Surface Area. (AM-35), Volume 35 9781400882328

Table of contents :
CONTENTS
PREFACE
CHAPTER I. INTRODUCTORY CONSIDERATIONS
§1. The Main Theorems
§2. Some Basic Definitions for Curves
§3. Some Definitions for Non-Parametric Surfaces
§4. The Example of Schwarz and Peano
CHAPTER II. LEBESGUE AREA
§5. The Lebesgue Area L
§6. Some Alternate Definitions of Lebesgue Area
§7. Some Critical Considerations on Area
CHAPTER III. THE GEÖCZE AREAS V AND U AND THE PEANO AREA P
§8. The Topological Index
§9. The Geocze and Peano Areas V, U, P
§10. Continuous Mappings and Semicontinuous Collections
§11. Some Properties of the Euclidean Plane E2
CHAPTER IV. BV AND AC PLANE MAPPINGS
§12. BV Plane Mappings
§13. AC Plane Mappings
§14. Local Properties of Plane Mappings
§15. A Characterization of AC Plane Mappings
CHAPTER V. THE FIRST THEOREM
§16. An Analytical Property of Continuous Mappings
§17. Some Properties of Homotopy for Continuous Curves in E3
§18. The First Theorem
CHAPTER VI. THE CAVALIERI INEQUALITY
§19. On the Boundary of Open Sets (Carathéodory Theory)
§20. Contours of a Continuous Surface and the Cavalieri Inequality
CHAPTER VII. IDENTIFICATION OF LEBESGUE, GEÖCZE, PEANO AREAS
§21. The Equality V = U
§22. Some Limit Theorems for the Functions U and V
§23. Some Analytical Properties of Continuous Mappings
§24. The Equality V = L = P
§25. The Lebesgue Area as a Measure Function
CHAPTER VIII. GEOMETRICAL PROPERTIES AND THE SECOND THEOREM
§26. Regular Approximate Differentials
§27. Interval Functions
§28. Generalized Jacobians
§29. Formulas for the Transformation of Areas and Double Integrals
§30. The Second Theorem
CHAPTER IX. THE REPRESENTATION PROBLEM
§31. Fréchet Equivalence
§32. Mean Value Integrals of L-Integrable Functions
§33. Some Particular Types of Surfaces
§34. Representation Theorems for Non-Degenerate Surfaces
CHAPTER X. THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM
§35. Generalized Conformal Representations
§36. A Retraction Process for Surfaces
§37. Representation of General Surfaces, The Third Theorem
APPENDIX A. A DIRECT PROOF OF A PROPERTY OF CONTINUOUS SURFACES
APPENDIX B. WEIERSTRASS INTEGRAL OVER A SURFACE
BIBLIOGRAPHY
SPECIAL SIGNS AND ABBREVIATIONS

Citation preview

Annals of Mathematics Studies Number 35

ANNALS OF MATHEMATICS STUDIES Edited by Emil Artin and Marston Morse 1. 3. 6. 7. 10. 11. 15.

Algebraic Theory of Numbers, by H e r m a n n W e y l Consistency of the Continuum Hypothesis, by K u r t G o d e l The Calculi of Lambda-Conversion, by A l o n z o C h u r c h Finite Dimensional Vector Spaces, by P a u l R. H a l m o s Topics in Topology, by S o l o m o n L e f s c h e t z Introduction to Nonlinear Mechanics, by N. K r y l o f f and N. B o g o l iu b o f f Topological Methods in the Theory of Functions of a Complex Variable,

by

M a r s to n M o r se

Transcendental Numbers, by C a r l L u d w ig S i e g e l Probleme General de la Stabilite du Mouvement, by M. A. L i a p o u n o f f Fourier Transforms, by S. B o c h n e r and K. C h a n d r a s e k h a r a n Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. L e f s c h e t z 21. Functional Operators, Vol. I, by J o h n von N e u m a n n 22. Functional Operators, Vol. II, by J o h n von N e u m a n n 23. Existence Theorems in Partial Differential Equations, by D o r o t h y L. 16. 17. 19. 20.

