will be completed by showing that 0 ∈ / Φ f2 (h (B)). Since f2 ∈ A1 , f2 (bn ) there exists α > 0 such that bn > 4α for almost all n. Let ε := cηα. Since 0 < h0 (0) = c < 1, we have h (bn ) h (an ) an − c < ε, bn − c < ε and h (bn ) < bn for sufficiently large n. Hence h (bn ) − h (an ) > (c − ε) bn − (c + ε) an > c (bn − an ) − 2εbn . Therefore λ ([0, bn ] \ h (B)) h (bn ) − h (an ) (bn − an ) bn ≥ ≥c − 2ε > f2 (bn ) f2 (bn ) f2 (bn ) f2 (bn ) cη 2ε = > 0, > cη − 4α 2 which gives 0 ∈ / Φ f2 (h (B)).

t u

Theorem 24.19 ([4]). The density topology Td is not homeomorphic to any topology T f 6= Td . Proof. If f ∈ A0 then, by Theorem 24.16, T f is not regular, so T f and Td are not homeomorphic. Assume that f ∈ A1 and T f 6= Td . Of course, T f \ Td 6= 0. / Suppose, contrary to our claim, that there is a homeomorphism h : (R, T f ) → (R, Td ). We can choose x0 ∈ R such that h0 (x0 ) = c 6= 1. We can also assume that h is increasing and h (x0 ) = x0 = 0. If c < 1 then repeating the proof of Theorem 24.18 we obtain a contradiction. Suppose that c > 1 and set g (x) := h(x) 2c . Since Td is invariant under multiplication by nonzero numbers, 1 for any U ∈ T f we have g (U) = 2c h (U) ∈ Td , and for any V ∈ Td we have −1 −1 g (V ) = h (2cV ) ∈ T f . Hence g is a homeomorphism. As g0 (x0 ) = 12 , we can repeat the proof of Theorem 24.18 for g. t u

24.5 f -density topologies and (∆2 ) condition In the theory of Orlicz spaces there is often use of the condition called (∆2 ). W. Orlicz says that a continuous, nondecreasing and unbounded function

24. Density type topologies generated by functions. Properties of f -density

425

ϕ : [0, ∞) → [0, ∞), with ϕ(0) = 0 and ϕ(x) > 0 for x > 0, satisfies (∆2 ) condition if lim supx→∞ ϕ(2x) ϕ(x) < ∞ (see [12] or [13]). In the consideration of ψdensity, the analogous condition is used for functions belonging to the family b (compare [8] and chapter 22). We will consider this condition for functions C from the family A. We say that a function f ∈ A fulfils (∆2 ) condition ( f ∈ ∆2 ) if lim sup x→0+

f (2x) < ∞. f (x)

We will use (∆2 ) condition to compare f -density topologies and to study their algebraic properties. It is useful to observe: Proposition 24.20 ([7]). For any f ∈ A the following conditions are equivalent: (1) f ∈ ∆2 , (2) for any positive number β , lim supx→0+ (3) there exists β > 1 such that

f (β x) f (x) < ∞, lim supx→0+ f f(β(x)x) < ∞.

Proof. (1)⇒(2). There is n ∈ N such that 2n > β . From  f (β x) f (2n x) f (2n x) f 2n−1 x f (2x) 6 = · ·...· n−1 n−2 f (x) f (x) f (2 x) f (2 x) f (x) n  < ∞. The implication we obtain lim supx→0+ f f(β(x)x) 6 lim supx→0+ ff(2x) (x) (2)⇒(3) is obvious. The proof of (3)⇒(1) is analogous to the first one. t u Note that for any α > 1, the function xα fulfils (∆2 ) condition and Txα ⊂ Td . α If α ∈ (0, 1) then the function xα does not belong to A, because limx→0+ xx = ∞. To obtain a function f ∈ ∆2 , generating topology T f bigger than Td (or incomparable with Td ), we will "glue together" square functions with various coefficients and constant ones. Such a construction is presented in the following lemma. It is worth to observe that the construction works not only for square functions. We can use for example xα with α > 1. Lemma 24.21 ([7]). If (an )n≥0 is a decreasing sequence tending to zero and √ bn := an an−1 for n ∈ N, then the functions  2  2 x   for x ∈ [b2n , b2n−1 ] ,  axn for x ∈ [an , bn ] ,  a2n−1 f (x) := an−1 for x ∈ [bn , an−1 ] , g (x) := a2n for x ∈ [b2n+1 , b2n ] ,   a a for x > a0 , for x > b1 0 0

426

Małgorzata Filipczak, Tomasz Filipczak

are continuous and fulfil (∆2 ) condition. Proof. Obviously, the functions f and g are continuous and belong to A. Observe that x2 for x ≥ an . f (x) ≤ an Indeed, if x ∈ [ak , bk ] for some k ≤ n, then f (x) = [bk , ak−1 ], k ≤ n, we have f (x) = f (bk ) =

b2k ak

Let x > 0. If x ∈ [an , bn ] for some n, then

x2 ak



x2 an ,

≤ an . f (2x) (2x)2 /an f (x) ≤ x2 /an

belong to [bn , an−1 ) then

f (2x) f (x)

f (2x) f (x)

≤ 4. Thus f fulfils (∆2 ) condition.

=

f (2x) f (an−1 )



f (2an−1 ) f (an−1 )

whereas for x ∈

x2

= 4. If x and 2x

= 1. Finally, if x ∈ [bn , an−1 ] and 2x > an−1 then

Similarly we show that g (x) ≤

x2 a2n−1

Example 24.22 ([7]). Let an :=

for x ≥ b2n , and hence

1 (n+1)! ,

g(2x) g(x)

≤ 4.

t u

n = 0, 1, . . . and let f , g be the func-

tions defined in Lemma 24.21. Then f , g ∈ ∆2 . Since lim infx→0+ f (x) x = 1 f (x) and lim supx→0+ x = ∞, Theorem 23.32 implies Td $ T f . Similarly, from g(x) lim infx→0+ g(x) x = 0 and lim supx→0+ x = ∞, we conclude that the topologies Tg and Td are not comparable (Tg * Td and Td * Tg ). Recall that the density topology Td is invariant under multiplication by nonzero numbers and any f -density topology is invariant under multiplication by numbers α ≥ 1 and α ≤ −1 (compare Theorem 23.40). Theorem 23.12 states that hsi-density topologies different from Td are not invariant under multiplication by numbers α ∈ (−1, 1). Since any hsi-density topology is equal to the topology T f for some f ∈ A1 , there are f -density topologies which are not invariant under multiplication by numbers α ∈ (−1, 1). There is a natural question if there exists an f -density topology T f ' Td which is invariant under multiplication by nonzero numbers. The following theorem gives a straightforward answer. Theorem 24.23 ([7]). If f ∈ ∆2 then T f is invariant under multiplication by nonzero numbers. Proof. According to Theorem 23.29, it is enough to show that if α ∈ (0, 1) and + 0 ∈ Φ+ f (A), then 0 ∈ Φ f (αA). If f ∈ ∆ 2 then we have

24. Density type topologies generated by functions. Properties of f -density

427

    αλ 0, αx \ A f αx λ ([0, x] \ αA)  = lim sup ≤ lim sup · f (x) f (x) f αx x→0+ x→0+     λ 0, αx \ A f αx  ≤ α · lim sup · lim sup = 0. f αx x→0+ x→0+ f (x) t u Now we are in the position to give a simple example of a function f ∈ A1 such that T f 6= Thsi for hsi ∈ Se (much nicer than Example 23.41). Example 24.24. Let f be the function from Example 24.22. Then f ∈ ∆2 and, by Theorem 24.23, topology T f is invariant under multiplication by nonzero e numbers. Thus Theorem 23.12 implies T f 6= Thsi for hsi ∈ S. In the paper [15] it was shown that, for ψ-density topologies, the invariantness of Tψ under multiplication by nonzero numbers implies lim supx→0+ ψ(2x) ψ(x) < ∞. This result can be generalized to f -density topologies contained in Td , i.e. the invariantness of T f under multiplication by nonzero numbers implies f ∈ ∆2 . Moreover, if T f * Td then the considered implication is untrue. Theorem 24.25 ([7]). Let f ∈ A. The following conditions are equivalent. (1) T f ⊂ Td . (2) The topology T f is invariant under multiplication by nonzero numbers if and only if f ∈ ∆2 . Proof. (1)⇒(2). By Theorem 24.23, if f ∈ ∆2 then T f is invariant under multiplication by nonzero numbers. Suppose that f ∈ / ∆2 . There exists a decreasing 1 n) sequence (bn ) such that bn+1 < 2 min {bn , f (bn )} and ff(2b (b ) > n for n ∈ N. n n o S∞ f (bn ) bn Let an := max 2 , bn − 2 and A := R \ n=1 [an , bn ]. We will show that A∈ / T f but 4 · A ∈ T f . Since T f ⊂ Td , Theorem 23.32 implies f (x) x ≤ M for some positive M and any x from some interval (0, h). We can assume that this inequality holds for all x > 0. Thus     λ ([0, bn ] \ A) bn − an bn 1 1 1 > > min , ≥ min , > 0. f (bn ) f (bn ) 2 f (bn ) 2 2M 2 Consequently, 0 ∈ / Φ+ / Tf . f (A) and A ∈ Observe that 4an > 2bn . For x ∈ [4an , 4bn ] we have   f (bn ) f (bn ) 4 + 2 2 λ ([0, x] \ 4A) 4 (bn+1 + (bn − an )) 4 ≤ < < . f (x) f (4an ) f (2bn ) n

428

Małgorzata Filipczak, Tomasz Filipczak

Using the preceding inequality, for x ∈ [4bn+1 , 4an ] we obtain λ ([0, x] \ 4A) λ ([0, 4bn+1 ] \ 4A) λ ([0, 4bn+1 ] \ 4A) 4 = 6 < . f (x) f (x) f (4bn+1 ) n+1 Hence 0 ∈ Φ + f (4A), and consequently 4A ∈ T f . (2)⇒(1). It is sufficient to show that for any function f ∈ ∆2 such that T f * Td there is a function g ∈ / ∆2 for which T f = Tg . Since T f * Td , Theorem 23.32 f (x) implies lim supx→0+ x = ∞. From f ∈ ∆2 it follows that lim infx→0+ f (x) x n2 and

M 2 . Then  c  f (2c ) nb n n n ≥ > 2 and 2an ≤ cn , nbn ≥ f 2 M2 M because f (2an ) ≤ M f (an ) < M 2 an < nan < nbn . Let us define   f (bn ) for x ∈ c2n , bn , n > M 2 ,  S g (x) := f (x) for x ∈ (0, ∞) \ n>M2 c2n , bn . From

g(cn ) g( c2n )

=

f (bn ) f ( c2n )

>

n2 bn nbn

= n we obtain g ∈ / ∆2 . Since f ≤ g, T f ⊂ Tg . To

prove the inverse inclusion, we use Theorem 24.2. Let A := A1 f g = {x > 0 : f (x) < g (x)} and ε := ε1 f g = lim sup x→0+

Since A ⊂

S

n>M 2

 cn 2 , bn , for any x ∈

cn−1 

cn 2, 2

λ (A ∩ [0, x]) . f (x)

we have

λ (A ∩ [0, x]) bn M2  < < , f (x) n f c2n and consequently ε = 0. Thus Tg ⊂ T f .

t u

f (x) In Theorem 23.30 it is shown that lim supx→0+ g(x) < ∞ implies T f ⊂ Tg . h  1 Example 23.31 asserts that the functions f (x) = n!1 , x ∈ (n+1)! , n!1 and g(x) =  i f (x) 1 1 1 , x ∈ , n! (n+1)! n! generates the same topology, although lim supx→0+ g(x) =

24. Density type topologies generated by functions. Properties of f -density

429

∞. It turns out that, if we additionally assume that f ∈ ∆2 and T f ⊂ Td , then f (x) the condition T f ⊂ Tg is equivalent to lim supx→0+ g(x) < ∞. Theorem 24.26 ([7], Th. 7). Suppose that f ∈ ∆2 , g ∈ A and T f ⊂ Td . If T f ⊂ f (x) Tg then lim supx→0+ g(x) < ∞. Note that the assumption T f ⊂ Td cannot be omitted. In [7, Ex. 3] there are f (x) constructed the functions f , g ∈ ∆2 such that lim supx→0+ g(x) = ∞ and T f = Tg .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16]

A. Bruckner, Differentiation of Real Functions, Lect. Notes in Math. 659, SpringerVerlag, 1978. M. Filipczak, T. Filipczak, Remarks on f -density and ψ-density, Tatra Mt. Math. Publ., 34 (2006), 141-149. M. Filipczak, T. Filipczak, On f -density topologies, Topology Appl. 155 (2008), 1980-1989. M. Filipczak, T. Filipczak, On homeomorphisms of density type topologies generated by functions, Tatra Mt. Math. Publ. 46 (2010), 7-13. T. Filipczak, The comparison of f -density and ψ-density, in: Real Functions, Density Topology and Related Topics, Łód´z University Press 2011, 23-28. M. Filipczak, T. Filipczak, On the comparison of density type topologies genarated by functions, Real Anal. Exchange 36 (2011), 341-352. M. Filipczak, T. Filipczak, On ∆2 condition for density-type topologies generated by functions, Topology Appl. 159 (2012), 1838-1846. M. Filipczak, M. Terepeta, On ∆2 condition in density-type topologies, Demonstratio Math. 44 (2) (2011), 423-432. M. Filipczak, E. Wagner-Bojakowska, The interior operation in f -density topology, Tatra Mt. Math. Publ. 35 (2007), 51-64. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN, 1985. I.P. Natanson, Theory of functions of a real variable, 2nd rev. ed. Ungar, New York 1961. W. Orlicz, Über eine gewisse Klasse von Räumen vom typus B, Bull. Acad. Polonaise, Série A (1932), 207-220; reprinted in his Collected Papers, PWN, Warszawa 1988, 217-230. M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., 1991. S. J. Taylor, On strengthening the Lebesgue Density Theorem, Fund. Math. 46 (1959), 305-315. M. Terepeta, E. Wagner-Bojakowska, ψ-density topology, Rend. Circ. Mat. Palermo 48 (1999), 451-476. E. Wagner-Bojakowska, W. Wilczy´nski, Comparison of ψ-density topologies, Real Anal. Exchange 25 (1999-2000), 661-672.

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Małgorzata Filipczak, Tomasz Filipczak

[17] E. Wagner-Bojakowska, W. Wilczy´nski, The interior operation in a ψ-density topology, Rend. Circ. Mat. Palermo 49 (2000), 5-26.

M AŁGORZATA F ILIPCZAK Faculty of Mathematics and Computer Sciences, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

T OMASZ F ILIPCZAK Institute of Mathematics, Łód´z Technical University ul. Wólcza´nska 215, 93-005 Łód´z, Poland E-mail: [email protected]

Chapter 25

On the abstract density topologies generated by lower and almost lower density operators

JACEK HEJDUK, RENATA WIERTELAK

2010 Mathematics Subject Classification: 54A10, 28A05. Key words and phrases: lower density operator, density topology.

25.1 Introduction In the real analysis we often deal with the different kind of density topologies. The most fundamental one is the classical density topology in the family of Lebesgue measurable sets introduced by Haupt and Pauc (1952, see [8]), and intensively investigated by Goffman, Waterman (1961, see [6]), Goffman, Neugebauer, Nishiura (1961, see [5]) and Tall (1976, see [24]). However, the idea of density we can found in monograph of Hobson "The theory of functions of a real variable and the theory of Fourier’s series" where is considered so called metric density (see [14]). The interesting analogue of the classical density in measure turned out to be the density topology introduced in the family of Baire sets on the real line by Poreda, Wagner-Bojakowska and Wilczy´nski (1985, see [22]). The both density topologies introduced by the sets of density points require some properties of the operator of density points, which is a special case of the lower density operator usually defined on an abstract measurable space. Many interesting examples of density topologies introduced and investigated recently based on lower and almost lower density operators. At this moment it is worth mentioning fundamental paper of Maharam (1958,

432

Jacek Hejduk, Renata Wiertelak

see [20]) and Hamlett, Jankovi´c, Rose (1993, see [7]) concerning lower density operators in the general case for measure and category. We have decided to collect as much as possible different properties of topologies generated by lower and almost lower density operators on arbitrary measurable space. The ideas of proofs are taken from [2], [18], [29], [30], where they are presented in a special case of the classical density topology and I-density topology.

25.2 The case of lower density operators Let X be a nonempty set, S be a σ -algebra of sets from X and J ⊂ S be a proper σ -ideal. Definition 25.1. We shall say that an operator Φ : S → S is a lower density operator on a measurable space (X, S, J ) if (i) Φ(0) / = 0, / Φ(X) = X; (ii) ∀A,B∈S Φ(A ∩ B) = Φ(A) ∩ Φ(B); (iii) ∀A,B∈S (A M B ∈ J ⇒ Φ(A) = Φ(B)); (iv) ∀A∈S A M Φ(A) ∈ J . Definition 25.2. We shall say that a topology τ is an abstract density topology on X if there exists a lower density operator on (X, S, J ) such that τ = TΦ , where TΦ = {A ∈ S : A ⊂ Φ(A)}. Topology TΦ is called generated by a lower density operator on (X, S, J ). Theorem 25.3. Let τ be an abstract density topology on X generated by a lower density operator Φ on (X, S, J ). Then a) A ∈ J if and only if A is τ-closed and τ-nowhere dense; b) if A ∈ J , then A is τ-closed and τ-discrete; c) J = M(τ), where M(τ) is the family of meager sets with respect to τ; d) A ∈ S if and only if A is union of a τ-open and a τ-closed set; e) Bor(τ) = B(τ) = S, where Bor(τ) is the family of Borel sets and B(τ) is the family of Baire sets with respect to τ; f) hX, τi is a Baire space; g) τ = {Φ(A) \ B : A ∈ S, B ∈ J }. Moreover, if J contains all singletons, then h) A ∈ J if and only if A is τ-closed and τ-discrete; i) A is τ-compact if and only if A is finite;

25. On the abstract density topologies

433

j) hX, τi is neither a first countable, nor a second countable, nor a separable space; k) if J contains an uncountable set, then hX, τi is not a Lindelöf space; l) every sequence consisting of different terms of X does not contain τconvergent subsequences. Proof. a) If A ∈ J , then X \ A ∈ τ and intτ (A) = 0. / Thus A is τ-closed and τnowhere dense. Conversely, if A is τ-closed and τ-nowhere dense and A ∈ / J, then A ∈ S \ J and Φ(A) ∩ A ∈ τ \ {0}. / It contradicts the fact that A is τnowhere dense. Conditions b) and c) are a consequence of a). d) It is sufficient to observe that if A ∈ S, then A = (A ∩ Φ(A)) ∪ (A \ Φ(A)). Obviously A ∩ Φ(A) ∈ τ and A \ Φ(A) is τ-closed. It is clear that condition d) implies e). f ) If A ∈ τ \ {0}, / then A ∈ / J . By condition c) A ∈ / M(τ). Hence hX, τi is a Baire space. g) If A ∈ S and B ∈ J , then Φ(A) \ B ∈ τ. Let A ∈ τ. Then A ⊂ Φ(A) so A = Φ(A)\(Φ(A)\A). Since Φ(A)\A ∈ J , we have τ ⊂ {Φ(A) \ B : A ∈ S, B ∈ J } and the proof of g) is completed. h) In virtue of b) it is sufficient to show that if A is τ-closed and τ-discrete, then A ∈ J . Suppose that A 6= 0. / Obviously A ∈ S and for every x ∈ A there exists a set Vx ∈ τ, such that x ∈ Vx and Vx ∩ A = {x}. Thus x ∈ Vx ⊂ Φ(Vx ) = Φ(Vx \ {x}) ⊂ Φ(X \ A). Hence A ⊂ Φ(X \ A). It follows that A = Φ(X \ A) \ (X \ A) ∈ J . i) Assume that a set A is compact and infinite. Let B ⊂ A be a countable infinite set. Then the family {(X \ B) ∪ {x}}x∈B is a τ-open cover of A without a finite subcover. The converse implication in condition i) is obvious. j) Let x ∈ X and {En }n∈N be a sequence of τ-open sets containing x. Let xn ∈ En \ {x} for every n ∈ N. Putting E = E1 \ {xn : n ∈ N}, we have E ∈ τ, x ∈ E and E does not contain set En for n ∈ N. So the first countability axiom is not satisfied. Also hX, τi is not a second countable space. At the same time hX, τi is not a separable space. Indeed, taking into account a countable and dense set A ⊂ X we have A ∈ J and therefore X = clτ (A) = A ∈ J , so X ∈ J . It contradicts the fact that J is a proper σ -ideal. k) Let D ∈ J be an uncountable set. Then the family {(X \ D) ∪ {x}}x∈D is a τ-open cover of X without a countable subcover. So that hX, τi is not a Lindelöf space.

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l) If {xn }n∈N is a sequence consisting of different terms of X, then by h) for S every subsequence {xnk }k∈N and x ∈ X we get that x ∈ (X \ ∞ k=1 {xnk }) ∪ {x} ∈ τ. It means that there is no τ-convergent subsequence of {xn }n∈N . t u Theorem 25.4. Let τ be a topology on X. Then the following conditions are equivalent: (i) τ is an abstract density topology; (ii) τ has the following properties: (a) A ∈ J if and only if A is τ-closed and τ-nowhere dense; (b) B(τ) = S. Proof. Implication (i) ⇒ (ii) follows from the previous theorem. (ii) ⇒ (i) Notice that hX, τi is a Baire space. Let A ∈ S = B(τ). By Theorem 4.6 from [21] the set A has the unique representation in the form G M P, where G is regular τ-open (i.e. G = intτ (clτ (G))) and P ∈ J . Put Φ(A) = G. Obviously Φ(0) / = 0, / Φ(X) = X, and A M Φ(A) ∈ J for A ∈ S. Moreover, if A M B ∈ J , then Φ(A) = Φ(B). Now let A, B ∈ S and A = G1 M P1 , B = G2 M P2 , where G1 , G2 are regular τ-open and P1 , P2 ∈ J . There exists P3 ∈ J such that A ∩ B = (G1 ∩ G2 ) M P3 . It follows that Φ(A) ∩ Φ(B) = G1 ∩ G2 = Φ(G1 ∩ G2 ) = Φ(A ∩ B). Therefore Φ is a lower density operator on (X, S, J ). We shall prove that TΦ = τ. If A is τ-open, then by Theorem 4.5 from [21] we have A = G \ P, where G is regular τ-open and P is τ-closed and τ-nowhere dense. Thus A ∈ S and A ⊂ G = Φ(A). Hence τ ⊂ TΦ . Suppose now that A ∈ TΦ , so A ∈ S, A ⊂ Φ(A) and A M Φ(A) ∈ J . It follows that A = Φ(A) \ P, where P ∈ J . Since P is τ-closed, A is τ-open. t u Corollary 25.5. If J contains all singletons, then the operator Φ described in the proof of the above theorem has the following form: ∀A∈S

Φ(A) = intτ {x ∈ X : x ∈ intτ (A ∪ {x})} .

Proof. Let A ∈ S and Φ1 (A) = intτ {x ∈ X : x ∈ intτ (A ∪ {x})}. First we show that if A, B ∈ S and A M B ∈ J , then Φ1 (A) = Φ1 (B). It is clear that A = B M C, where C ∈ J . We demonstrate that Φ1 (A) ⊂ Φ1 (B). Suppose that x ∈ Φ1 (A). Hence x ∈ Φ1 (B M C) and x ∈ intτ ((B M C) ∪ {x}). There exists a τ-open set Wx 3 x such that Wx ⊂ ((B M C) ∪ {x}). Therefore Wx \ (C \ {x}) ⊂ (B ∪ {x})

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and x ∈ Wx \ (C \ {x}) which is a τ-open set. It follows that x ∈ intτ (B ∪ {x}) and Φ1 (A) ⊂ Φ1 (B). Similarly we can show that Φ1 (B) ⊂ Φ1 (A). Now let A = V M P, where V is regular τ-open and P ∈ J . Then Φ(A) = V and Φ1 (A) = Φ1 (V ). It is sufficient to prove that V = Φ1 (V ). Since V ⊂ Φ1 (V ) it remains to show that Φ1 (V ) ⊂ V . Suppose that x ∈ Φ1 (V ) and x ∈ W ∈ τ. Since x ∈ intτ (V ∪ {x} ), there exists a set Wx ∈ τ such that x ∈ Wx ⊂ V ∪ {x}. It is clear that x ∈ (W ∩ Wx ) 6= 0. / Moreover, (W ∩ Wx ) \ {x} 6= 0, / because J contains all singletons and (W ∩Wx ) ⊂ Φ(W ∩Wx ). Consequently (W ∩Wx ) \ {x} ⊂ V and W ∩ V 6= 0. / It implies that the set {x ∈ X : x ∈ intτ (V ∪ {x})} is τ-open. Therefore Φ1 (V ) = intτ Φ1 (V ) ⊂ intτ (clτ V ) = V . t u Theorem 25.6. Let Φ be a lower density operator on (X, S, J ). Then the family TΦ = {A ∈ S : A ⊂ Φ(A)} is a topology on X if and only if the pair (S, J ) has the hull property. Proof. Sufficiency. Let Φ be a lower density operator on (X, S, J ). Obviously 0, / X ∈ TΦ and the family TΦ is closed under finite intersections. Let {At }t∈T ⊂ S TΦ . We shall prove that t∈T At ∈ TΦ . Let B be a S-measurable kernel of the S set t∈T At . Hence for every t ∈ T we have (At ∩ B) M At ∈ J and B⊂

[ t∈T

At ⊂

[

Φ(At ) =

t∈T

[

Φ(At ∩ B) ⊂ Φ(B).

t∈T

Since Φ(B) \ B ∈ J , we have t∈T At ∈ S. Moreover, it is obvious that S S t∈T At ⊂ Φ( t∈T At ). Therefore t∈T At ∈ TΦ . Necessity. Obviously TΦ is an abstract density topology on (X, S, J ). By Theorem 25.3 we get B(TΦ ) = S and M(TΦ ) = J . The well known fact that the pair (B(TΦ ), M(TΦ )) has the hull property (see [17]) completes the proof. t u S

S

Corollary 25.7. If hX, τi is a topological space such that X ∈ / M(τ), then every lower density operator Φ on (X, B(τ), M(τ)) generates TΦ topology on X. Proof. Since the pair (B(τ), M(τ)) has the hull property, then by Theorem 25.6 any lower density operator Φ on (X, B(τ), M(τ)) generates topology on X. t u Remark 25.8. If Φ(A) = A for every set A ∈ Bor, then Φ is the lower density operator on (R, Bor, J ), where J = {0}, / but it is clear that TΦ is not a topology on R. The reason is that the pair (Bor, J ) does not have the hull property .

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By the idea implemented in Theorem 25.6 we get the following topologies on the real line: • density topology (O. Haupt, C. Pauc (1952), see [8]); • I-density topology (W. Poreda, E. Wagner-Bojakowska, W. Wilczy´nski (1985), see [22]); • topology involving measure and category (W. Wojdowski (1989), see [33]); • hsi-density topology with respect to category (J. Hejduk, G. Horbaczewska (2003), see [12], [15]); • hsi-density topology with respect to measure (M. Filipczak, J. Hejduk (2004), see [4]); • density topology related to category with respect to a sequence tending to zero (R. Wiertelak (2006), see [28]); • hsi-simple density topology with respect to category (V. Aversa, W. Wilczy´nski (2004), see [1]); • ΨI -density topology (E. Łazarow, A. Vizvary (2010), see [19]); • category ψ-density topology (W. Wilczy´nski, W. Wojdowski (2011), see [31]). Abstract density topologies allow us to find a clear and useful representation of the interior of any set. Namely, we have the following theorem which proof for the case of density topology can be found in [30]. Theorem 25.9. If τ is an abstract density topology on X generated by a lower density operator Φ on (X, S, J ), then for any set A ⊂ X we have intτ A = A ∩ Φ(B), where B is an S-measurable kernel of A. Proof. Let A ⊂ X. By Theorem 25.6 the pair (S, J ) possesses the hull property. Let B ⊂ A be an S-measurable kernel of A. Observe that A ∩ Φ(B) ∈ τ. Indeed, A ∩ Φ(B) = (B ∩ Φ(B)) ∪ ((A \ B) ∩ Φ(B)). Since Φ(B) ∩ (A \ B) ⊂ (Φ(B) \ B) ∈ J , we have A ∩ Φ(B) ∈ S. Moreover, Φ(A ∩ Φ(B)) = Φ(B ∩ Φ(B)) = Φ(B). It follows that A ∩ Φ(B) ∈ τ. Let us assume that V ∈ τ and V ⊂ A. We show that V ⊂ A ∩ Φ(B). Since V = (V ∩ B) ∪ (V ∩ (A \ B)) and V ∩ (A \ B) ⊂ V \ B ⊂ A \ B, we have V \ B ∈ J and also V ∩ (A \ B) ∈ J . Therefore Φ(V ) = Φ(V ∩ B) = Φ(V ) ∩ Φ(B) which implies that V ⊂ Φ(V ) ⊂ Φ(B). Thus V ⊂ A ∩ Φ(B). t u Corollary 25.10. If τ is an abstract density topology on X generated by a lower density operator Φ on (X, S, J ), then Φ(A) ⊂ clτ (A) for every A ∈ S.

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Proof. Let A ∈ S. Then clτ (A) = X \ intτ (X \ A) = X \ [(X \ A) ∩ Φ(X \ A)] = A ∪ (X \ Φ(X \ A)). From Φ(A) ⊂ X \ Φ(X \ A), it follows that Φ(A) ⊂ clτ (A).

t u

Theorem 25.11. If τ is an abstract density topology on X generated by a lower density operator Φ on (X, S, J ), then A ⊂ X is a regular τ-open set if and only if A = Φ(A). Proof. Necessity. Let A be a regular τ-open set, i.e. A = intτ (clτ (A)). Then, by Theorem 25.9, A = clτ (A) ∩ Φ(clτ (A)). In virtue of Corollary 25.10, Φ(A) ⊂ clτ (A). Moreover, Φ(A) ⊂ Φ(clτ (A)). It follows that Φ(A) ⊂ A. Evidently A ∈ τ, so that A ⊂ Φ(A) and finally A = Φ(A). Sufficiency. Let A = Φ(A). Then A ∈ τ and clτ (A) \ A ∈ J . Hence intτ (clτ (A)) = clτ (A) ∩ Φ(clτ (A)) = clτ (A) ∩ Φ(A) = Φ(A) = A. Therefore A is a τ-regular open set.

t u

Theorem 25.12. Let Φ1 , Φ2 be the lower density operators on (X, S, J ) generating TΦ1 , TΦ2 topologies, respectively. Then TΦ1 = TΦ2 if and only if Φ1 = Φ2 . Proof. Sufficiency is obvious. Necessity. Let A ∈ S. Then Φ1 (A) ∈ TΦ1 = TΦ2 . It follows that Φ2 (Φ1 (A)) ⊃ Φ1 (A). Since Φ1 (A) M A ∈ J , we obtain Φ2 (Φ1 (A)) = Φ2 (A) and consequently Φ2 (A) ⊃ Φ1 (A). Similarly we show that Φ1 (A) ⊃ Φ2 (A). t u Corollary 25.13. If τ is an abstract density topology on X, then there exists a unique lower density operator Φ on (X, S, J ) such that TΦ = τ. Definition 25.14. We shall say that operators Φ1 , Φ2 : S → 2X are equivalent on (X, S, J ) if Φ1 (A) M Φ2 (A) ∈ J for every set A ∈ S. It will be denoted by Φ1 ≈ Φ2 . It is evident that lower density operators Φ1 , Φ2 on (X, S, J ) are equivalent. By Theorem 25.3 and Theorem 25.12 we have the following property. Property 25.15. If Φ1 , Φ2 are lower density operators on (X, S, J ) and Φ1 generates topology TΦ1 on X, then Φ2 generates topology TΦ2 on X such that B(TΦ1 ) = B(TΦ2 ) = S and M(TΦ1 ) = M(TΦ2 ) = J .

438

Jacek Hejduk, Renata Wiertelak

Let us assume that hX, τi is an arbitrary topological space and TΦ is a topology generated by a lower density operator Φ on (X, S, J ) such that τ ⊂ TΦ . In real analysis two kinds of continuity are considered: topological and restrictive. Definition 25.16. We shall say that f : hX, TΦ i → hR, Tnat i is topologically continuous at x0 ∈ X if ∀ ε>0



V ∈TΦ ,x0 ∈V

V ⊂ {x ∈ X : | f (x) − f (x0 )| < ε}.

Definition 25.17. We shall say that f : hX, TΦ i → hR, Tnat i is restrictively continuous at x0 ∈ X if there exists a set A ∈ S such that x0 ∈ Φ(A) ∩ A and f|A is τ-continuous at x0 . Property 25.18. If f : hX, TΦ i → hR, Tnat i is restrictively continuous at x0 ∈ X, then f is topologically continuous at x0 . Proof. Let x0 ∈ X and A ∈ S be a set such that x0 ∈ Φ(A) and f|A is τ-continuous at x0 . Fix ε > 0. There exists a set V ∈ τ such that x0 ∈ A ∩ V and A ∩ V ⊂ {x ∈ X : | f (x) − f (x0 )| < ε}. Putting W = A ∩ Φ(A) ∩ V we obtain W ∈ TΦ , x0 ∈ W and W ⊂ {x ∈ X : | f (x) − f (x0 )| < ε}. t u It is worth mentioning that in the case of real line there exists characterization of equivalence of restrictive and topological continuity. Theorem 25.19 (cf. [16]). Let TΦ be a topology generated by a lower density operator Φ on (R, S, J ) such that Tnat ⊂ TΦ , f : hR, TΦ i → hR, Tnat i and x0 ∈ R. Then the following conditions are equivalent: a) f is TΦ -topologically continuous at x0 if and only if it is restrictively continuous at x0 ; T b) for every descending sequence {En }n∈N ⊂ S such that x0 ∈ ∞ n=1 Φ(En ) there exists a sequence {rn }n∈N ⊂ R+ , rn & 0 such that ! x0 ∈ Φ

∞ [

(En \ (x0 − rn , x0 + rn )) .

n=1

Remark 25.20 (cf. [29]). In the topological space hR, TI i, where TI is I-density topology, the topological and restrictive continuity are not equivalent. Theorem 25.21. Let hX, τi be a topological space and TΦ be a topology generated by a lower density operator Φ on (X, B(τ), M(τ)) such that τ ⊂ TΦ . For every function f : X → R the following conditions are equivalent:

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a) f has the Baire property with respect to τ; b) there exists a set A ∈ M(τ) such that for every x ∈ X \ A function f is TΦ -restrictively continuous at x; c) there exists a set A ∈ M(τ) such that for every x ∈ X \ A function f is TΦ -topologically continuous at x. Proof. a) ⇒ b) Let f be a function with the Baire property. There exists a set A ∈ M(τ) such that f|X\A is τ-continuous. Then for every x ∈ X \ A we have x ∈ Φ(X \ A) = X and f is restrictively continuous at x. By Property 25.18 implication b) ⇒ c) holds. c) ⇒ a) We show that E = f −1 ((a, b)) ∈ B(τ) for every a, b ∈ R, such that a < b. Let C be the set of TΦ -topological continuity points of f . Obviously E = (E ∩C) ∪ (E \C) and E \C ∈ M(τ). Let z ∈ E ∩C and ε > 0 be such that ε < min{b − f (z), f (z) − a}. Then there exists a set Vz ∈ TΦ such that z ∈ Vz ⊂ {x ∈ X : | f (x) − f (z)| < ε}. Putting Vz0 = Vz ∩C we obtain z ∈ Vz0 and Vz0 ∈ TΦ . S Hence E ∩C = z∈E∩C Vz0 ∈ TΦ ⊂ B(τ). Therefore E = f −1 ((a, b)) ∈ B(τ). t u

25.3 The case of almost lower density operators Let (X, S, J ) be a measurable space, where X is a nonempty set, S be a σ -algebra of subsets of X and J ⊂ S a proper σ -ideal. Definition 25.22. We shall say that an operator Φ : S → 2X is an almost lower density operator on measurable space (X, S, J ) if (i) Φ(0) / = 0, / Φ(X) = X; (ii) ∀A,B∈S Φ(A ∩ B) = Φ(A) ∩ Φ(B); (iii) ∀A,B∈S A M B ∈ J ⇒ Φ(A) = Φ(B); (iv) ∀A∈S Φ(A) \ A ∈ J . It is worthwhile noting that in the above definition instead of a σ -algebra S we can consider a family S closed under finite intersections such that 0/ ∈ S, X ∈ S and J ⊂ S. The next theorem follows by the same method as in proof of sufficient condition of Theorem 25.6. Theorem 25.23. Let Φ be an almost lower density operator on (X, S, J ). If the pair (S, J ) has the hull property, then the family TΦ = {A ∈ S : A ⊂ Φ(A)} is a topology on X.