B e r n s t e in

24. Contributions to the Theory of Games, Vol. I, edited by H. W. K u h n and A. W. T u c k e r 25. Contributions to Fourier Analysis, edited by A. Z y g m u n d , W. T r a n s u e , M. M o r s e , A. P. C a l d e r o n , and S. B o c h n e r 26. A Theory of Cross-Spaces, by R o b e r t S c h a t t e n 27. Isoperimetric Inequalities in Mathematical Physics, by G. P o l y a and G. S zeg o 28. Contributions to the Theory of Games, Vol. II, edited by H. K u h n and A. W. T u c k e r 29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L e f s c h e t z 30. Contributions to the Theory of Riemann Surfaces, edited by L. A h l f o r s

et al. 31. Order-Preserving Maps and Integration Processes, by E d w a r d J. M c S h a n e 32. Curvature and Betti Numbers, by K. Y an o and S. B o c h n e r 33. Contributions to the Theory of Partial Differential Equations, edited by L. B e r s , S. B o c h n e r , and F. J oh n 34. Automata, edited by C . E. S h a n n o n and J. M c C a r t h y . In press 35. Surface Area, by L a m b e r t o C e s a r i 36. Contributions to the Theory of Nonlinear Oscillations, Vol. Ill, edited by S. L e f s c h e t z . In press 37. Lectures on the Theory of Games, by H a r o ld W. K u h n . In press 38. Linear Inequalities and Related Systems, edited by H. W. K u h n and A. W. T u c k e r . In press 39. Contributions to the Theory of Games, Vol. Ill, edited by M. D r e s h e r and A. W. T u c k e r . In press

SURFACE AREA BY LA M BERTO CESARI

Princeton, New Jersey Princeton University Press

1956

Copyright, 1956, by Princeton University Press London: Geoffrey Cumberlege, Oxford University Press L. C. Card 5^-9745

Printed in the United States of America

TO ISOTTA

PREFACE Recent studies have shown that parametric surfaces in the ordinary space under the sole hypothesis that they be continuous and have finite area, have remarkable analytical and geometrical properties which have already proved to be essential in the discussion of questions of analysis and of the cal­ culus of variations. Analogous results may be expected for k-dimensional parametric varieties in the n-dimensional space En or in any metric space E. Such studies seem to suggest a basis for differential geometry and the cal­ culus of variations for continuous parametric varieties. Since no ordinary differential element necessarily exists, the basic elements like Jacobians, tangent planes and normals, are introduced by topological and set-theoretical methods. A trend toward an essential reduction of smoothness hypotheses can be observed also in recent research in differential geometry. This book is a study of the concepts connected with parametric surfaces and area and of the related main theorems with complete proofs. Formally, these theorems extend the analogous theorems for curves and length; intrinsically, however, they involve a much larger field of mathematics and offer a deep­ er insight. The material is organized around three main theorems. Theorems I and III appear for the first time in book form. The same holds for the identity between Peano, Lebesgue and Geocze areas P = L = V, the proof of which, here published for the first time, does not involve the problem of repre­ sentation. Thus it was possible to prove the three main theorems in their natural order. Theorem I and the equality P = L = V, both purely analy­ tical in character, are proved before the concepts of Frechet equivalence and generalized Jacobians are introduced. Then the concept of generalized Jacobian is introduced (Chapter VTII) and the second main theorem, in­ volving this concept, is proved. Finally, in the last two chapters (IX and X), the concept of Frechet equivalence is introduced and the third theo­ rem, or representation theorem, is given. Great generality is also achieved in this book by proving all the main theo­ rems for continuous mappings (surfaces) from admissible sets, in particular from open plane sets and finitely connected Jordan regions. Such generality could have been further increased by considering mappings from compact 2-manifolds or from open sets on such manifolds. The basic reasonings con­ cerning area do not present difficulties. Since research is being done on some of the topological arguments (contour, retraction) such a further vii