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We will say that TΦ is a topology generated by the almost lower density operator Φ on (X, S, J ). The following example shows that the inverse of the last theorem does not hold. Example 25.24. Let (R, Bor, J ) be a measurable space, where J denotes the σ -ideal of countable sets. Putting  R, if R \ A ∈ J , ∀A∈S Φ(A) = 0, / if R \ A ∈ / J, we obtain the topology TΦ = {A ∈ Bor : A = 0/ ∨ R \ A ∈ J } but the pair (Bor, J ) does not possesses the hull property. Example 25.25. Let Φ1 , Φ2 be the almost lower density operators on (R, L, N ) defined in the following way  R, if R \ A ∈ N , ∀A∈L Φ1 (A) = 0, / if R \ A ∈ / N,  R, if R \ A ∈ N , ∀A∈L Φ2 (A) = Φd (A) ∩ B, if R \ A ∈ / N, where B is a Bernstein set. Then Φ1 6= Φ2 but TΦ1 = TΦ2 = {A ∈ L : A = 0/ ∨ R \ A ∈ N }. It means that the analogue of Theorem 25.12 in the case of almost lower density operators is not true. It turns out that Φ1 and Φ2 are not equivalent. However, we have the following property. Property 25.26. Let Φ1 , Φ2 be the almost lower density operators on (X, S, J ). Then the equality TΦ1 = TΦ2 implies that Φ1 (A) M Φ2 (A) ∈ J for every A ∈ TΦ1 . Proof. Let A ∈ TΦ1 = TΦ2 . Then Φ1 (A) = A ∪ (Φ1 (A) \ A) and Φ2 (A) = A ∪ (Φ2 (A) \ A). Hence Φ1 (A) M Φ2 (A) ∈ J for every A ∈ TΦ1 . t u Below there are examples of topologies generated by the almost lower density operators on the real line: • ψ-density topology (M. Terepeta, E. Wagner-Bojakowska, (1999), see [25]);

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• density topology with respect to the O’Malley points (W. Poreda, W. Wilczy´nski (2001), see [23]); • density topology with respect to measure and category (J. Hejduk (2002), see [10]); • complete density topology (W. Wilczy´nski, W. Wojdowski (2007), see [32]); • f -density topology (M. Filipczak, T. Filipczak, (2008), see [3]); • f -symmetrical density topology (J. Hejduk (2008), see [9]); • simple density topology (V. Aversa, W. Wilczy´nski (2004), see [1]); • density topology in the aspect of measure with respect to a sequence tending to zero (J. Hejduk, R. Wiertelak (2012), see [13]). The same ideas as the ones used in proofs of Theorem 25.3 and Theorem 25.9 allow us to prove the next theorem. Theorem 25.27. If Φ is an almost lower density operator on (X, S, J ) generating topology TΦ , then the following conditions are satisfied: a) if A ∈ J , then A is TΦ -closed and TΦ -nowhere dense; b) if A ∈ J , then A is TΦ -closed and TΦ -discrete; c) J ⊂ M(TΦ ); d) if J = M(TΦ ), then B(TΦ ) ⊂ S; e) intTΦ (A) ⊂ A ∩ Φ(A) for every A ∈ S. Moreover, if J contains all singletons, then f) A ∈ J if and only if A is TΦ -closed and TΦ -discrete; g) A is TΦ -compact if and only if A is finite; h) hX, TΦ i is neither a first countable, nor a second countable, nor a separable space; i) if J contains an uncountable set, then hX, TΦ i is not a Lindelöf space; j) every sequence consisting of different terms of X does not contain TΦ -convergent subsequence. Remark 25.28. Topology TΨ obtained by the almost lower density operator ΦΨ on measurable space (R, L, N ) contains TΨ -closed set and TΨ -nowhere dense set from L \ N (see [25]). Hence the inverse properties to a) and the equality in c) do not hold. Moreover, M(TΨ ) = 2R (see [26]). Therefore hR, TΨ i is not a Baire space. Simultaneously it is not true that M(TΨ ) = N nor B(TΨ ) ⊂ L. So, assumption in condition d) is necessary. There is an example (in [27]) of a set A ∈ L such that intTΨ (A) 6= A ∩ ΦΨ (A), so inclusion in e) can be proper.

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Remark 25.29. If X 6= 0, / S = 2X , J = {0}, / then the operator Φ(A) = A is the almost lower density operator on (X, S, J ) such that TΦ is the discrete topology. Then it is clear that inverse of b) is not satisfied. Remark 25.30. Let us consider a lower density operator Φ on (R, L, N ) given by formula:  R, if R \ A ∈ N , ∀A∈L Φ(A) = 0, / if R \ A ∈ / N. Then TΦ = {A ⊂ R : A = 0/ ∨ R \ A ∈ N } is a topology generated by Φ such that M(TΦ ) = N . Evidently B(TΦ ) ⊂ L and the inverse inclusion is not true. Hence inclusion in d) can be proper. Theorem 25.31. If Φ1 , Φ2 are equivalent almost lower density operators on (X, S, J ) and TΦ1 , TΦ2 topologies on X generated by Φ1 , Φ2 respectively, then 1◦ M(TΦ1 ) = M(TΦ2 ); 2◦ B(TΦ1 ) = B(TΦ2 ). First we need the following lemma. Lemma 25.32. If Φ1 , Φ2 are equivalent almost lower density operators on (X, S, J ), then for every set A ∈ TΦ1 there exists a set E ∈ J such that A \ E ∈ TΦ2 . Proof. Let A ∈ TΦ1 , hence A ⊂ Φ1 (A). Put E = Φ1 (A) M Φ2 (A). Then E ∈ J and A ⊂ Φ2 (A) M E. Thus A ⊂ Φ2 (A) ∪ E. This implies that A \ E ⊂ Φ2 (A) = Φ2 (A \ E). Therefore A \ E ∈ TΦ2 . t u Proof of Theorem 25.31. First we show that the families of nowhere dense sets with respect to topologies TΦ1 and TΦ2 are equal. Let A be a nowhere dense set with respect to TΦ1 and V2 nonempty TΦ2 -open set. By Lemma 25.32 there exists a set E1 ∈ J such that V2 \ E1 ∈ TΦ1 . Moreover, there exists a nonempty set V1 ∈ TΦ1 such that V1 ⊂ V2 \E1 and A∩V1 = 0. / By Lemma 25.32 there exists a set E2 ∈ J such that V1 \ E2 ∈ TΦ2 . Obviously V1 \ E2 6= 0. / Thus V1 \ E2 ⊂ V2 and A ∩ (V1 \ E2 ) = 0. / Hence A is a nowhere dense set with respect to TΦ2 . In a similar way we prove the inverse inclusion. Condition 1◦ is now an immediate consequence of the equality of the families of nowhere dense sets with respect to topologies TΦ1 and TΦ2 . Now we prove that B(TΦ1 ) = B(TΦ2 ). Suppose that A ∈ B(TΦ1 ). Then A = V M Y , where V ∈ TΦ1 and Y ∈ M(TΦ1 ). By Lemma 25.32 there exists a set E1 ∈ J such that V \ E1 ∈ TΦ2 . Hence A = [(V \ E1 ) ∪ (V ∩ E1 )] M Y .

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Since V \ E1 ∈ TΦ2 and (V ∩ E1 ) ∈ J ⊂ M(TΦ2 ) and Y ∈ M(TΦ2 ), we have A ∈ B(TΦ2 ). In a similar way we prove that B(TΦ2 ) ⊂ B(TΦ1 ). t u In the further consideration we apply the following proposition. Proposition 25.33 (cf. 1.14 in [10]). Let Φ be the almost lower density operators on (X, S, J ). Then the following conditions are equivalent: 1. ∀A∈S 2. ∀A∈S

A \ Φ(A) ∈ J ; A M Φ(A) ∈ J .

Theorem 25.34. If Φ is an almost lower density operator on (X, S, J ) generating topology TΦ , then (i) M(TΦ ) = J if and only if there exists a σ -algebra S 0 ⊂ S such that Φ 0 = Φ|S 0 is a lower density operator on (X, S 0 , J ) and TΦ = TΦ 0 ; (ii) M(TΦ ) = J and B(TΦ ) = S if and only if Φ is a lower density operator on (X, S, J ). Proof. (i) Necessity. Let S 0 = B(TΦ ). Then J ⊂ S 0 ⊂ S. It is sufficient to prove that Φ 0 = Φ|S 0 is a lower density operator on (X, S 0 , J ). For every A ∈ S 0 we have A = B M C, where B ∈ TΦ , C ∈ J . Hence Φ 0 (A) = Φ(A) = Φ(B) ⊃ B. It follows that A \ Φ 0 (A) ⊂ A \ B ∈ J . By Proposition 25.33 we conclude that Φ 0 (A) \ A ∈ J . So Φ 0 (A) M A ∈ J and Φ 0 is a lower density operator on (X, S 0 , J ). The equality TΦ = TΦ 0 is obvious. Sufficiency. If Φ 0 is a lower density operator on (X, S 0 , J ) generating topology TΦ 0 , then by Theorem 25.3, M(TΦ 0 ) = J . Since M(TΦ ) = M(TΦ 0 ), we get J = M(TΦ ). (ii) Necessity. If M(TΦ ) = J , then Φ|S 0 , where S 0 = B(TΦ ), is a lower density operator on (X, S 0 , J ). Since S = S 0 , we have Φ is a lower density operator on (X, S, J ). Sufficiency. If Φ is a lower density operator on (X, S, J ), then by Theorem 25.3 we obtain M(TΦ ) = J and B(TΦ ) = S. t u Example 25.35. Let Y ⊂ (0, 1) and Y ∈ / L. If we put  R, if R \ A ∈ N , ∀A∈L Φ(A) = Φd (A) ∩Y, if R \ A ∈ / N, then we obtain an almost lower density operator on (R, L, N ) generating topology TΦ = {A ⊂ R : R \ A ∈ N } ∪ (Td ∩ 2Y ) but Φ((0, 1)) = Φd ((0, 1)) ∩Y ∈ / L.

444

Jacek Hejduk, Renata Wiertelak

In the context of the above observation we see the range of an almost density operator can be wider then the σ -algebra S. However we have the following theorem. Theorem 25.36. If Φ is an almost lower density operator on (X, S, J ), then there exists a subfamily R ⊂ S such that J ⊂ R, X ∈ R, R is closed under finite intersections and an almost lower density operator Φ 0 : R → S on (X, R, J ) such that TΦ 0 = {A ∈ R : A ⊂ Φ 0 (A)} = TΦ . Proof. Let R = {A ∈ S : Φ(A) ∈ S}. Then 0, / X ∈ R, J ⊂ R and R is closed 0 under finite intersections. Let Φ = Φ|R be the restriction of Φ to the family R. It is clear that Φ 0 : R → S is an almost lower density operator on (X, R, J ). We show that TΦ 0 = TΦ . It is sufficient to show that TΦ ⊂ TΦ 0 . If A ∈ TΦ , then A ∈ S and A ⊂ Φ(A). Since Φ(A)\A ∈ J , we have Φ(A) = A∪(Φ(A)\A) ∈ S and A ∈ R. From the inclusion A ⊂ Φ 0 (A) we obtain A ∈ TΦ 0 . t u Theorem 25.37 (cf. [11]). Let hX, τi be a topological space such that X∈ / M(τ) and Φ be an almost lower density operator on (X, B(τ), M(τ)). Then Φ generates topology TΦ . Moreover, if there exists a τ-dense set D ∈ M(τ) and τ ⊂ TΦ , then the topological space hX, TΦ i is not regular. Proof. Since the pair (B(τ), M(τ)) has the hull property, then by Theorem 25.23 TΦ = {A ∈ B(τ) : A ⊂ Φ(A)} is a topology on X. Let D ∈ M(τ) be a τ-dense set and τ ⊂ TΦ . First we prove that if D ⊂ W ∈ TΦ , then X \W ∈ M(τ). Evidently W = B M C, where B ∈ τ, C ∈ M(τ). We show that B is τ-dense. Suppose that there exists a nonempty set E ∈ τ such that B ∩ E = 0. / By the assumption that τ ⊂ TΦ we have W ∩ E ∈ TΦ and thus W ∩ E ⊂ Φ(W ∩ E) = Φ(B ∩ E) = 0. / Hence W ∩ E = 0/ and it contradicts the fact that W is τ-dense. Thus B is τ-dense and τ-open. Therefore X \ B ∈ M(τ) and X \W ∈ M(τ). Taking a TΦ -closed set D and a point x0 ∈ X \ D we see that for TΦ -open and disjoint sets W and V such that D ⊂ W and x0 ∈ V we get X \W ∈ M(τ) and V ∈ / M(τ). So W ∩ V 6= 0. / This contradiction forces us to conclude that the topological space hX, TΦ i is not regular. t u Corollary 25.38. If Φ is an almost lower density operator on (R, Bor, M) such that Tnat ⊂ TΦ , then the topological space hR, TΦ i is not regular.

25. On the abstract density topologies

445

Now we will concentrate on the topologies generated by almost lower density operators invariant with respect to translation or multiplication. Definition 25.39. We shall say that a topology τ on R is invariant with respect to translation (multiplication) if  ∀A∈τ ∀t∈R A + t ∈ τ ∀A∈τ ∀α∈R\{0} αA ∈ τ . Definition 25.40. We shall say that a measurable space (R, S, J ) is invariant with respect to translation (multiplication) if  i) ∀A∈S ∀t∈R A + t ∈ S ∀A∈S ∀α∈R\{0} αA ∈ S , ii) ∀A∈J ∀t∈R A + t ∈ J ∀A∈J ∀α∈R\{0} αA ∈ J , Definition 25.41. We shall say that an almost lower density operator Φ on an invariant with respect to translation (multiplication) measurable space (R, S, J ) is invariant with respect to translation (multiplication) if  ∀A∈S ∀t∈R Φ(A+t) = Φ(A)+t ∀A∈S ∀α∈R\{0} Φ(αA) = αΦ(A) . Evidently, we have the following property. Property 25.42. If Φ is an almost lower density operator invariant with respect to translation (multiplication) on a measurable space (R, S, J ) invariant with respect to translation (multiplication) generating topology TΦ , then TΦ is invariant with respect to translation (multiplication). Property 25.43. Let (R, S, J ) be a measurable space invariant with respect to translation and multiplication. If Φ is an almost lower density operator on (R, S, J ) invariant with respect to translation generating topology TΦ , then TΦ is invariant with respect to translation and, moreover, TΦ is invariant with respect to multiplication if and only if ∀A∈TΦ

∀α∈R\{0}

(0 ∈ A =⇒ 0 ∈ Φ(αA)) .

Proof. By the previous property TΦ is invariant with respect to translation. Let α ∈ R \ {0} and A ∈ TΦ . We show that αA ∈ TΦ . Let y ∈ αA, then y/α ∈ A ⊂ Φ(A). Hence 0 ∈ A − y/α. By assumption 0 ∈ Φ(αA − y) = Φ(αA) − y. Therefore y ∈ Φ(αA) and it follows that αA ⊂ Φ(αA). Since αA ∈ S, we obtain αA ∈ TΦ . Now assume that TΦ is invariant with respect to multiplication. Suppose that ∃A∈TΦ

∃α∈R\{0}

(0 ∈ A ∧ 0 ∈ / Φ(αA)) .