PREFACE extension is left for discussion in special papers. I hope that the student will find the present volume a guide to the prob­ lems concerning continuous parametric surfaces and varieties. For the bene­ fit of the reader who wants to consult the original papers or more ex­ tensive treatises, I have added - aside from the bibliography at the end of the volume - various references to further results and parallel developments under different viewpoints. During my sojourn in the United States, particularly in the time I spent at the Institute for Advanced Study, I had the opportunity to contact many specialists in the field, and the occasional informal conversations with them have been highly stimulating. Also the joint seminars with T. Rado at the Ohio State University (19^8), C. B. Morrey at the University of California in Berkeley (1 9 ^9 ), L. C. Young at the University of Wisconsin (1950), and A. Rosenthal at Purdue University ( 1 9 5 1 ) have been an enrich­ ing experience for me. I am grateful to those who have given suggestions and read some chapters, particularly to M. C. Ayer, C. Goffman, R. E. Fullerton, Ch. J. Neugebauer, C. R. Putnam, A. Rosenthal, and E. Silverman.

Lamberto Cesari Purdue University, September 195 5

viii

CONTENTS Page PREFACE CHAPTER I.

CHAPTER II.

CHAPTER III.

CHAPTER IV.

CHAPTER V.

CHAPTER VI.

CHAPTER VII.

INTRODUCTORY' CONSIDERATIONS §1. The Main Theorems §2. Some Basic Definitions for Curves §3. Some Definitions for Non-Parametric Surfaces §4 . The Example of Schwarz and Peano LEBESGUE AREA §5 - The Lebesgue Area L §6. Some Alternate Definitions of Lebesgue Area §7 * Some Critical Considerations on Area THE GEOCZE AREAS V AND U AND THEPEANO AREA P §8. The Topological Index §9 * The Geocze and Peano Areas V, U, P §10. Continuous Mappings and Semicontinuous Collections §11. Some Properties of the Euclidean Plane E2 BV AND §12. 513. §1^. §1 5 *

AC PLANE MAPPINGS BV Plane Mappings AC Plane Mappings Local Properties of Plane Mappings A Characterization of AC Plane Mappings

THE FIRST THEOREM §16. An Analytical Property of Continuous Mappings § 1 7 * Some Properties of Homotopy for Continuous Curves in E^ §18. The First Theorem THE CAVALIERI INEQUALITY § 1 9 * On the Boundary of Open Sets (Caratheodory Theory) §2 0. Contours of a Continuous Surface and the Cavalieri Inequality IDENTIFICATION OF LEBESGUE,GEOCZE, PEANO AREAS §21. The Equality V = U §22. Some Limit Theorems for the Functions U and V §23. Some Analytical Properties of Continuous Mappings ix

1 1 10 21 2k 27

27 55

7k

83 83 116 *\k}

16k 173

17 3 2 15

219 236 2k 2 2^2 265 28^ 298 298 316 331 331

3^2 362

CONTENTS Page §2 4 . §25.

The Equality V = L = P The Lebesgue Area as a Measure Function

CHAPTER VIII.GEOMETRICALPROPERTIES AND THE SECOND THEOREM §26. Regular Approximate Differentials §2 7 * Interval Functions §28. Generalized Jacobians §2 9 * Formulas for the Transformation of Areasand Double Integrals §30. The Second Theorem CHAPTER IX. THE REPRESENTATION PROBLEM §31- Frechet Equivalence §32. Mean Value Integrals of L-Integrable Functions §33. Some Particular Types of Surfaces §3 ^* Representation Theorems for Non-Degenerate Surfaces CHAPTER X.

THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM §3 5 - Generalized Conformal Representations §36. A Retraction Process for Surfaces §3 7 - Representation of General Surfaces, The Third Theorem

390 395 407 407 410

419 430 437 449 449 46o 472 477

48-7 487

504 528

APPENDIX A. A DIRECT PROOF OF A PROPERTY OFCONTINUOUS SURFACES

550

APPENDIX B. WEIERSTRASS INTEGRAL OVER ASURFACE

564

BIBLIOGRAPHY

577

SPECIAL SIGNS

AND ABBREVIATIONS

595

x

CHAPTER I.

INTRODUCTORY- CONSIDERATIONS

***************************************************************************

§1. THE MAIN THEOREMS 1.1. Curves and Jordan Length While formal definitions, references and simpler notations will be given in the following sections, we shall first get acquainted with certain concepts and theorems concerning curves and surfaces, and determine the aim and the motivation of the theory discussed in this book. By curve we shall mainly understand a mapping from an interval [a < u < b] into any real Euclidean space, say E^j that is, any set of equations (1 )

C:

x = x(u),

y = y(u),

z = z(u),

a < u < b,

where x(u), y(u), z(u) are real single-valued functions defined in [a,b]. With this we have chosen one of the various concepts of curve which are discussed in the different branches of mathematics, namely the concept which is generally used, for instance, in Differential Geometry, Calculus of Variations, Functional Analysis, Integral Geometry. A curve defined as above is often said to be a path curve, or a parametric curve (§2). By length 1 (C) of a curve C we shall understand the familiar Jordan length, that is, the supremum of the elementary lengths of the polygonal lines c inscribed in C. Such a definition of length may be used for both continuous and discontinuous curves C. For continuous curves the Jordan length is also the limit of the elementary lengths of the polygonal lines c inscribed in C and approaching C (§2), and has various other alternate definitions and interpretations (2 .4 ). In consequence of the definition above the diameter of a curve C (i.e. the supremum of the lengths of the chords) is always < the length 1(C). The following known main theorems for Jordan length hold for both continu­ ous and discontinuous curves. They are expressed In terms of familiar con­ cepts of the theory of real variables, namely the Lebesgue integral (brief­ ly L-integral), and the concepts of total variation V[f] of a function f(u), a < u < b, of function f (u) of bounded variation (BV) and ab­ solutely continuous (AC) in [a,b]. THEOREM (1 * |) (Jordan,

1 8 8 4 ):

1

For every curve

C we have

2

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

V[x], V[y ], Viz] < 1 (C) < Vtx] + Vty] + V i z ] . Hence 1 (C) < + oo if and only if x(u), y(u), z(u) are all of bounded variation (BV) in [a,b] (see references in (2.5))* THEOREM ( 1 ) (Tonelli, 1908-12): For every curve C of finite Jordan length 1 (C) < +00, the func­ tions x(u), y(u), z(u) are BV in [a,b], have almost everywhere in [a,b] ordinary deriva­ tives xu, yu, zu which are L-integrable in [a,b], and (2)

The equality sign holds in (2) if and only if x(u), y(u), z(u) are all absolutely continuous (AC) in [a,b] (see references in (2.5))* Familiar examples (2.6) show that different sets of equations (1) correspond to the same intuitive curve. This happens in particular each time we change in (1 ) the parameter u into a new parameter u1 by means of a monotone continuous function u = cp(u1 ). The new set of equations, or mapping is a different curve according to the definition given above. This inconvenience is generally eliminated by introducing the concept of Frechet equivalence for mappings (1 ) (see 2.6). Then all mappings (1 ) which are Frechet equiva­ lent to a given one are said to be different representations of the same Frechet curve U, or, for the sake of simplicity, of the same curve C. Thus the Theorems (1 1 ), (1 ^) can be completed by the following simple and well known statement: THEOREM (I3): Every continuous curve C of finite Jordan length has at least one representation for which equality sign holds in (2); that is, for which the Jordan length is given by the classical integral (see 2.8).