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Since αA ∈ TΦ , then αA ⊂ Φ(αA). Moreover, 0 ∈ Φ(αA) because 0 ∈ αA. This contradiction completes the proof. t u

References [1] V. Aversa, W. Wilczy´nski, Simple density topology, Rend. Circ. Mat. Palermo, serie II, 53(3) (2004), 344-352. [2] K. Ciesielski, L. Larson, K. Ostaszewski, I -Density Continuous Functions, Mem. Amer. Math. Soc. 107(515), 1994. [3] M. Filipczak, T. Filipczak, On f -density topologies, Topology Appl. 155 (2008), 19801989. [4] M. Filipczak, J. Hejduk, On topologies associated with the Lebesgue measure, Tatra Mt. Math. Publ. 28 (2004), 187-197. [5] C. Goffman, C. J. Neugebauer, T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497-505. [6] C. Goffman, D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116-121. [7] T. R. Hamlett, D. Jankovi´c, D. A. Rose, Lower density topologies, Ann. New York Acad. Sci., Papers on General Topology and Applications 704 (1993), 309–321. [8] O. Haupt, C. Pauc, La topologie de Denjoy approximative envisagée comme vraie topologie, C. R. Acad. Sci. Paris 234 (1952), 390-392. [9] J. Hejduk, On density topologies generated by functions, Tatra Mt. Math. Publ. 40 (2008), 133-141. [10] J. Hejduk, On density topologies with respect to invariant σ -ideals, J. Appl. Anal. 8(2) (2002), 201-219. [11] J. Hejduk, On the regularity of topologies in the family of sets having the Baire property, Filomat (submitted). [12] J. Hejduk, G. Horbaczewska, On I -density topologies with respect to a fixed sequence, Reports on Real Analysis (Rowy), (2003), 78-85. [13] J. Hejduk, R. Wiertelak, On the generalization of the density topology on the real line, Mathematica Slovaca (to appear). [14] E. W. Hobson, The theory of functions of a real variable and the theory of Fourier’s series, vol. I. Dover Publications, Inc., New York, 1958. [15] G. Horbaczewska, The family of I -density type topologies, Comment. Math. Univ. Carolinae 46(4) (2005), 735-745. [16] J. M. Je¸drzejewski, On limit numbers of real functions, Fund. Math. 83(3) (1973/1974), 269-281. [17] K. Kuratowski, Topology, vol 1, Polish Scientific Publications, Warsaw, 1966. [18] J. Lukeš, J. Malý, L. Zajiˇcek, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Math. 1189, Springer-Verlag, Berlin 1986. [19] E. Łazarow, A. Vizvary, ΨI -density topology, Pr. Nauk Akad. Jana Długosza Cze¸st. Mat. 15 (2010), 67-80. [20] D. Maharam, On a theorem of von Neumann, Proc. Amer. Math. Soc. 9 (1958), 987–994. [21] J. C. Oxtoby, Measure and category, Springer-Verlag, Berlin 1980.

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[22] W. Poreda, E. Wagner-Bojakowska, W. Wilczy´nski, A category analogue of the density topology, Fund. Math. 125(2) (1985), 167-173. [23] W. Poreda, W. Wilczy´nski, Topology similar to the density topology, Bull. Soc. Sci. Lett. Łód´z, 34 (2001), 55-60. [24] F. D. Tall, The density topology, Pacific. J. Math. 62(1) (1976), 275-284. [25] M. Terepeta, E. Wagner-Bojakowska, ψ-density topology, Rend. Circ. Mat. Palermo, Serie II 48(3) (1999), 451-476. [26] E. Wagner-Bojakowska, Remarks on ψ-density topology, Atti Sem. Mat. Fis. Univ. Modena 49(1) (2001), 79-87. [27] E. Wagner-Bojakowska, W. Wilczy´nski, The union of ψ-density topologies, Atti Sem. Mat. Fis. Univ. Modena 50(2) (2002), 313-326. [28] R. Wiertelak, A generalization of density topology with respect to category, Real Anal. Exchange 32(1) (2006/2007), 273-286. [29] W. Wilczy´nski, A category analogue of the density topology, approximate continuity and the approximate derivative, Real Anal. Exchange 10(2) (1984-85), 241-265. [30] W. Wilczy´nski, Density topologies, Chapter 15 in Handbook of Measure Theory, Ed. E. Pap. Elsevier (2002), 675-702. [31] W. Wilczy´nski, W. Wojdowski, A category ψ-density topology, Cent. Eur. J. Math. 9(5) (2011), 1057-1066. [32] W. Wilczy´nski, W. Wojdowski, Complete density topology, Indag. Math., New series, 18(2) (2007), 295-303. [33] W. Wojdowski, Density topologies involving measure and category, Demonstratio Math. 22(3) (1989), 797-812.

JACEK H EJDUK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

R ENATA W IERTELAK Faculty of Mathematics and Computer Science, Łód´z University ul. Banacha 22, 90-238 Łód´z, Poland E-mail: [email protected]

Chapter 26

Path continuity connected with the notion of density

STANISŁAW KOWALCZYK, KATARZYNA NOWAKOWSKA

2010 Mathematics Subject Classification: 26A15, 54C30. Key words and phrases: density of a set at a point, lower density, upper density, continuous functions, approximately continuous functions, path continuity.

26.1 Preliminaries A. M. Bruckner, R. J. O’Malley and B. S.Thomson in [4] investigated the notion of a system of paths and studied a number of generalized derivatives. Properties of path continuous functions was intensively studied in [2], [6], [7], [8], [9], [10], [11], [13]. Similar approach to the notion of continuity was used in [5], [15]. We use this idea of path continuity for studying some notions of generalized continuity connected with density of a set at a point. Some basic properties of these classes of functions are presented. First, we shall collect some of the notions and definitions which appear frequently in the sequel. The symbol λ ∗ (E) denotes the Lebesgue outer measure of E ⊂ R. In the whole paper we consider only real-valued functions defined on an open interval I = (a, b). Let E be a measurable subset of R and let x ∈ R. According to [3], the numbers λ (E∩[x,x+t]) d + (E, x) = lim inf t + t→0

450

and

Stanisław Kowalczyk, Katarzyna Nowakowska

+

d (E, x) = lim sup λ (E∩[x,x+t]) t t→0+

are called the right lower density of E at x and right upper density of E at x. The left lower and upper densities of E at x are defined analogously. If +

d + (E, x) = d (E, x)

or



d − (E, x) = d (E, x),

then we call these numbers the right density and left density of E at x, respectively. The numbers d(E, x) = lim sup t→0+ k→0+

and d(E, x) = lim inf + t→0 k→0+

λ (E ∩ [x − t, x + k]) k +t

λ (E ∩ [x − t, x + k]) k +t

are called the upper and lower density of E at x respectively. If d(E, x) = d(E, x), we call this number the density of E at x and denote it by d(E, x). It is clear that E has the density at x if and only if all four one-sided densities are equal. When d(E, x) = 1, we say that x is a point of density of E. Definition 26.1. [7] Let E be a measurable subset of R and x ∈ R. 1. For 0 < ρ < 1 we say that x is a point of ρ-type upper density of E if d(E, x) > ρ. 2. We say that x is a point of 1-type upper density of E if d(E, x) = 1. Definition 26.2. [7] A real-valued function f defined on an open interval I is called ρ-upper continuous at x provided that there is a measurable set E ⊂ I such that the point x is a point of ρ-type upper density of E, x ∈ E and f|E is continuous at x. If f is ρ-upper continuous at every point of I we say that f is ρ-upper continuous. We will denote the class of all ρ-continuous functions defined on an open intervals I by UC ρ . Definition 26.3. [13] Let E be a measurable subset of R and x ∈ R. 1. For 0 < Λ ≤ ρ < 1 we say that x is a point of [Λ , ρ]-density of E if d(E, x) > ρ and d(E, x) > Λ .

26. Path continuity connected with the notion of density

451

2. For 0 < Λ < 1 we say that x is a point of [Λ , 1]-density of E if d(E, x) = 1 and d(E, x) > Λ . 3. We say that x is a point of [1, 1]-density of E if d(E, x) = d(E, x) = 1. Definition 26.4. [8], [13] Let 0 < λ ≤ ρ ≤ 1. A real-valued function f defined on an open interval I is called [Λ , ρ]-continuous at x ∈ I provided that there is a measurable set E ⊂ I such that x is a point of [Λ , ρ]-density of E, x ∈ E and f|E is continuous at x. If f is [Λ , ρ]-continuous at each point of I we say that f is [Λ , ρ]-continuous. We will denote the class of all [Λ , ρ]-continuous functions by C[Λ ,ρ] . It is clear that C[1,1] is exactly the class of approximately continuous functions. Sometimes density of a set at a point is defined in other, symmetric, way. According to, for example [14], the lower density of E at x and upper density of λ (E∩[x−t,x+t]) and lim sup λ (E∩[x−t,x+t]) , respectively. E at x are defined as lim inf 2t 2t + t→0

t→0+

+

+

We will denote these densities by s-d (E, x) and s-d (E, x), respectively. If s-d(E, x) = s-d(E, x) then we call this number the symmetric density of E at x and denote it by s-d(E, x). Corollary 26.5. For each measurable E ⊂ R and x ∈ R we have 1. s-d(E, x) ≤ d(E, x), 2. s-d(E, x) ≥ d(E, x), 3. x is a point of the density of E, if and only if x is a point of symmetric density of E. Definition 26.6. Let E be a measurable subset of R and x ∈ R. 1. For 0 < ρ < 1, then we say that x is a point of sρ-type upper density of E if s-d(E, x) > ρ. 2. We say that x is a point of s-1-type upper density of E if s-d(E, x) = 1. Definition 26.7. A real-valued function f , defined on an open interval I, is called sρ-upper continuous at x, provided that there is a measurable set E ⊂ I such that the point x is a point of sρ-type upper density of E, x ∈ E and f|E is continuous at x. If f is sρ-upper continuous at every point of I, we say that f is sρ-upper continuous. We will denote the class of all ρ-continuous functions defined on an open intervals I by sUC ρ . Definition 26.8. Let E be a measurable subset of R and x ∈ R.

452

Stanisław Kowalczyk, Katarzyna Nowakowska

1. For 0 < Λ ≤ ρ < 1 we say that x is a point of s-[Λ , ρ]-density of E if s-d(E, x) > ρ and s-d(E, x) > Λ . 2. For 0 < Λ < 1 we say that x is a point of s-[Λ , 1]-density of E if s-d(E, x) = 1 and s-d(E, x) > Λ . 3. We say that x is a point of s-[1, 1]-density of E if s-d(E, x) = s-d(E, x) = 1. Definition 26.9. Let 0 < Λ ≤ ρ ≤ 1. A real-valued function f defined on an open interval I is called s-[Λ , ρ]-continuous at x ∈ I, provided that there is a measurable set E ⊂ I such that x is a point of s-[Λ , ρ]-density of E, x ∈ E and f|E is continuous at x. If f is s-[Λ , ρ]-continuous at each point of I, we say that f is s-[Λ , ρ]-continuous. We will denote the class of all s-[Λ , ρ]-continuous functions by sC[Λ ,ρ] . Corollary 26.10. Cap ⊂ C[Λ ,ρ] ∩ sC[Λ ,ρ] for each 0 < Λ ≤ ρ ≤ 1 and Cap = C[1,1] = sC[1,1] .

26.2 Basic properties Lemma 26.11. Let 0 < Λ ≤ ρ ≤ 1. The following inclusions are obvious. 1. C[Λ ,ρ] ⊂ U C ρ , 2. sC[Λ ,ρ] ⊂ sUC ρ , 3. sC[Λ ,ρ] ⊂ U C ρ , 4. sUC ρ ⊂ U C ρ . Theorem 26.12. Let 0 < ρ ≤ 1. Then each function f from UC ρ is measurable. Proof. Assume that there exists f ∈ U C ρ which is not measurable. Then we can find a number a ∈ R for which at least one of the sets {x ∈ I : f (x) < a}, {x ∈ I : f (x) > a} is nonmeasurable. Without loss of generality we may assume that the set {x ∈ I : f (x) < a} is nonmeasurable. Denote A = {x ∈ I : f (x) < a},

B = {x ∈ I : f (x) ≥ a}.

It is obvious that B = I \ A is also nonmeasurable. Consider a measurable sets A1 ⊂ A, B1 ⊂ B such that A \ A1 and B \ B1 do not contain a measurable set of positive measure. Therefore A \ A1 and B \ B1 are nonmeasurable sets. Let F = (A \ A1 ) ∪ (B \ B1 ) = I \ (A1 ∪ B1 ).

26. Path continuity connected with the notion of density

453

Then F is measurable. Let Φd (F) be a set of all density points of F. By the well-known Lebesgue Density Theorem, λ (F \ Φd (F)) = 0. Therefore there exists x0 ∈ (A \ A1 ) ∩ Φd (F). Since f is ρ-upper continuous at x0 , it follows that there exists a measurable set E ⊂ R such that x0 ∈ E, d(E, x0 ) > ρ and f|E is continuous at x0 . Since x0 ∈ A, we have f (x0 ) < a. Therefore it is possible to find δ > 0 such that E ∩ (x0 − δ , x0 + δ ) ⊂ A. Let E 0 = E ∩ (x0 − δ , x0 + δ ). Hence x0 ∈ E 0 , f|E 0 is continuous at x0 , E 0 ⊂ A and d(E 0 , x0 ) = d(E, x0 ) > ρ > 0.

(26.1)

We have E 0 = (E 0 ∩ A1 ) ∪ (E 0 ∩ (A \ A1 )). Since E 0 and E 0 ∩ A1 are measurable, E 0 ∩ (A \ A1 ) is measurable, too. Hence λ (E 0 ∩ (A \ A1 )) = 0. Moreover, d(E 0 ∩ A1 , x0 ) = 1 − d(R \ (E 0 ∩ A1 ), x0 ) ≤ 1 − d(F, x0 ) = 1 − 1 = 0, because (E 0 ∩ A1 ) ∩ F = 0. / It follows that d(E 0 , x0 ) ≤ d(E 0 ∩ A, x0 ) + d(E 0 ∩ (A \ A1 ), x0 ) = 0 + 0 = 0, contradicting (26.1). Thus the assumption that f may be nonmeasurable is false. t u Corollary 26.13. All considered classes of functions C[Λ ,ρ] , sC[Λ ,ρ] , UC ρ and sUC ρ consist of Lebesgue measurable functions. Lemma 26.14. Let 0 < ρ ≤ 1, x ∈ R and let {En : n ∈ N} be a decreasing family of measurable sets such that d(En , x) ≥ ρ for n ≥ 1. Then there exists a measurable set E such that d(E, x) ≥ ρ and for each n ∈ N there exists δn > 0 for which E ∩ [x − δn , x + δn ] ⊂ En . +



Proof. By assumptions, d (En , x) ≥ ρ or d (En , x) ≥ ρ for each n. Hence + there exists an infinite sequence {Enk : k ∈ N} such that d (Enk , x) ≥ ρ for all − k ≥ 1 or d (Enk , x) ≥ ρ for all k ≥ 1. Without loss of generality we may as+ sume that the first possibility occurs. Then d (En , x) ≥ ρ for all n ≥ 1, because {En : n ∈ N} is a decreasing family.  1 ]) Let x1 > x be any point for which λ (Ex11∩[x,x > ρ 1 − 12 and x1 − x < 1. −x Next, we can find x < x2 < x1 such that     λ (E1 ∩ [x2 , x1 ]) 1 λ (E2 ∩ [x, x2 ]) 1 1 >ρ 1− , >ρ 1− and x2 −x < . x1 − x 2 x2 − x 4 2

454

Stanisław Kowalczyk, Katarzyna Nowakowska

∩[xi ,xi−1 ]) Assume that points x1 , x2 , . . . , xn are chosen, x < xn < . . . < x1 , λ (Ei−1 > xi−1 −x   λ (En ∩[x,xn ]) 1 1 1 ρ 1 − 2i−1 for i = 2, . . . , n, > ρ 1 − 2n and xn − x < n . Then there xn −x exists x < xn+1 < xn such that     1 1 λ (En ∩ [xn+1 , xn ]) λ (En+1 ∩ [x, xn+1 ]) > ρ 1− n , > ρ 1 − n+1 xn − x 2 xn+1 − x 2 1 . and xn+1 − x < n+1 We have constructed inductively a decreasing sequence {xn }n≥1 such that

 λ (En ∩ [xn+1 , xn ]) 1 > ρ 1− n for n ≥ 1. xn − x 2 Let E =

∞ S

(26.2)

 En ∩ [xn+1 , xn ] ∪ {x}. Since

n=1

lim sup n→∞

  λ (E ∩ [x, xn ]) λ (En ∩ [xn+1 , xn ]) 1 ≥ lim sup ≥ lim ρ − n = ρ, n→∞ xn − x xn − x 2 n→∞

we have d(E, x) ≥ ρ. By definition of E, for each n there exists δn = xn − x > 0 such that E ∩ [x − δn , x + δn ] = E ∩ [x, xn ] ⊂ En . t u

The proof is complete. We will give a condition equivalent to ρ-upper continuity at a point x.

Theorem 26.15. Let 0 < ρ ≤ 1 and let f : I → R be a measurable function. Then f is ρ-upper continuous at x ∈ I if and only if  lim+ d {y ∈ I : | f (x) − f (y)| < ε}, x > ρ if 0 < ρ < 1 ε→0

or  d {y ∈ I : | f (x) − f (y)| < ε}, x = 1 for all ε > 0

if ρ = 1.

Proof. Assume that f is ρ-upper continuous at x. Let E ⊂ I be a measurable set such that x ∈ E, f|E is continuous at x and d(E, x) > ρ for ρ < 1, or d(E, x) = 1 if ρ = 1. Since f|E is continuous at x, for each ε > 0 we can find δ > 0 such that [x − δ , x + δ ] ∩ E ⊂ {y : | f (x) − f (y)| < ε}. Hence   d {y ∈ I : | f (x) − f (y)| < ε}, x ≥ d {y ∈ E : | f (x) − f (y)| < ε}, x = d(E, x)

26. Path continuity connected with the notion of density

455

for each ε > 0. Therefore  lim+ d {y ∈ I : | f (x) − f (y)| < ε}, x ≥ d(E, x) > ρ

if ρ < 1

ε→0

or  d {y ∈ I : | f (x) − f (y)| < ε}, x ≥ d(E, x) = 1

for each ε > 0,

if ρ = 1.

Finally, assume that  ρ1 = lim+ d {y ∈ I : | f (x) − f (y)| < ε}, x > ρ

if ρ < 1

ε→0

or  d {y ∈ I : | f (x) − f (y)| < ε}, x = 1 for each ε > 0

if ρ = 1.

Applying Lemma 26.14 for sets En = {y ∈ I : | f (x) − f (y)| < n1 }, we can construct a measurable set E such that x ∈ E, d(E, x) ≥ ρ1 > ρ

if

0 ρ 1 − 12 and δ1 < 1. Next, 2δ1 we can find δ2 ∈ (0, δ1 ) such that   1 λ (E1 ∩ ([x − δ1 , x − δ2 ] ∪ [x + δ2 , x + δ1 ])) > ρ 1− , 2δ1 2   λ (E2 ∩ [x − δ2 , x + δ2 ]) 1 > ρ 1− 2δ2 4 and δ2 < 12 . Assume that real positive numbers δ1 , δ2 , . . . , δn are chosen, δn < δn−1 < . . . < δ1 ,

456

Stanisław Kowalczyk, Katarzyna Nowakowska

  λ (Ei−1 ∩ ([x − δi−1 , x − δi ] ∪ [x + δi , x + δi−1 ]) 1 > ρ 1 − i−1 2δi−1 − x 2  n ,x+δn ]) > ρ 1 − 21n and δn < 1n . Then there exists for i = 2, . . . , n, λ (En ∩[x−δ 2δn  1 n+1 ]∪[x+δn+1 ,x+δn ]) > ρ 1 − δn+1 ∈ (0, δn ) such that λ (En ∩([x−δn ,x−δ2δ n , 2 n  λ (En+1 ∩[x−δn+1 ,x+δn+1 ]) 1 1 > ρ 1 − and δ < n+1 n+1 n+1 . 2δn+1 2 We have constructed inductively a decreasing sequence {δn }n≥1 of positive numbers such that for n ≥ 1  1 λ (En ∩ ([x − δn , x − δn+1 ] ∪ [x + δn+1 , x + δn ])) > ρ 1− n . (26.3) 2δn 2 Let E = {x} ∪

∞ S

 En ∩ ([x − δn , x − δn+1 ] ∪ [x + δn+1 , x + δn ]) . Since

n=1

λ (E ∩ [x − δn , x + δn ]) ≥ 2δn n→∞   λ (En ∩ ([x − δn , x − δn+1 ] ∪ [x + δn+1 , x + δn ])) 1 ≥ lim sup ≥ lim ρ − n = ρ, n→∞ 2δn 2 n→∞ lim sup

we have s-d(E, x) ≥ ρ. By definition of E, we have E ∩ [x − δn , x + δn ] = E ∩ [x, xn ] ⊂ En for each n. The proof is complete. t u Now, we can give a condition equivalent to sρ-upper continuity at a point x. Theorem 26.18. Let 0 < ρ ≤ 1 and let f : I → R be a measurable function. Then f is sρ-upper continuous at x ∈ I if and only if  lim+ s-d {y ∈ I : | f (x) − f (y)| < ε}, x > ρ if 0 < ρ < 1 ε→0

or  s-d {y ∈ I : | f (x) − f (y)| < ε}, x = 1 for all ε > 0

if ρ = 1.

Proof. The proof is analogous to the proof of Theorem 26.15. The unique difference is that we use Lemma 26.17 instead of Lemma 26.14. t u Corollary 26.19. \

sUC ρ = sUC 1 .

0 Λ − 1n b

for each 0 < a < b < εn . Proof. Fix any n ∈ N. Since d + (E, x) ≥ Λ , there exists εn ∈ (0, 1) such that λ (E∩[x,x+c]) 1 > Λ − 2n for each c ∈ (0, εn ). If 0 < a < b < εn , then c a a λ (E ∩ [x + 2n , x + b]) = λ (E ∩ [x, x + b]) − λ (E ∩ [x, x + 2n ]) ≥ 1 a ≥ b(1 − 2n ) − 2n > bΛ − nb .

Hence

a λ (E∩[x+ 2n ,x+b]) b

> Λ − 1n .

t u

Lemma 26.22. Let 0 < ρ ≤ 1 and let x ∈ R. Assume that E ⊂ R is measurable + and d (E, x) ≥ ρ. For every n ∈ N there exists decreasing sequence {αm }m∈N of positive reals converging to 0 such that λ (E ∩ [x + α2nm , x + αm ]) > ρ − 1n αm for each m ∈ N. +

Proof. Fix n ∈ N. Since d (E, x) ≥ ρ, there exists a decreasing sequence {βm }m∈N of positive reals such that lim

m→∞

λ (E ∩ [x, x + βm ]) + = d (E, x) ≥ ρ. βm

Then we can find m0 such that

λ (E∩[x,x+βm ]) βm

> ρ − n1 for all m ≥ m0 . Hence,

βm βm λ (E ∩ [x + 2m , x + βm ]) λ (E ∩ [x, x + βm ]) λ (E ∩ [x, x + 2m ]) ≥ − > ρ − 1n βm βm βm

for each m ≥ m0 . Then the sequence {αm }m∈N , where αm = βm+m0 for m ∈ N, has all the required properties. t u Lemma 26.23. Let 0 < Λ ≤ ρ ≤ 1 and let {En }n∈N be a decreasing sequence of measurable sets such that x ∈

∞ T

En , d + (En , x) ≥ Λ and d(En , x) ≥ ρ for all

n=1

n ∈ N. Then there exists a measurable set E such that d(E, x) ≥ Λ , d(E, x) ≥ ρ, x ∈ E and for each n ∈ N there exists δn > 0 for which E ∩ [x − δn , x + δn ] ⊂ En

458

Stanisław Kowalczyk, Katarzyna Nowakowska +

Proof. As in the proof of Lemma 26.14, we can assume that d (En , x) ≥ ρ for all n. By Lemma 26.21, for each n ∈ N there exists εn > such that a , x + b]) λ (En ∩ [x + 2n 1 >Λ− b n

for all 0 < a, b < εn . By Lemma 26.22, for each n ∈ N we can find decreasing sequence {αmn }m∈N such that n

λ (En ∩ [x + α2nm , x + αmn ]) 1 >ρ− n αm n for all m, n ∈ N. We will construct inductively a sequence of positive reals {an }n∈N such that for each n ∈ N 1. an < εn , an 2. an+1 < 2n , 3. there exists mn ∈ N for which

h

αmn n n 2n , αmn

i

⊂ [an+1 , an ].

Choose any a1 < ε1 . Assume that we have chosen a1 , . . . , an satisfying conditions 1) − 3). Then  we can find mn ∈ Nsuch that αmn n < an . Now, we can take an arbitrary an+1 ∈ 0, min{εn+1 , F=

∞ [

αmn n 2n }

. Put

(En ∩ [x + an+2 , x + an+1 ]).

n=1

Let y ∈ [x, x + a2 ], y = x + c . Then c ∈ [an+1 , an ] for some n. Since an+2 , we have

an+1 2n

F ∩ [an+2 , x + c] ⊃ (En ∩ [x + an+2 , x + an+1 ])∪   n+1 ∪ (En−1 ∩ [x + an+1 , x + c]) ⊃ En ∩ x + a2n ,x+c . Hence    n+1 λ (F ∩ [x, x + c]) ≥ λ En ∩ x + a2n , x + c > c(Λ − 1n ) and λ (F∩[x,x+c]) > Λ − 1n . Therefore d + (F, x) ≥ Λ . c On the other hand,   n+1 F ∩ [x, x + αmn n ] ⊃ En ∩ [x + an+2 , x + αmn n ] ⊃ En ∩ x + a2n , x + αmn n . Therefore

λ (F∩[x,x+αmn n ]) αmn n

> ρ − n1 for n ∈ N and

>

26. Path continuity connected with the notion of density +

d (F, x) ≥ lim sup n→∞

459

 λ (F ∩ [x, x + αmn n ]) ≥ lim sup ρ − n1 = ρ. n αmn n→∞

Finally, we can easily see that for each n ∈ N we can choose δn = an+1 for which E ∩ [x, x + δn ] ⊂ En . Similarly, we can construct a measurable set G ⊂ (−∞, x) such that d − (G, x) ≥ λ and for each n ∈ N, G ∩ [x − δn , x] ⊂ En for some positive δn . Then the set E = F ∪ G ∪ {x} has all the required properties. t u Theorem 26.24. Let 0 < Λ ≤ ρ < 1. A measurable function f : I → R is [Λ , ρ]continuous at x0 if and only if lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > ρ.

ε→0+

Proof. asasAssume that a measurable f is [Λ , ρ]-continuous at x0 . Then there exists measurable E ⊂ R such that x0 ∈ E, d(E, x0 ) > Λ , d(E, x0 ) > ρ and f|E is continuous at x0 . By the continuity of f at x0 , for each ε > 0 we can find δ > 0 for which [x0 − δ , x0 + δ ] ⊂ {x : | f (x) − f (x0 )| < ε}. Hence d({x : | f (x) − f (x0 )| < ε}, x0 ) ≥ d({x ∈ E : | f (x) − f (x0 )| < ε}, x0 ) = d(E, x0 ) and d({x : | f (x)− f (x0 )| < ε}, x0 ) ≥ d({x ∈ E : | f (x)− f (x0 )| < ε}, x0 ) = d(E, x0 ). Therefore lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > ρ.

ε→0+

Now assume that Λ1 = lim+ d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ ε→0

and ρ1 = lim+ d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > ρ. ε→0

Applying Lemma 26.23 for En = {x ∈ I : | f (x) − f (x0 )| < 1n }, we can find a measurable set E ⊂ R such that x0 ∈ E, d(E, x0 ) ≥ Λ1 > Λ , d(E, x0 ) ≥ ρ1 > ρ

460

Stanisław Kowalczyk, Katarzyna Nowakowska

and for each n ∈ N there exists δn > 0 such that E ∩ [x0 − δn , x0 + δn ] ⊂ En . Hence, f|E is continuous at x0 and f is [Λ , ρ]-continuous. t u Theorem 26.25. Let 0 < Λ < 1. A measurable function f : I → R is [Λ , 1]continuous at x0 if and only if lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) = 1 for each ε > 0. Proof. Assume that a measurable f is [Λ , 1]-continuous at x0 . Then there exists a measurable E ⊂ R such that x0 ∈ E, d(E, x0 ) > Λ , d(E, x0 ) = 1 and f|E is continuous at x0 . By the continuity of f at x0 , for each ε > 0 we can find δ > 0 for which E ∩ [x0 − δ , x0 + δ ] ⊂ {x : | f (x) − f (x0 )| < ε}. Hence d({x : | f (x) − f (x0 )| < ε}, x0 ) ≥ d({x ∈ E : | f (x) − f (x0 )| < ε}, x0 ) = d(E, x0 ) and d({x : | f (x)− f (x0 )| < ε}, x0 ) ≥ d({x ∈ E : | f (x)− f (x0 )| < ε}, x0 ) = d(E, x0 ). Therefore lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) = 1. Now assume that Λ1 = lim+ d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ ε→0

and lim d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) = 1

ε→0+

for each ε > 0. Applying Lemma 26.23 for En = {x ∈ I : | f (x) − f (x0 )| < n1 }, we can find a measurable set E ⊂ R such that x0 ∈ E, d(E, x0 ) ≥ λ1 > λ , d(E, x0 ) = 1 and for each n ∈ N there exists δn > 0 such that E ∩ [x0 − δn , x0 + δn ] ⊂ En . Hence f|E is continuous at x0 and f is [Λ , ρ]-continuous. t u Corollary 26.26. \ 0 Λ − 1n 2b

for each 0 < a < b < εn . Proof. Fix any n ∈ N. There exists εn ∈ (0, 1) such that for each c ∈ (0, εn ). If 0 < a < b < εn , then

λ (E∩[x−c,x+c]) 2c

1 > Λ − 2n

a a λ (E ∩ (([x − b, x − 2n ] ∪ [x + 2n , x + b])) = λ (E ∩ [x − b, x + b])− a a 1 b − λ (E ∩ [x − 2n , x + 2n ]) ≥ 2b(1 − 2n ) − 2a 2n > bΛ − n .

t u Lemma 26.28. Let 0 < ρ ≤ 1 and let x ∈ R. Assume that E ⊂ R is measur+ able and s-d (E, x) ≥ ρ. For every n ∈ N there exists a decreasing sequence {αm }m∈N of positive reals converging to 0 such that λ (E ∩ ([x − αm , x − α2nm ] ∪ [x + α2nm , x + αm ])) > ρ − 1n 2αm for each m ∈ N. Proof. Fix n ∈ N. There exists a decreasing sequence {βm }m∈N of positive reals such that lim

m→∞

λ (E ∩ [x − βm , x + βm ]) + = s-d (E, x) ≥ ρ. 2βm

Then we can find m0 such that

λ (E∩[x−βm ,x+βm ]) 2βm

> ρ − 1n for all m ≥ m0 . Hence

βm βm λ (E ∩ ([x − βm , x − 2m ] ∪ [x + 2m , x + βm ])) ≥ 2βm β

β

λ (E ∩ [x − βm , x + βm ]) λ (E ∩ [x − 2mm , x + 2mm ]) − > ρ − 1n ≥ 2βm 2βm for each m ≥ m0 . Then the sequence {αm }m∈N , where αm = βm+m0 for m ∈ N, has all the required properties. t u Lemma 26.29. Let 0 < Λ ≤ ρ ≤ 1 and let {En }n∈N be a decreasing sequence of measurable sets such that x ∈

∞ T n=1

En , s-d + (En , x) ≥ Λ and s-d(En , x) ≥ ρ

462

Stanisław Kowalczyk, Katarzyna Nowakowska

for all n ∈ N. Then there exists a measurable set E such that s-d(E, x) ≥ Λ , s-d(E, x) ≥ ρ, x ∈ E and for each n ∈ N there exists a positive real δn > 0 for which E ∩ [x − δn , x + δn ] ⊂ En Proof. By Lemma 26.27, for each n ∈ N there exists εn > such that a a λ (En ∩ ([x − b, x − 2n ] ∪ [x + 2n , x + b])) 1 >Λ− 2b n

for all 0 < a < b < εn . By Lemma 26.28, for each n ∈ N we can find decreasing sequence {αmn }m∈N such that n

n

λ (En ∩ ([x − αmn , x − α2nm ] ∪ [x + α2nm , x + αmn ])) 1 >ρ− 2αmn n for all m, n ∈ N. As in the proof of Lemma 26.23, we will construct inductively a sequence of positive reals {an }n∈N such that for each n ∈ N 1. an < εn , an 2. an+1 < 2n , 3. there exists mn ∈ N for which

h

αmn n n 2n , αmn

Let c ∈ [an+1 , an ] for some n. Since

an+1 2n

i

⊂ [an+1 , an ].

> an+2 , we have

F ∩ ([x − c, x − an+2 ] ∪ [x + an+2 , x + c]) ⊃  ⊃ En ∩ ([x − an+1 , x − an+2 ] ∪ [x + an+2 , x + an+1 ]) ∪  ∪ En−1 ∩ ([x − c, x − an+1 ] ∪ [x + an+1 , x + c]) ⊃     n+1 n+1 ⊃ En ∩ x − c, x − a2n ∪ x + a2n ,x+c . Hence       n+1 n+1 ∪ x + a2n ,x+c > c(Λ − n1 ) λ (F ∩[x−c, x+c]) ≥ λ En ∩ x − c, x − a2n and λ (F∩[x−c,x+c]) > Λ − n1 . Therefore s-d + (F, x) ≥ Λ . c On the other hand, F ∩ [x − αmn n , x + αmn n ] ⊃ En ∩ ([x − αmn n , x − an+2 ] ∪ [x + an+2 , x + αmn n ]) ⊃     n+1 n+1 , x − αmn n ∪ x + αmn n , x + α2n ⊃ En ∩ x − α2n . It follows that

λ (F∩[x−αmn n ,x+αmn n ]) 2αmn n

> ρ − n1 for n ∈ N and

26. Path continuity connected with the notion of density +

s-d (F, x) ≥ lim sup n→∞

463

 λ (F ∩ [x − αmn n , x + αmn n ]) ≥ lim sup ρ − n1 = ρ. n 2αmn n→∞

Finally, we can easily see that for each n ∈ N we can choose δn = an+1 for which E ∩ [x − δn , x + δn ] ⊂ En . t u Theorem 26.30. Let 0 < Λ ≤ ρ < 1. A measurable function f : I → R is s[Λ , ρ]-continuous at x0 , if and only if lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > ρ.

ε→0+

Proof. Assume that a measurable f is s[Λ , ρ]-continuous at x0 . Then there exists a measurable E ⊂ I such that x0 ∈ E, s-d(E, x0 ) > Λ , s-d(E, x0 ) > ρ and f|E is continuous at x0 . By the continuity of f at x0 , for each ε > 0 we can find δ > 0 for which E ∩ [x0 − δ , x0 + δ ] ⊂ {x : | f (x) − f (x0 )| < ε}. Hence s-d({x : | f (x) − f (x0 )| < ε}, x0 ) ≥ s-d({x ∈ E : | f (x) − f (x0 )| < ε}, x0 ) = = s-d(E, x0 ) and s-d({x : | f (x) − f (x0 )| < ε}, x0 ) ≥ s-d({x ∈ E : | f (x) − f (x0 )| < ε}, x0 ) = = s-d(E, x0 ). Therefore lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > ρ.

ε→0+

Now assume that Λ1 = lim+ s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ ε→0

and ρ1 = lim+ s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > ρ. ε→0

Applying Lemma 26.29 for En = {x ∈ I : | f (x) − f (x0 )| < 1n }, we can find a measurable set E ⊂ I such that x0 ∈ E, s-d(E, x0 ) ≥ λ1 > λ , s-d(E, x0 ) ≥ ρ1 > ρ

464

Stanisław Kowalczyk, Katarzyna Nowakowska

and for each n ∈ N there exists δn > 0 such that E ∩ [x0 − δn , x0 + δn ] ⊂ En . Hence f|E is continuous at x0 . Thus f is s[Λ , ρ]-continuous at x0 . t u Theorem 26.31. Let 0 < Λ < 1. A measurable function f : I → R is s-[Λ , 1]continuous at x0 , if and only if lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) = 1 for each ε > 0. Proof. Assume that a measurable f is s-[Λ , 1]-continuous at x0 . Then there exists measurable E ⊂ R such that x0 ∈ E, s-d(E, x0 ) > Λ , s-d(E, x0 ) = 1 and f|E is continuous at x0 . By the continuity of f at x0 , for each ε > 0 we can find δ > 0 for which E ∩ [x0 − δ , x0 + δ ] ⊂ {x : | f (x) − f (x0 )| < ε}. Hence s-d({x : | f (x) − f (x0 )| < ε}, x0 ) ≥ s-d({x ∈ E : | f (x) − f (x0 )| < ε}, x0 ) = = s-d(E, x0 ) and s-d({x : | f (x) − f (x0 )| < ε}, x0 ) ≥ s-d({x ∈ E : | f (x) − f (x0 )| < ε}, x0 ) = = s-d(E, x0 ). Therefore lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ

ε→0+

and s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) = 1. Now assume that Λ1 = lim+ s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) > Λ ε→0

and lim s-d({x ∈ I : | f (x) − f (x0 )| < ε}, x0 ) = 1

ε→0+

for each ε > 0. Applying Lemma 26.29 for En = {x ∈ I : | f (x) − f (x0 )| < 1n }, we can find a measurable set E ⊂ R such that x0 ∈ E, d(E, x0 ) ≥ Λ1 > Λ , d(E, x0 ) = 1 and for each n ∈ N there exists δn > 0 such that E ∩ [x0 − δn , x0 + δn ] ⊂ En . Hence, f|E is continuous at x0 and f is s[Λ , ρ]-continuous. t u

26. Path continuity connected with the notion of density

465

Corollary 26.32. \

sC[Λ ,ρ] = Cap .