1.2.

Parametric Surfaces and Lebesgue Area

The definition of curve we have chosen in (1.1) suggests the corresponding definition of surface (and of variety). By surface we shall understand indeed any mapping from some plane set A into any Euclidean space, say E^; that is any set of equations

§ 1. THE MAIN THEOREMS (1 .3 ) (3)

S:

x = x(u, v),

y = y(u, v),

z = z(u, v),

(u, v)eA

where for A we may take for instance any closed Jordan region (simple or not), or any open set (not necessarily connected), etc. (see 5 - 1 )• A sur­ face S defined as above is often said to be a path surface, or a para­ metric surface. Already in the last century mathematicians tried to define the area of S by approaching S by means of polyhedral surfaces s and by taking for area of S either the supremum of the elementary areas of the surfaces s inscribed in S, or the limit of the same areas as the surfaces s (in­ scribed in S) approach S. A criticism of Schwarz and Peano (see 4.1 ) showed that such a supremum may be + «> and that such a limit need not exist even for very simple elementary surfaces. As proposed by Lebesgue (1902) the area L(S) of a continuous surface S can be defined as the lower limit of the elementary areas of the polyhedral surfaces s (in­ scribed or not inscribed in S) approaching S (5-8). Such a definition corresponds to one of the alternate definitions of Jordan length (2 .4 ). We shall return to the concept of area in (1 .5) and later on in §§5, 6, 7.

1.3. The Non-parametric Case A particular case of surfaces are the so called non-parametric surfaces S: x = u, y =.v, z = z(u, v), (u, v)€A; i.e., (4 )

S:

z = z(x, y),

(x, y)eA,

where we shall suppose for a moment that A is a square (see 3.2). The familiar concepts of functions of one real variable of bounded variation and absolutely continuous can be extended in many different ways to func­ tions of two, or more real variables, and some of the corresponding con­ cepts have a remarkably wide range of applications. For the study of the area of (4 ) the concepts of function z(x, y) of bounded variation in the sense of Tonelli (BVT) and absolutely continuous in the sense of Tonelli (ACT) shall be given in (3* 1 )• By making use of the total variations Vx [z], Vy[z] with respect to x and y of z(x, y) in the sense of Tonelli (3* 1 ), and the concept of Lebesgue area, the following theorems hold, which extend to continuous non-parametric surfaces the Theorems (1 1 ) and ( 1 ) for curves and Jordan length. THEOREM (T1 ) (Tonelli, 1926): For every continuous non-parametric surface S we have

CHAPTER I. z]'

INTRODUCTORY CONSIDERATIONS

|A|,

V

Vy [z] - L(S) - |A| + Vx [z] + Vy [z]A *

Hence in A

L(S) < + ° ° if and only if (see references in (3-1 ))•

z(x, y) is

BVT

THEOREM (T2 ) (Tonelli, 1926): For every continuous non-parametric surface S of finite Lebesgue area L(S) < + 00 the function z(x, j), (x, y)eA, is BVT, the first partial derivatives zx .z y exist almost everywhere (a.e.) in A and are L-integrable functions in A, and we have (5 )

L(S) > (A) ; ; v r + z2 +

dx dy

The equality sign holds in (5 ) if and only if is ACT in A (see references in (3-1 ))•

z(x, y)

Remaining in the class of non-parametric surfaces a change of representation does not occur, but if we take into consideration representations representations of S ofin Sin detail by by introducing introducing the general parametric form (3) as we shall do inindetail Frechet equivalence (6.3; 31*2), then, because ofofaa theorem theorem of C. of B. C. Morrey B.Morrey (1936) (see 3^*6), the following statement can bebeadded: added: THEOREM (T^): Any continuous non-parametric surface S of finite Lebesgue area has at least one parametric representation (3) where x, y, z are ACT functions in A, the first partial derivatives xu , x , yu , yy , zu , zy , exist a.e. in A and are L 2-integrable func­ tions in A , and the Lebesgue area of S is given by the classical integral L(S) = (A)JJj2 + J2 +

du dv •

where J 1, J2, J^ are the ordinary Jacobians vJuz v - v v zu , zux v - zv x . x uy v - x vy u of the couples of functions (y, z), (z, x), (x, y). Furthermore the representation is co n f o m a l in a con­ venient generalized sense (see 3I4-.6). Thus all Theorems (l1 ), (1 2 ), (1 ^) for curves and Jordan length have their analogue for non-parametric continuous surfaces and Lebesgue area. In ad­ dition T. Rad6 has proved (1928) that for each non-parametric surface S the Lebesgue area can be obtained as a regular limit process involving the

§1. THE MAIN THEOREMS (1 .4 ) function

z(x, y)

(see 3.1).

The Theorems (T1), (T2 ) have been also extended to non-continuous non-para­ metric surfaces by L. Cesari and C. Goffman (see 3 -3 ) by making use of a convenient modified form of the Lebesgue area, of a generalization of the concept of function of two real variables of bounded variation in the sense of Tonelli, and of the concept of absolute continuity for discontinuous func­ tions introduced by G. C. Evans since 1920 in potential theory.

1 .4

. The Parametric Case: Aim of The Book

The above sketch of known results shows a remarkable analogy between the main theorems for Jordan length of a parametric curve (continuous, as well as discontinuous) and the main theorems for Lebesgue area of a non-para­ metric surface (continuous, as well as discontinuous). This fact suggests, as a program, that, by introducing convenient concepts, in particular total variation, absolute continuity, etc., analogous statements could be proved for parametric surfaces — continuous as well as discontinuous — in or­ dinary space, or in E^, or in abstract spaces, and that analogous state­ ments may hold for a k-dimensional area of a k-dimensional variety. We shall discuss in the present book the problem of the area for parametric continuous surfaces in E^, from the concept of area to the complete ex­ tension of the Theorems (1 ) and (T) above to all continuous parametric surfaces (3)

S:

x = x(u, v),

y = y(u, v),

z = z(u, v),

(u, v)eA

.

We shall limit ourselves to surfaces in E^ mainly because this book is intended to be a guide for the student, but most of the considerations and proofs need only a few changes for the corresponding case in E^. The problem of area for parametric continuous surfaces has involved the analysis of the basic underlying concepts, namely of area, total variation, absolute continuity, and Jacobian, as we shall see later in the book. These concepts are related with concepts purely topologic in character, and first of all with the concept of topological (or Kronecker) index (§8). A first difficulty which arises in the extension of the Theorems (T) to general continuous parametric surfaces (3) is due to a simple, often over­ looked, fact. While for curves, as we have observed in (1.1), the diameter is always < the length of the curve itself, and so for any small arc, the diameter approaches zero with its length, we have, for surfaces and area, a different situation. Namely, no relation exists between diameter of the

6

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

surface and its area and we can have surfaces of a very small area with a diameter as large as we want; for instance, this situation can be realized by taking a thin ribbon or a cone whose areas can be made as small as we want. It is essentially a consequence of this fact that while for curves the finiteness of the length implies bounded variation for the functions x(u), y(u), z(u) (Theorem (l1) in 1.1 ), here for parametric surfaces (3) the finiteness of the area implies nothing about the single functions x(u, v), y(u, v), z(u, v). Indeed, for instance, the surface S: x = y = z = g(u, v), (u, v)eA, where g(u, v) is any arbitrary con­ tinuous function, has all its points on a segment of the straight line x = y = z in and, at least now intuitively, later on rigorously (5.9, note) we shall ascribe to it area zero. As 3 . Banach pointed out, thefiniteness of the area of a surface (3) has important consequences on theproperties of the couples (y, z), (z, x), (x, y) of functions x, y, z, that is, on the properties of the plane mappings T 1 : x =0,

y = y(u, v),

z = z(u, v),

T2 : x = x (u, v),

y = 0,

T3 : x = x(u,

y = y(u, v),

v),

which are the projections of

S

z = z(u, v),

(u, v)eA,

z = 0,

on the coordinate

yz-, zx-, and xy-planes.