00

λ ({x ∈ (x0 − δ , x0 + δ ) : | f (x) − f (x0 )| < ε}) > 0.

We say that f has Denjoy property, if it has Denjoy property at each point x ∈ I. We will denote the class of all functions with Denjoy property by Den. By Theorem 26.12 and definition of UC ρ we have. Corollary 26.34. Let 0 < ρ < 1. If f ∈ U C ρ then f has Denjoy property. Corollary 26.35. By Lemma 26.11, all defined classes of functions have Denjoy property. The diagram shows the relations among the considered classes of functions. sC[Λ ,ρ] 

-

sUC ρ @ @

@

@ R @

@ @

Cap

Den

@ @ @

@ @ @ R @

 R @

-

C[Λ ,ρ]

?

UC ρ

No other implication can be stated as a few examples show. Example 26.36. There exists f : R → R such that f ∈ (sUC ρ ∩UC ρ )\(sC[Λ ,ρ] ∪ C[Λ ,ρ] ) for all 0 < ρ ≤ 1, Λ ∈ (0, ρ]. Let {[an , bn ]}n∈N and {[cn , dn ]}n∈N be two sequences of closed intervals  ∞  + S such that 0 < . . . dn+1 < cn < an < bn < dn < . . . < d1 , d [an , bn ], 0 = 1,  ∞   n=1  ∞ + S + S [an , bn ], 0 = 0 and d ([cn , an ] ∪ [bn , dn ]), 0 = 0. Define f : R → d n=1

n=1

R letting  ∞ S  ([an , bn ] ∪ [−bn , −an ]) ∪ {0},  0 for x ∈   n=1 ∞  f (x) = 1 for x ∈ R \ {0} ∪ S ([cn , dn ] ∪ [−dn , −cn ]) ,    n=1  linear in intervals [cn , an ], [bn , dn ], [−an , −cn ], [−dn , −bn ], n = 1, . . . .

466

Stanisław Kowalczyk, Katarzyna Nowakowska

It is clear that f is continuous at each point except at 0. Let E =

∞ S

([an , bn ] ∪

n=1

[−bn , −an ]) ∪ {0}. Then f|E is constant and d(E, 0) = s-d(E, 0) = 1. Therefore f ∈ sUC ρ ∩ UC ρ for ρ ∈ (0, 1]. On the other hand, d({x : | f (x)| < 1}, 0) = s-d({x : | f (x)| < 1}, 0) = 1 − d(E, 0) = 0. Thus f ∈ / sC[Λ ,ρ] ∪ C[Λ ,ρ] for any 0 < ρ ≤ 1, Λ ∈ (0, ρ]. Example 26.37. For each 0 < Λ < ρ ≤ 1 there exists f : R → R such that f ∈ (UC ρ ∩ C[Λ ,ρ] ) \ (sC[Λ ,ρ] ∪ sUC ρ ). Let 0 < Λ < ρ ≤ 1. There exists α, β ∈ (0, 1] such that α > Λ , α+β 2 < ρ and β > ρ if ρ < 1 or β = 1 if ρ = 1. Let {[an , bn ]}n∈N , {[cn , dn ]}n∈N , {[a0n , b0n ]}n∈N , {[c0n , dn0 ]}n∈N be four sequences of closed intervals such that . . . < c0n < a0n < b0n < dn0 < c0n+1 < . . . < 0 < . . . dn+1 < cn < an < bn < dn < . . . , d+

∞ [

∞  [  [an , bn ], 0 = β , d − [a0n , b0n ], 0 = α,

n=1

d

 ∞ + S

n=1

  ∞  − S ([cn , an ] ∪ [bn , dn ]), 0 = 0 and d ([c0n , a0n ] ∪ [b0n , dn0 ]), 0 = 0.

n=1

n=1

Define f : R → R letting  ∞ S  0 for x ∈ ([an , bn ] ∪ [a0n , b0n ]) ∪ {0},    n=1 ∞ f (x) = 1 for x ∈ R \ ({0} ∪ S ([cn , dn ] ∪ [c0n , dn0 ])),    n=1  linear in intervals [cn , an ], [bn , dn ], [c0n , a0n ], [b0n , dn0 ], n = 1, . . . . The function f is continuous at each point except at 0. Let E =

∞ S

([an , bn ] ∪

n=1

[a0n , b0n ]) ∪ {0}. Then f|E is constant, d(E, 0) = β and d(E, 0) ≥ α. Therefore f ∈ U C ρ ∩ C[Λ ,ρ] . But, s-d({x : | f (x)| < 1}, 0) = α+β / sC[Λ ,ρ] ∪ 2 < ρ. Thus f ∈ sUC ρ . Example 26.38. For each 0 < Λ ≤ ρ ≤ 1, Λ < 1 there exists f : R → R such that f ∈ sC[Λ ,ρ] \ C[Λ ,ρ] . Fix 0 < Λ < ρ ≤ 1, Λ < 1. There exists α, β ∈ (0, 1] such that α+β 2 Λ > β . Let {[an , bn ]}n∈N , {[cn , dn ]}n∈N , {[a0n , b0n ]}n∈N , {[c0n , dn0 ]}n∈N be four sequences of closed intervals such that . . . < c0n < a0n < b0n < dn0 < c0n+1 < . . . < 0 < . . . dn+1 < cn < an < bn < dn < . . . ,

26. Path continuity connected with the notion of density

−a0n

d

+

= bn for

S ∞

+

S ∞





467

S ∞



[an , bn ], 0 = d [a0n , b0n ], 0) all n, d n=1 n=1 ∞ ∞ [  [  d+ [an , bn ], 0 = α, d − [a0n , b0n ], 0 = n=1 n=1

= 1, β,

  ∞  − S ([cn , an ] ∪ [bn , dn ]), 0 = 0 and d ([c0n , a0n ] ∪ [b0n , dn0 ]), 0 = 0.

n=1

n=1

Define f : R → R letting  ∞ S  0 for x ∈ ([an , bn ] ∪ [a0n , b0n ]) ∪ {0},    n=1 ∞ f (x) = 1 for x ∈ R \ ({0} ∪ S ([cn , dn ] ∪ [c0n , dn0 ])),    n=1  linear in intervals [cn , an ], [bn , dn ], [c0n , a0n ], [b0n , dn0 ], n = 1, . . . . Obviously, f is continuous except at 0. Let E =

∞ S

([an , bn ] ∪ [a0n , b0n ]) ∪ {0}.

n=1

Then f|E is constant, s-d(E, 0) = 1 and s-d(E, 0) = α+β 2 > Λ . Therefore f ∈ sC[Λ ,ρ] . But d({x : | f (x)| < 1}, 0) = β < λ . Thus f ∈ / C[Λ ,ρ] .

26.3 Relation between considered classes of functions for different values λ and ρ The following proposition is obvious. Proposition 26.39. 1. Let 0 < ρ1 ≤ ρ2 ≤ 1. Then UC ρ2 ⊂ U C ρ1 and sUC ρ2 ⊂ sUC ρ1 . 2. Let 0 < Λ1 ≤ ρ1 ≤ 1, 0 < Λ2 ≤ ρ2 ≤ 1, Λ1 ≤ Λ2 and ρ1 ≤ ρ2 . Then C[Λ2 ,ρ2 ] ⊂ C[Λ1 ,ρ1 ] and sC[Λ2 ,ρ2 ] ⊂ sC[Λ1 ,ρ1 ] . Example 26.40. For each 0 < ρ1 < ρ2 ≤ 1 there exists f ∈ (UC ρ1 ∩ sUC ρ1 ) \ (UC ρ2 ∪ sUC ρ2 ). Moreover, for each 0 < Λ2 ≤ ρ2 , we have f ∈ (C[ρ1 ,ρ1 ] ∩ sC[ρ1 ,ρ1 ] ) \ (C[Λ2 ,ρ2 ] ∪ sC[Λ2 ,ρ2 ] ). Let {[an , bn ]}n∈N and {[cn , dn ]}n∈N be two sequences of closed intervals  S ∞ such that 0 < . . . dn+1 < cn < an < bn < dn < . . . < d1 , d + [an , bn ], 0 = n=1  ∞  + S ρ1 +ρ2 and d ([cn , an ] ∪ [bn , dn ]), 0 = 0. Define f : R → R letting 2 n=1

468

f (x) =

Stanisław Kowalczyk, Katarzyna Nowakowska

 ∞ S  0 for x ∈ ([an , bn ] ∪ [−bn , −an ]) ∪ {0},    n=1

∞  S 1 for x ∈ R \ {0} ∪ ([cn , dn ] ∪ [−dn , −cn ]) ,    n=1  linear in intervals [cn , an ], [bn , dn ], [−an , −cn ], [−dn , −bn ], n = 1, . . . .

It is clear that f is continuous at each point except at 0. Let E = {0} ∪ ∞ S

[an , bn ]. Then f|E is constant and d(E, 0) = s-d(E, 0) =

n=1

ρ1 +ρ2 2

> ρ1 . There-

fore f ∈ sUC ρ ∩ U C ρ ∩ C[ρ1 ,ρ1 ] ∩ sC[ρ1 ,ρ1 ] . On the other hand, d({x : | f (x)| < 1}, 0) =s-d({x : | f (x)| < 1}, 0) ≤ d(

∞ [

([cn , dn ] ∪ [−dn , −bn ]) =

n=1

=

ρ1 + ρ2 < ρ2 . 2

Thus f ∈ / U C ρ2 ∪ sUC ρ2 ∪ C[Λ2 ,ρ2 ] ∪ sC[Λ2 ,ρ2 ] for any Λ2 ∈ (0, ρ2 ]. From Proposition 26.39 and Example 26.40 we have. Theorem 26.41. 1. Let ρ1 , ρ2 ∈ (0, 1]. Then UC ρ2 ⊂ UC ρ1 if and only if ρ1 ≤ ρ2 . Moreover, if ρ1 < ρ2 then UC ρ2 $ UC ρ1 . 2. Let ρ1 , ρ2 ∈ (0, 1]. Then sUC ρ2 ⊂ sUC ρ1 if and only if ρ1 ≤ ρ2 . Moreover, if ρ1 < ρ2 then sUC ρ2 $ sUC ρ1 . Example 26.42. Let 0 < Λ1 < ρ1 < 1. For each Λ1 < Λ2 ≤ 1 there exists f ∈ (C[Λ1 ,ρ1 ] ∩ sC[Λ1 ,ρ1 ] ) \ (C[Λ2 ,Λ2 ] ∪ sC[Λ2 ,Λ2 ] ). Let {[an , bn ]}n∈N and {[cn , dn ]}n∈N be two sequences of closed intervals S  ∞ [an , bn ], 0 = such that 0 < . . . dn+1 < cn < an < bn < dn < . . . < d1 , d + n=1  ∞   ∞  + S + S Λ1 +Λ2 [a , b ], 0 = 1 and ([c , a ] ∪ [b , d ]), 0 = 0. Define , d d n n n n n n 2 n=1

n=1

f : R → R letting  ∞ S  0 for x ∈ ([an , bn ] ∪ [−bn , −an ]) ∪ {0},    n=1 ∞  f (x) = 1 for x ∈ R \ {0} ∪ S ([cn , dn ] ∪ [−dn , −cn ]) ,    n=1  linear in intervals [cn , an ], [bn , dn ], [−an , −cn ], [−dn , −bn ], n = 1, . . . . It is clear that f is continuous at each point except at 0. Let E = {0} ∪ ∞ S

([an , bn ] ∪ [−bn , −an ]). Then f|E is constant d(E, 0) = s-d(E, 0) =

n=1

Λ1 +Λ2 2

>

26. Path continuity connected with the notion of density

469

Λ1 and d(E, 0) = s-d(E, 0) = 1. Hence f ∈ C[Λ1 ,ρ1 ] ∩ sC[Λ1 ,ρ1 ] . On the other hand, d({x : | f (x)| < 1}, 0) =s-d({x : | f (x)| < 1}, 0) ≤ ≤(

∞ [

([cn , dn ] ∪ [−dn , −cn ]), 0) =

n=1

Λ1 + Λ2 < Λ2 . 2

Thus f ∈ / C[Λ2 ,Λ2 ] ∪ sC[Λ2 ,Λ2 ] . From Proposition 26.39 and Examples 26.42 and 26.40 we have. Theorem 26.43. 1. Let 0 < Λ1 ≤ ρ1 ≤ 1 and 0 < Λ2 ≤ ρ2 ≤ 1. Then C[Λ2 ,ρ2 ] ⊂ C[Λ1 ,ρ1 ] if and only if Λ1 ≤ Λ2 and ρ1 ≤ ρ2 . Moreover, if Λ1 < Λ2 or ρ1 < ρ2 then C[Λ2 ,ρ2 ] $ C[Λ1 ,ρ1 ] . 2. Let 0 < Λ1 ≤ ρ1 ≤ 1 and 0 < Λ2 ≤ ρ2 ≤ 1]. Then sC[Λ2 ,ρ2 ] ⊂ sC[Λ1 ,ρ1 ] if and only if Λ1 ≤ Λ2 and ρ1 ≤ ρ2 . Moreover, if Λ1 < Λ2 or ρ1 < ρ2 then sC[Λ2 ,ρ2 ] $ sC[Λ1 ,ρ1 ] . Example 26.44. Let 0 < Λ1 ≤ ρ1 ≤ 1] and 0 < ρ2 ≤ 1. Then there exists f ∈ (UC ρ2 ∩ sUC ρ2 ) \ (C[Λ1 ,ρ1 ] ∪ sC[Λ1 ,ρ1 ] ) Let {[an , bn ]}n∈N and {[cn , dn ]}n∈N be two sequences of closed intervals S  ∞ such that 0 < . . . dn+1 < cn < an < bn < dn < . . . < d1 , d + [an , bn ], 0 = 0, n=1  ∞  ∞   + S + S d [an , bn ], 0 = 1 and d ([cn , an ] ∪ [bn , dn ]), 0 = 0. Define n=1

n=1

f : R → R letting  ∞ S  0 for x ∈ ([an , bn ] ∪ [−bn , −an ]) ∪ {0},    n=1 ∞  f (x) = 1 for x ∈ R \ {0} ∪ S ([cn , dn ] ∪ [−dn , −cn ]) ,    n=1  linear in intervals [cn , an ], [bn , dn ], [−an , −cn ], [−dn , −bn ], n = 1, . . . . Clearly, f is continuous at each point except at 0. Let E = {0} ∪

∞ S

([an , bn ] ∪

n=1

[−bn , −an ]). Then f|E is constant d(E, 0) = s-d(E, 0) = 0 and d(E, 0) = s-d(E, 0) = 1. Hence f ∈ UC ρ2 ∩ sUC ρ2 . On the other hand,

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Stanisław Kowalczyk, Katarzyna Nowakowska

d({x : | f (x)| < 1}, 0) =s-d({x : | f (x)| < 1}, 0) ≤ ≤d(

∞ [

([cn , dn ] ∪ [−dn , −cn ]), 0) = 0.

n=1

Thus f ∈ / C[Λ1 ,ρ1 ] ∪ sC[Λ1 ,ρ1 ] . Example 26.45. Let 0 < ρ2 < ρ1 ≤ 1. Then for each 0 < Λ ≤ ρ2 there exists f ∈ (C[Λ ,ρ2 ] ∩ sC[Λ ,ρ2 ] ) \ (UC ρ1 ∪ sUC ρ1 ) Let {[an , bn ]}n∈N and {[cn , dn ]}n∈N be two sequences of closed intervals S  ∞ such that 0 < . . . dn+1 < cn < an < bn < dn < . . . < d1 , d + [an , bn ], 0 = n=1  ∞  + S ρ1 +ρ2 and d ([cn , an ] ∪ [bn , dn ]), 0 = 1. Define f : R → R letting 2 n=1

f (x) =

 ∞ S  0 for x ∈ ([an , bn ] ∪ [−bn , −an ]) ∪ {0},    n=1

∞  S 1 for x ∈ R \ {0} ∪ ([cn , dn ] ∪ [−dn , −cn ]) ,    n=1  linear in intervals [cn , an ], [bn , dn ], [−an , −cn ], [−dn , −bn ], n = 1, . . . .