We may sketch already the form of the theorems, say (L1), (Lg ), (L3), which extend the Theorems (1 ) and (T) above. By introducing a convenient concept of total variation V(Tp ) of the mapping T , r = 1 ,2 ,3 * (§12), we shall prove the following statement (L1), which extends the statements (11) and (T^ ): (L1 ) THE FIRST THEOREM: S we have

For every continuous surface

V(Tp ) < L(S) < V(T1 )+ V(T2 ) + V(T3),

r =

1 ,2 , 3 •

Hence L(S) < + °o if and only if the plane mappings T.j, T2, T^ are all of bounded variation in A (§18). Since, as we have observed, the finiteness of the area does not imply any property of differentiability for the functions x, y, z, we cannot define, for general continuous parametric surfaces, the Jacobians J1, J2, . Thus a convenient concept of generalized Jacobian shall be introduced only

§1. THE MAIN THEOREMS

(1 . 5

)

7

where the continuity of the functions x, y, z is presupposed (see §28). Needless to say such generalized Jacobians 3 p coincide a.e. in A with the ordinary ones Jp provided the functions x, y, z have first partial ordinary derivatives a.e. in A. By further definition of a concept of absolute continuity for the mappings Tp (§1 3 )* we shall prove the follow­ ing Theorem (L2 ) which extends the statements (1 2) and (T2 ): (L2 ) THE SECOND THEOREM: For every continuous surface S of finite Lebesgue area the plane mappings T , r = 1 f 3 > are of bounded variation, have a.e. in A generalized Jacobians Sp(u, v) which are L-integrable functions in A, and (6 )

The equality sign holds in (6) if and only if the plane mappings T ^ T2, T^ are all absolutely continuous in A. (§30). Finally by making use of the same concept of Frechet equivalence for sur­ faces quoted above, we shall prove also the following Theorem (L^) which extends the statements (1^) and (T^): (L^) THE THIRD THEOREM: Every continuous surface S of finite Lebesgue area has at least one representa­ tion (3) where x, y, z have ordinary first partial derivatives, the ordinary Jacobians Jp are L-integrable functions in A and the Lebesgue area L(S) is given by the classical area integral

Furthermore the representation is conformal in a con­ venient generalized sense (§37)*

1.5 • General Considerations on the Concept of Area For some decades after the criticism of Schwarz and Peano (1.2; ), many definitions of area had been proposed and the Lebesgue area is just one of them. The various definitions, as the Lebesgue area, were motivated by intuitive considerations and, of course, they agreed on the value of the area for all elementary surfaces and even, as it was proved later, in the large class of all surfaces (3) represented by functions x, y, z Lipschitzian in A, but not necessarily beyond this level, for 1 .4

8

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

instance, in the class of all continuous surfaces. On the other hand, the modern tendency of Calculus of Variations to enlarge the class of elements among which to seek for extremals has had an important weight in forcing re­ search toward the goal of a theory for area valid at least in the large class of all continuous surfaces. This is the class with which this book deals. An analysis conducted for decades — and by many authors — has shown that all proposed definitions of area do agree in this class provided the under­ lying concepts are defined with sufficient care and insight. In this book — besides Lebesgue area — we shall especially discuss two of these areas, P(S) and V(S). The area P(S), or Peano area, is obtained by a natural refinement of the definition proposed by Peano in 1890. The area V(S), sometimes called Geocze area, is the form due to C. B. Morrey and E. J. McShane (1936) to which previous definitions have been reduced (Z. de Geocze, W. H. Young, S. Banach) (§7 )- We shall prove the following statement: THEOREM (L*): For every continuous surface have L(S) = V(S) = P(S) (§2 4 ).