The function f is continuous at each point except at 0. Let E = {0} ∪ ∞ S

([an , bn ] ∪ [−bn , −an ]). Then f|E is constant d(E, 0) = s-d(E, 0) =

n=1 ρ2 . Hence

ρ1 +ρ2 2

>

f ∈ C[Λ ,ρ2 ] ∩ sC[Λ ,ρ2 ] . On the other hand,

d({x : | f (x)| < 1}, 0) =s-d({x : | f (x)| < 1}, 0) = =d(

∞ [

([cn , dn ] ∪ [−dn , −cn ]), 0) =

n=1

ρ1 + ρ2 < ρ1 . 2

Thus f ∈ / U C ρ1 ∪ sUC ρ1 . From Proposition 26.39 and Examples 26.44 and 26.45 we have. Theorem 26.46. 1. Let 0 < Λ1 ≤ ρ1 ≤ 1 and 0 < ρ2 ≤ 1. Then UC ρ2 ⊂ C[Λ1 ,ρ1 ] if and only if ρ1 ≤ ρ2 . Moreover, C[Λ1 ,ρ1 ] $ UC ρ2 . 2. Let 0 < Λ1 ≤ ρ1 ≤ 1 and 0 < ρ2 ≤ 1. Then sUC ρ2 ⊂ sC[Λ1 ,ρ1 ] if and only if ρ1 ≤ ρ2 . Moreover, sC[Λ1 ,ρ1 ] $ sUC ρ2 .

26. Path continuity connected with the notion of density

471

References [1] A. Alikhani, Borel measurability of extreme path derivatieves, Real Anal. Exchange 12 (1986-87), 216-246. [2] K. Banaszewski, Funkcje ciagłe ˛ wzgl˛edem systemu s´cie˙zek, Doctoral Thesis, Łód´z, 1995. [3] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Mathematics, Vol. 659, Springer-Verlag Berlin Heidelberg New York, 1978. [4] A. M. Bruckner, R. J. O’Malley, B. S. Thomson, Path Derivatives: A Unified View of Certain Generalized Derivatives, Trans. Amer. Math. Soc. 283 (1984), 97–125. [5] J. Je¸drzejewski, On limit numbers of real functions, Fund. Math. 83(3) (1973/74), 269–281. [6] A. Karasi`nska, E. Wagner-Bojakowska, Some remarks on ρ-upper density, Tatra Mt. Math. Publ. 46 (2010), 85–89. [7] S. Kowalczyk, K. Nowakowska, A note on ρ-upper continuous functions, Tatra Mt. Math. Publ. 44 (2009), 153–158. [8] S. Kowalczyk, K. Nowakowska, Maximal classes for the family of [λ , ρ]-continuous functions, Real Anal. Exchange 36 (2010-11), 307–324. [9] S. Kowalczyk, K. Nowakowska, Maximal classes for ρ-upper continuous functions, Journal of Applied Analysis 19 (2013), 69—89. [10] M. Marciniak, R. Pawlak, On the restrictions of functions. Finitely continuous functions and path continuity, Tatra Mt. Math. Publ. 24 (2002), 65–77. [11] M. Marciniak, On path continuity, Real Anal. Exchange 29(1) (2003-2004), 247–255. [12] J. Masterson, A nonstandard result about path continuity, Acta Math. Hungar. 59, Issue 1-2, 1992, 147–149. [13] K. Nowakowska, On the family of [Λ , ρ]-continuous functions, Tatra Mt. Math. Publ. 44 (2009), 129–138. [14] F. D. Tall, The density topology, Pacific Journal of Mathematics 62(1) (1976), 275–284. [15] B. S. Thomson, Real Functions, Lecture Notes in Mathematics, Vol. 1170, SpringerVerlag Berlin Heidelberg New York, 1985. [16] W. Wilczy`nski, Density topologies, Handbook of Measure Theory, chapter 15, Elsevier 2012, 307–324.

S TANISŁW KOWALCZYK Institute of Mathematics, Pomeranian Academy ul. Arciszewskiego 22d, 76-200 Słupsk, Poland E-mail: [email protected]

K ATARZYNA N OWAKOWSKA Institute of Mathematics, Pomeranian Academy ul. Arciszewskiego 22d, 76-200 Słupsk, Poland E-mail: [email protected]

Chapter 27

Decompositions of permutations of N with respect to divergent permutations

ROMAN WITUŁA

2010 Mathematics Subject Classification: 40A05, 05A99. Key words and phrases: divergent permutations, convergent permutations, sum-preserving permutations.

27.1 Basic technical notions A bijection of set A ⊂ N onto itself is called here a permutation of A. If p is a permutation of N then the symbol pk denotes the product (composition is an equivalent term) of k permutations p, i.e., p1 := p, pk+1 := p ◦ pk , k ∈ N. A permutation p of N will be called almost identity on N if there exists k = k(p) ∈ N such that p(n) = n for each n ∈ N, n > k. We note that any permutation p of N can be written as a product of distinct (meaning "disjoint" in this paper) cycles:  – finite cycles: a = pn (a), p(a), p2 (a), . . . , pn−1 (a) where a ∈ N and n is called the length of this cycle,  – infinite cycles: . . . , p−2 (a), p−1 (a), a, p(a), p2 (a), . . . where a ∈ N. Also any product of distinct cycles represents a permutation. A cycle of length 2 is called a transposition. Family of permutations of N will be denoted by P. Let p ∈ P. The p-order of element a ∈ N is defined to be the smallest positive integer k satis-

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Roman Wituła

fying relation pk (a) = a. If such an integer does not exist we say that p-order of a is infinite. It is obvious that in this case the p-cycle generated by a has the form (. . . , p−2 (a), p−1 (a), a, p(a), p2 (a), . . .), i.e. it is the infinite cycle and pk (a) 6= pl (a) for any two different integers k and l. Set G ⊂ N will be called a minimal set of generators of p if G is a set of values of any choice function on the family of sets of members of all distinct cycles of p. The p-cycle generated by a will be denoted by cycle(p, a). Moreover, we say that (a, b, c, . . .) or (. . . , β , α, a, b, c, . . .) are p-cycles if all a, b, c, α, β , . . . are positive integers and b = p(a), c = p2 (a), α = p−1 (a), β = p−2 (a) and so on. Finally the order of permutation p ∈ P is defined to be the smallest positive integer k such that pk = id(N), where the symbol id(A) denotes the identity function on A for every nonempty A ⊂ N. If such an integer does not exist we say that order of p is infinite. In this paper the inclusion will be denoted by ⊆. Sign ⊂ is reserved for the proper inclusion, i.e., A ⊂ B if A ⊆ B and A 6= B. Finite set I ⊂ N will be called an interval or an interval of N if there exist m, n ∈ N, m 6 n, such that I = {k ∈ N : m 6 k 6 n}. Only this type of intervals will be discussed in the paper. We say that a nonempty set A ⊂ N is a union of n MSI (or of at most n MSI or of at least n MSI) if there exists a family F of n (or of at most n or of at least n, respectively) intervals of N forming a partition of A and such that dist(I, J) > 2 for any two different elements I, J of F. MSI is the abbreviated form of the notion of mutually separated intervals. A countable family {An } of nonempty and finite subsets of N will be called an increasing sequence if max An < min An+1 for every n ∈ N. Let a, b ∈ N and 0/ 6= A ⊂ N. Then we will write a < A < b if a < α < b for every α ∈ A.

27.2 Main notions and results Permutation p ∈ P rearranging some convergent real series ∑ an into divergent series ∑ a p(n) is called a divergent permutation. Family of all permutations of that kind will be denoted by D. Whereas the subset of D composed of all permutations p such that p−1 is also a divergent permutation, will be denoted by DD and called, after Kronrod [2] and myself, the family of two-sided divergent permutations. For contrast, permutation p ∈ C := P \ D is called a convergent permutation because it rearranges each convergent real series ∑ an into a series which is convergent as well.

27. Decompositions of permutations of N with respect to divergent permutations

475

In the paper we will use permanently the following combinatorial characterization of divergent permutations (dual to the Agnew’s combinatorial characterization of convergent permutations [1], [5], [6]). Let p ∈ P. Then p ∈ D if and only if for every n ∈ N there exists an interval I ⊂ N such that set p(I) is a union of at least n MSI. There exist also many other characterizations of divergent permutations (see [7], [12], [8]) but they will be not used in this paper. We say that a nonempty family A ⊂ P is algebraically small if P \ G(A) 6= 0, / where G(A) denotes the group of permutations generated by A. Similarly we say that a family A ⊂ P is algebraically big if A ◦ A := {p ◦ q : p, q ∈ A} = P. Henceforward the symbol ◦ of composition of permutations will be also used with regard to the superposition of any nonempty sets A, B ⊂ P, i.e.  A ◦ B := p ◦ q : p ∈ A and q ∈ B . It is known that C is algebraically small. This fact was proven by Pleasants [3], [4]. Remark 27.1. There exist subsets of P which are neither algebraically small nor algebraically big. For example, any set G of generators of P such that G ◦ G 6= P possesses this property. In the following example the construction of such set of generators of P will be presented. Example 27.2. Using transfinite induction we can construct a transfinite sequence {Gω }ω∈Ω of subsets of P such that Gend =

[



ω∈Ω

is a set of generators of P and for every ω0 ∈ Ω if G is a group of permutations S generated by set ω card I

and

|b − b∗ | > card J

for any two different a, a∗ ∈ A and b, b∗ ∈ B, respectively.

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Roman Wituła

Lemma 27.6. Each permutation p ∈ P is a product of two permutations q1 and q2 of N having the following form  (27.1) ∏ a2n−1 , a2n , n∈N

where {an } is some one-to-one sequence of all positive integers. In other words, both q1 and q2 are products of infinitely many distinct transpositions. Theorem 27.7. Let p ∈ P. (i) If there exists an infinite set of generators of p, which is minimal with respect to inclusion, then for each k = 2, 3, . . . there exist permutations φi = φi (k) ∈ DD and ψi = ψi (k) ∈ DD, i = 1, 2, all having infinite order, such that φ2 φ1k = ψ2k ψ1 = p. (ii) If there exist a finite set of generators of p, then for every k = 2, 3, . . . there exist permutations φi = φi (k) ∈ CC and ψi = ψi (k) ∈ P for i = 1, 2, such that ψ1k φ1 = φ2 ψ2k = p. Moreover, if p belongs to D or DD then ψ1 and ψ2 can be chosen to belong also to D or DD, respectively. This result follows at once from the relations (see [8], [10], [11]): C ◦ C = C,

DC ◦ DC = DC,

DC ◦ CC = CC ◦ DC = DC.

Theorem 27.8. Let us denote by I the family of all almost identity permutations on N. Then for any k = 2, 3, . . . , ∞ there exists a group of permutations Gk ⊂ I ∪ DD with the following properties: (i) qk is the identity function of N for every element q ∈ Gk , (ii) the set of all elements q ∈ Gk ∩ DD, whose order is precisely equal to k, has the cardinality of the continuum, (iii) if k is a prime number or k = ∞ then the order of any element q ∈ Gk ∩ DD is precisely equal to k.

Final remark The following relation holds as well ∞ [ k=2

DDk 6= P,

(27.2)

27. Decompositions of permutations of N with respect to divergent permutations

479

where DDk := {pk : p ∈ DD}, k = 2, 3, . . .. A reason for this relation is given by the following fact. If p ∈ DDk , where k > 2 and in the decomposition of permutation p into cycles there are only finitely many infinite cycles , then number of these cycles is divisible by k. Whereas we know that there exist permutations p ∈ DD which are infinite cycles (see Example 27.9). Thus DD \

∞ [

DDk 6= 0. /

(27.3)

k=2

Simultaneously it means that relations (27.2) and (27.3) are of algebraic nature. By the way we would like to notice that we do not know whether ∞ [

DDk = P.

k=1

We do not know either if there exists a permutation p ∈ DC which is an infinite cycle. Example 27.9. In this example we present a permutation q ∈ (DD ∩ S) which is an infinite cycle. Let {In } be an increasing sequence of intervals of positive integers forming a partition of N and satisfying the condition n ∈ N.

card I2n−1 = card I2n = 3n, Then permutation q is given by relation

 . . . , c5 , c3 , c1 , c2 , c4 , c6 , . . . , where cγn is "a finite cycle" of the form γ

γ

γ

γ

γ

n n n n , in−2 ,... , i3n−4 cγn = i3nn , iγnn , i3n−2 , in−1

γn γ γn γn γn γn  . . . , in+2 , i1n , i3n−1 , i3n−3 , i3n−5 , . . . in+1 , γ

whereas Gγn = {isn : s = 1, 2, . . . , 3n} – here γn denotes the upper index – is the increasing sequence of all elements of interval Iγn for γn ∈ {2n − 1, 2n} for each n ∈ N. Since each of two following sets γ

q([i1n , inγn ])

and

q−1 ([i1n , iγnn ]) γ

is a union of n MSI for each n ∈ N, therefore q ∈ DD.

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Next, from the relations given below  2n−1 q(G2n+1 ) = G2n+1 \ {i2n+1 3n+3 } ∪ {i3n },  q(G1 ) = G1 \ {i13 } ∪ {i23 },  2n+2 q(G2n ) = G2n \ {i2n 3n } ∪ {i3n+3 }, which hold for any n ∈ N, we get that q ∈ S.

27.3 Proofs Proof of Proposition 27.5. Let us fix k ∈ N, k > 1. Suppose that the sets (n) (n) (n) {a1 , a2 , . . . , akn }, n ∈ N, form the partition of G ∩ O. We can assume that  (n) |u − v| > max ai : i = 1, 2, . . . , n

(27.4)

(n)

for any two different u, v ∈ {ai+n : i = 1, 2, . . . , n}. Let r be the q-order common for each element of O and let the permutation p be given by following formula n

p=

∏ ∏ p(i, n)

n∈N i=1

in case when r = ∞ and by the formula n

p=

∏ ∏ q(i, n)

n∈N i=1

in the case when r < ∞. Notations p(i, n) and q(i, n) designate here the cycles defined in the following way   (n)  (n)  (n)  (n) p(i, n) := . . . , q−1 bi , q−1 bi+n , q−1 ai+2n , . . . , q−1 ai+(k−1)n , (n)

(n)

(n)

(n)

bi , bi+n , ai+2n , . . . , ai+(k−1)n ,   (n)  (n)  (n)  (n) q bi , q bi+n , q ai+2n , . . . , q ai+(k−1)n , . . . and

27. Decompositions of permutations of N with respect to divergent permutations

481

 (n) (n) (n) (n) q(i, n) := bi , bi+n , ai+2n , . . . , ai+(k−1)n ,  (n) (n)  (n)  (n)  q bi , q bi+n , q ai+2n , . . . , q ai+(k−1)n , . . . ....................................  (n) (n)  (n)  (n)  qr−1 bi , qr−1 bi+n , qr−1 ai+2n , . . . , qr−1 ai+(k−1)n , (n)

(n)

(n)

(n)

where bi = τ n (ai ), bi+n = τ n (ai+n ) and τ = τ(i, n) denotes the transposition (n)

(n)

of elements ai and ai+n for every i = 1, . . . , n and n ∈ N. Then we easily verify that pk = q and O = {n ∈ N : p(n) 6= n}. As a result we have the following inclusion (n) (n) (n) {a1+n , a2+n , . . . , a2n } ⊂ γ(In ),   (n) where In = 1, max{ai : i = 1, . . . , n} and γ = p if n ∈ 2N and γ = p−1 if n ∈ 2N − 1. By (27.4) we conclude that each of the sets p(In ), n ∈ 2N, and p−1 (In ), n ∈ 2N − 1, is a union of at least n MSI. Thus q ∈ DD as desired. t u Proof of Theorem 27.3 (i). This assertion follows immediately from Proposition 27.5 applied to Lemma 27.6. t u Proof of Lemma 27.6. Let p ∈ P. If p is a finite cycle having one of the following forms  p = b−n , b−n+1 , . . . , b−1 , b0 , b1 , . . . , bn−1 , bn or  p = b−n , b−n+1 , . . . , b−1 , b1 , . . . , bn−1 , bn , then

  p = q2 q1 ,

where

n−1

 q1 = ∏ (bk , b−k−1 )

n

and

k=i

q2 = ∏ (bk , b−k )

(27.5)

k=1

for i = 0 or 1, respectively. Next, if p is an infinite cycle of the form  p = . . . , b−2 , b−1 , b0 , b1 , b2 , . . . , then (

p = q2 q1 , where q1 = ∏ (bk , b−k−1 ) k∈N0

and

q2 = ∏ (bk , b−k ). k∈N

(27.6)

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In general case, if set {n ∈ N : p(n) 6= n} is infinite then we can apply decompositions (27.5) and (27.6) to all the finite and infinite p-cycles, respectively. On the other hand, if p is almost identity on N, then by applying decomposition (27.5) to all the nontrivial p-cycles and, additionally, by using the following decomposition the identity function of f is equal to qq, where f := { fn : n ∈ N} is a sequence of all fixed points of p and q = ∏k∈N ( f2k−1 , f2k ), we may express p as a composition of two permutations of form (27.1). This completes the proof of lemma. t u Proof of Theorem 27.3 (ii). Let us fix p ∈ P. In the sequel we will construct the permutations q1 , q2 ∈ DD, both having infinite order, and such that q2 q1 = p. First we choose inductively an increasing sequence {In } of intervals satisfying the conditions card In = 5n, (27.7) sets

[ n∈N

I2n−1 and

[

p−1 (I2n ) are disjoint,

(27.8)

n∈N

S

complements of the following two sets: n∈N In and S −1 n∈N I2n−1 ∪ p (I2n ) in N are infinite.

(27.9)

S

Next we define permutation φ of set n∈N In by the following formula  φ (i + a) = 2i + 1 + a, (27.10) φ (i + n + a) = 2i + a,  φ (2i + 1 + 2n + a) = i + 2n + a, (27.11) φ (2i + 2n + a) = i + 3n + a, φ (i + 4n + a) = (i + 1)(mod n) + 4n + a,

(27.12)

where a = min In for every i = 0, 1, . . . , n − 1 and n ∈ N. Hence φ (In ) = In for every n ∈ N. Now we can define the permutations q1 and q2 . S Let q1 be an increasing map of the complement of set n∈N I2n−1 ∪ p−1 (I2n ) S in N onto the complement of set n∈N In in N. Suppose also that q1 is equal S to the restriction of φ to n∈N I2n−1 . On the other hand, let q2 be equal to the S restriction of φ to n∈N I2n . The values of q1 and q2 corresponding to all the other elements of N are defined by the equation p = q2 q1 . The main properties of q1 and q2 , required to be verified, are as follows: q1 , q2 ∈ DD,

(27.13)

27. Decompositions of permutations of N with respect to divergent permutations

orders of q1 and q2 are infinite.

483

(27.14)

To check (27.13) we observe that, by (27.10) and (27.11), each of the following sets   φ [min In , n − 1 + min In ] and φ −1 [2n + min In , 3n − 1 + min In ] is a union of n MSI. Now, if we use the definitions of q1 and q2 , the assertion follows. To prove (27.14) it is sufficient to use the definitions of q1 and q2 in the same manner as above, together with an observation that, by (27.12), for every n ∈ N the permutation φ has a cycle of length n and the domain of which is contained in In . t u Proof of Theorem 27.7 (i). Let k ∈ N, k > 1. We aim to construct permutations φi , ψi ∈ DD, i = 1, 2, satisfying condition φ2 φ1k = ψ2k ψ1 = p and all having the infinite order. We shall distinguish two cases. First, let us suppose that p has infinitely many infinite cycles. Let G ⊂ N denote the family of generators of all infinite p-cycles which is minimal with respect to inclusion. Next, suppose that the infinite sets G1 and G2 form a par (n) (n) (n) tition of G and in turn that the sets a1 , a2 , . . . , an , n ∈ N, form a partition of G1 . Let s(i, n) be positive integers for every i = 1, . . . , n, chosen so that  (n) |u − v| > max ai : i = 1, . . . , n

(27.15)

for every two different u and v from the set  εs(i,n) (n)  p ai : i = 1, . . . , n and ε = ±1 . (n)

(n)

(n) 

Then the cycle . . . , p−1 (ai ), ai , p ai (n)

ξi (n)

where ξi (n)

ξi

(n)

and ζi

 , . . . can be written in the form

(n) k

ζi

,

are the cycles defined as follows

= . . . , p−2−s (a), p−1−s (a), p−s (a), a, ps (a), p2s (a), . . . , p(k−1)s (a), p1+(k−1)s (a), p2+(k−1)s (a), . . .

and



484

Roman Wituła (n)

ζi

= p−s (a), a, ps (a), p2s (a), . . . , p(k−2)s (a), p1−s (a), p(a), p1+s (a), p1+2s (a), . . . , p1+(k−2)s (a), ....................................... −2

p (a), ps−2 (a), p2s−2 (a), p3s−2 (a), . . . , p(k−1)s−2 (a),  p−1 (a), ps−1 (a), p2s−1 (a), p3s−1 (a), . . . , p(k−1)s−1 (a) . (n)

Here we have s = s(i, n) and a = ai Let us put

for i = 1, . . . , n and n ∈ N. n

(n)

φ1 = q ∏ ∏ ζi n∈N i=1

and φ2 =





a∈G3

   n (n) cycle(p, a) ◦ ∏ ∏ ξi , n∈N i=1

where G3 ⊂ N is a family of generators of all finite p-cycles, which is minimal with respect to the inclusion, and q denotes a permutation of N such that  qk = ∏ . . . , p−1 (a), a, p(a), . . . a∈G2

and   n ∈ N : q(n) 6= n = pm (a) : m ∈ Z and a ∈ G2 .

(27.16)

The existence of q results from Proposition 27.5. Additionally, we require that q has the property from Proposition 27.5. This and (27.16) yield that φ1 ∈ DD. From the definition of φ2 it follows that  εs(i,n) (n)  p ai : i = 1, . . . , n ⊂ φ2ε (In ), (n)

where In := [1, max{ai : i = 1, . . . , n}], for ε = ±1 and for every n ∈ N. Henceforth and from (27.15) we conclude that any of the sets φ2ε (In ), ε = ±1, is a union of at least n MSI, so that φ2 ∈ DD, as required. A trivial verification shows that φ2 φ1k = p and that the orders of φ1 and φ2 are infinite. Construction of permutations ψ1 and ψ2 may be carried out in the similar (n) way. Then it is sufficient to set ψ1 = φ2 and ψ2 = φ1 and to define cycles ζi in the following way

27. Decompositions of permutations of N with respect to divergent permutations (n)

ζi

485

= p1−s (a), p(a), ps+1 (a), p2s+1 (a), . . . , p(k−2)s+1 (a), p2−s (a), p2 (a), ps+2 (a), p2s+2 (a), . . . , p(k−2)s+2 (a), ....................................... −1

p (a), ps−1 (a), p2s−1 (a), p3s−1 (a), . . . , p(k−1)s−1 (a),  a, ps (a), p2s (a), p3s (a), . . . , p(k−1)s (a) . (n)

Definition of ξi is the same as above. Now we consider the case in which permutation p has infinitely many of finite cycles. Let F ⊂ N denote a family of generators of all finite p-cycles but such that any two different elements a, b ∈ F generate different p-cycles. Suppose that the infinite sets Fn , n ∈ N, form a partition of F. Now we fix (n) a one-to-one sequence ai , i ∈ Z, of all elements of family Fn for each n ∈ N. The choice of these sequences is such that the following inequality holds  (n)  |u − v| > max γ a2i : i ∈ Z and |i| 6 n + 1 (27.17) for any two different elements u and v of set  (n)  γ a2i−1 : i ∈ Z and |i| 6 n + 1 , where γ = p or γ is the identity function on N, for every n ∈ N. Let us denote (n) by s(i, n) p-order of element ai for all indices i ∈ Z and n ∈ N. Now we define the auxiliary cycles  (n)  (n)  (n)  (n)  (n)  σn = . . . , p a2 , p a1 , p a0 , p a−1 , p a−2 , . . . , (n)  (n)  (n)  δn = . . . , p a−1 , p2 a−1 , . . . , ps(−1,n) a−1 , (n)  (n)  (n)  p a0 , p2 a0 , . . . , ps(0,n) a0 ,  (n)  (n)  (n)  p a1 , p2 a1 , . . . , ps(1,n) a1 , . . . ,

(n) 

ζn = . . . , p a1

and

(n) 

(n)  , . . . , ps(1,n) a1 , (n)  (n)  (n)  p a0 , p2 a0 , . . . , ps(0,n) a0 ,  (n)  (n)  (n)  p a−1 , p2 a−1 , . . . , ps(−1,n) a−1 , . . . ,

, p2 a1

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 (n) (n) (n) (n) (n) ξn = . . . , a−2 , a−1 , a0 , a1 , a2 , . . . , for every n ∈ N. (n)  (n) Since ps(i,n) ai = ai for all i ∈ Z and n ∈ N, a trivial verification shows that   (n)  (n)  (n)  σn δn = ζn ξn = ∏ p ai , p2 ai , . . . , ps(i,n) ai (27.18) i∈Z

for every n ∈ N. Let us define φ2 =

∏ σn

and

ψ1 =

n∈N

∏ ξn .

n∈N

Then the following inclusions follow immediately     (n)  (n)  (n) (n)  p A1 ⊆ φ 1, max p A2 and A1 ⊆ ψ 1, max A2 ,  (n) (n) where φ ∈ {φ2 , φ2−1 }, ψ ∈ {ψ1 , ψ1−1 }, A1 = a2i−1 : i ∈ Z and |i| 6 n and  (n) (n) A2 = a2i : i ∈ Z and |i| 6 n + 1 , n ∈ N. This forces, by (27.17), that any   (n)  (n)  of the following sets φ 1, max p A2 and ψ 1, max A2 is a union of at least (2n + 1) MSI for every φ ∈ {φ2 , φ2−1 }, ψ ∈ {ψ1 , ψ1−1 } and for every n ∈ N. Thus φ2 , ψ1 ∈ DD. It remains to define the permutations φ1 and ψ2 . To this aim let us observe that, by Proposition 27.5, there exist solutions φ , ψ ∈ DD of the following equations φk = ∏ ω and ψ k = ∏ ω, ω∈Γ1

ω∈Γ2

where  Γ1 = ω : ω = δn for some n ∈ N or ω is an infinite p-cycle ,  Γ2 = ω : ω = ζn for some n ∈ N or ω is an infinite p-cycle . Put φ1 = φ and ψ2 = ψ. Hence, from the fact that any of permutations φ2 and ψ1 has an infinite cycle, we see that all four permutations φi , ψi , i = 1, 2, are of infinite order. Moreover, relation (27.18) makes it obvious that φ2 φ1k = ψ2k ψ1 = p. This completes the proof.

t u

Proof of Theorem 27.7 (ii). Let us fix k ∈ {2, 3, . . .}. Let G ⊂ N be a minimal set of generators of p (with respect to inclusion). Suppose that H is the sub-

27. Decompositions of permutations of N with respect to divergent permutations

487

set of G of all elements having the infinite p-order. Since, by hypothesis, G is finite, therefore set H is nonempty. Let a ∈ H and let n(s), s ∈ Z, be an increasing sequence of integers corresponding to such choice of a. The required properties of n(s), s ∈ Z, on this occasion are the following n(s) < 0 iff s < 0, s ∈ Z, and n(0) = 0,

n(s)

p

(27.19)

n(s + 1) ≡ n(s)(mod k), s ∈ Z, (27.20) n(s+3) (a) < p (a) : t ∈ Z and n(s + 1) 6 t 6 n(s + 2) < p (a), (27.21) 

t

for every s ∈ Z, s > 0, and  pn(s−3) (a) > pt (a) : t ∈ Z and n(s − 2) 6 t 6 n(s − 1) > pn(s) (a), (27.22) for every s ∈ Z, s 6 0. Now define three auxiliary cycles. We put (a)

ξ1 = ∏ pn(s)−w(s)+1 (a), pn(s)−2w(s)+1 (a), . . . , pn(s)−kw(s)+1 (a), s∈Z

pn(s)−w(s)+2 (a), pn(s)−2w(s)+2 (a), . . . , pn(s)−kw(s)+2 (a), .................................... n(s)−1

p

(a), pn(s)−w(s)−1 (a), . . . , pn(s)−(k−1)w(s)−1 (a),  pn(s) (a), pn(s)−w(s) (a), . . . , pn(s)−(k−1)w(s) (a) ,

(a)

ξ2 = ∏ pn(s) (a), pn(s)+v(s) (a), . . . , pn(s)+(k−1)v(s) (a), s∈Z

pn(s)+1 (a), pn(s)+v(s)+1 (a), . . . , pn(s)+(k−1)v(s)+1 (a), ....................................  pn(s)+v(s)−1 (a), pn(s)+2v(s)−1 (a), . . . , pn(s)+kv(s)−1 (a) , and ζ (a) = . . . , pn(−1) (a), pn(−1)+v(−1) (a), . . . , pn(−1)+(k−1)v(−1) (a), pn(0) (a), pn(0)+v(0) (a), . . . , pn(0)+(k−1)v(0) (a),  pn(1) (a), pn(1)+v(1) (a), . . . , pn(1)+(k−1)v(1) (a), . . .  for every a ∈ G, where v(s) = k−1 n(s + 1) − n(s) and w(s) = v(s − 1), s ∈ Z. Since the sequence n(s), s ∈ Z, is increasing, we obtain from (27.20) that all

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indices v(s), s ∈ Z, are positive integers. Verification of the following equalities may be then carried out immediately. We have  (a) k (a) (a) k ξ1 ζ = ζ (a) ξ2 = . . . , p−1 (a), a, p(a), . . . for each a ∈ H, and consequently ψ1k φ1 = φ2 ψ2k = p, where φi :=



∏ a∈G\H

   and cycle(p, a) ◦ ∏ ζ (a) a∈H

ψi :=

(a)

∏ ξi

a∈H

for i = 1, 2. To see that φ1 , φ2 ∈ CC we just have to show that ζ (a) ∈ CC for each a ∈ H. For this the following suffices. Let a ∈ H and let I be an interval such that a 6∈ I and J := I ∩ {pn (a) : n ∈ Z} = 6 0. / Set n(s) = min J and n(t) = max J. Then, in view of conditions (27.19), (27.21) and (27.22), the following inclusion is fulfilled ε  ζ (a) (I) ⊃ I \ pl (a) : l = n(τ) − iw(τ) or l = n(τ) + iv(τ) where τ = s or t and for i = 0, 1, . . . , k , where ε = −1 or 1. Hence we check at once that set ζ (a) (I) is a union of at most 4(k + 1) MSI. Thus we have ζ (a) ∈ CC as claimed. Let us notice additionally that if p belongs to D or to DD then the above constructions of permutations ψ1 and ψ2 imply that ψ1 and ψ2 can be then selected such that they belong also to D or DD, respectively. This result follows at once from the relations (see [8], [10], [11]): C ◦ C = C,

DC ◦ DC = DC,

DC ◦ CC = CC ◦ DC = DC.

This completes also the proof of theorem.

t u

Proof of Theorem 27.8. Let us fix k ∈ {2, 3, . . . , ∞}. Let pn , n ∈ N, be a sequence of prime numbers whose range is infinite. This sequence does not necessarily contain all prime numbers and may not be a one-to-one sequence. Assume that the increasing sequence {In } of intervals is a partition of N and that we have  (2k − 1)n for every n ∈ N whenever k ∈ N, card In = (2pn − 1)n for every n ∈ N when k = ∞.

27. Decompositions of permutations of N with respect to divergent permutations

489

Now we define an auxiliary permutation qn of In for each n ∈ N. We set  qn sn + t = sn + n + 2t,  qn sn + (2i − 1)n + 2t = sn + (2i + 1)n + 2t,  qn sn + (2l − 3)n + 2t = sn + t, for t = 0, 1, . . . , n − 1, and for i = 1, 2, . . . , l − 2, where l = k whenever k ∈ N, or l = pn when k = ∞. For the remaining t ∈ In we put qn (t) = t. Then a trivial verification shows that qln = id(In ), (27.23)   qin [sn , sn + n − 1] = sn + (2i − 1)n + 2t : t = 0, 1, . . . , n − 1 (27.24) and qin

−1

  [sn , sn + n − 1] = sn + (2(l − i) + 1)n + 2t : t = 0, 1, . . . , n − 1 , (27.25) i i −1 i.e. each of two sets qn ([sn , sn + n − 1]) and (qn ) ([sn , sn + n − 1]) is a union of n MSI for each i = 1, 2, . . . , l − 1, where l = k whenever k ∈ N, or l = pn when k = ∞. Let M(k) denote the family of all sequences {an } such that an ∈ {1, 2, . . . , k} for all n ∈ N whenever k ∈ N or an ∈ {1, 2, . . . , pn } for all n ∈ N when k = ∞. Now we are ready to define family Gk . We put n o Gk = q : q = ∏ qann and {an } ∈ M(k) , n∈N

where permutation q = ∏n∈N qann is defined as follows q(t) = qann (t) for every t ∈ In and n ∈ N. Observe that intervals In , n ∈ N, are pairwise disjoint and hence this definition is correct. It is obvious that if k is finite then qk = id(N). Let q ∈ Gk , q = ∏n∈N qann . If the inequality an < ln holds for infinitely many indices n ∈ N, where  k for all n ∈ N whenever k ∈ N, ln := pn for all n ∈ N when k = ∞, then, by (27.24) and (27.25), we get q ∈ DD. Furthermore, if k is a prime number or k = ∞ then the order of this q is precisely equal to k as required. t u

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Acknowledgement Author would like to thank the Referee for the constructive comments and suggestions.

References [1] R. P. Agnew, Permutations preserving convergence of series, Proc. Amer. Math. Soc. 6 (1955), 563–564. [2] A. S. Kronrod, On permutation of terms of numerical series, Mat. Sbornik 60(2) (1946), 237–280 (in Russian). [3] P. A. B. Pleasants, Rearrangements that preserve convergence, J. London Math. Soc. 15, II. Ser. (1977), 134–142. [4] P. A. B. Pleasants, Addendum: Rearrangements that preserve convergence, J. Lond. Math. Soc. 18, II. Ser. (1978), 576. [5] P. Schaefer, Sum-preserving rearrangemments of infinite series, Amer. Math. Monthly 88 (1981), 33–40. [6] P. Schaefer, Sums of rearranged series, College Math. J. 17 (1986), 66–70. [7] R. Wituła, On the set of limit points of the partial sums of series rearranged by a given divergent permutation, J. Math. Anal. Appl. 362 (2010), 542–552. [8] R. Wituła, Permutations Preserving Sums of Rearranged Real Series, Cent. Eur. J. Math. 11 (2013), 956–965. [9] R. Wituła, The family F of permutations of N, Math. Slovaca (in press). [10] R. Wituła, Algebraic and set-theoretical properties of some subsets of families of convergent and divergent permutations, Tatra Mountains Math. Publ. 55 (2013), 27–36. [11] R. Wituła, Algebraic properties of the convergent and divergent permutations, Filomat (in review). [12] R. Wituła, D. Słota, R. Seweryn, On Erd˝os’ theorem for monotonic subsequences, Demonstratio Math. 40 (2007), 239–259.

ROMAN W ITUŁA Institute of Mathematics, Silesian University of Technology ul. Kaszubska 23, 44-100 Gliwice, Poland E-mail: [email protected]

List of denotations

card(A) - cardinality of the set A P (A) - the family of all subsets of A Z - the set of integers N - the set of positive integers Q - the set of rational numbers R - the set of real numbers C - the set of complex numbers Tnat - the natural topology on the real line Bor - the σ -algebra of Borel subsets of R B - the σ -algebra of sets having the Baire property M - the σ -ideal of first category sets int (A) (cl (A)) - the interior (closure) of A L (Ln )- the σ -algebra of Lebesgue measurable subsets of R (Rn ) λ (A) (λn (A)) - the Lebesgue measure of A ∈ L (A ∈ Ln ) λ ∗ (A) (λn∗ (A)) - the outer Lebesgue measure of A ⊂ R (A ⊂ Rn ) N - the σ -ideal of null subsets of R C - the family of continuous functions Cap - the family of approximately continuous functions B1 (Bα ) - the family of functions of Baire class one (Baire class α) DB 1 - the family of Darboux functions of Baire class one

491