S we

Reference shall be made to other proposed areas which have been made to coin­ cide with the areas above through an analogous process as for Geocze and Peano areas. In such a way the area which is discussed in this book appears to be the point of encounter of many different ideas, and the various definitions of L, P, V, etc. are only different interpretations of a unique concept of area, which, following the common usage, we denote as Lebesgue area. It is possible that the reason for the affinity of the Theorem (1 ) for curves and Jordan length, and (T), (L) for surfaces and Lebesgue area, and of the equalities L = V = P etc., lies in an important common property of Jordan length and Lebesgue area. Namely, Jordan length and Lebesgue area are both lower semicontinuous functionals in the class of all curves and continuous surfaces respectively. In any case it shall be pointed out that this is the property which makes them so valuable in Calculus of Variations (in the direct method as well as in the Marston Morse topological theory). It may be added that it is precisely this property that led Lebesgue to his concept of area. Finally the coincidences of V and P with Lebesgue area L have been obtained by modifying the original concepts P, V, etc. in such a way to make them lower semicontinuous. The question arises now whether it is possible to give a convenient set of axioms such that one and only one area satisfy them; namely, the area L = V = P etc. quoted above. We should require, of course, that the area coincide with elementary area for polyhedral surfaces and that it be a lower

§ 1. THE MAIN THEOREMS

(1 . 5

)

9

semicontinuous functional (5*12; 7•5)• A more general question is whether it is possible to axiomatize the concepts of length, area, volume, etc. for mappings in a way analogous to the one which has been so successfully developed since Caratheodory (1 9 1 4 ) for gen­ eral measure functions for classes of sets in general spaces. We shall prove ( § 2 5 ) that the Lebesgue area of S defines an outer Caratheodory measure in A, but the question in the general form above is beyond the scope of this book. The author has intended to free the area theory from difficulties with which all previous expositions (the author1s previous papers included) have been confronted. One of these difficulties was the restriction in the proof of some main theorems that the basic set A is a simple Jordan region. In the book the three main theorems and the equalities L = V = P between Lebesgue, Geocze, and Peano areas are proved for continuous mappings from any admissible set A, where' the class of the admissible sets is large enough to include all open plane sets and all finitely connected Jordan regions. Another difficulty was that the proofs of the second theorem and of the equalities L = V = P were obtained by the consistent use of a par­ ticular solution of the representation problem; namely, Morrey's important theorem ( 3^ . 6 , ii) on the representation of non-degenerate mappings (sur­ faces) from a 2-cell (or a 2-sphere). The present greater generality con­ cerning the basic set A (mapping from open plane sets, or from finitely connected closed Jordan regions) makes particularly difficult the use of this device. On the other hand, since all concepts involved in the first and second theorems and in Lebesgue, Geocze, and Peano areas are defined in terms of a given mapping and are properties of that given mapping, there is no evident reason to use Frechet equivalence in the proof of the same theorems, with the particular methods and technique pertaining to the rep­ resentation problem. Thus in this book first and second theorem and the equalities L = V = P are proved directly (Chapters I-VIII) without any recourse to Frechet equivalence and representation problem. The last two chapters (IX, X) are exclusively dedicated to F-equivalence, representation problem and the technique pertaining to these questions. The last two chapters therefore are somewhat independent of the previous ones. Here the various tools (Dirichlet integral, closure theorem, equicontinuity, retraction) are introduced and used in the proof of Morrey's theorem and the third theorem. Since all these tools are the same as the ones which are being used in recent developments of calculus of variations for surfaces in parametric forms, the author has tried to obtain in the limited space a presentation of the material as general as possible and not related only to the problem of representation. MorreyTs theorem and the subsequent third theorem follow therefore from results concerning the Dirichlet integral, equicontinuity and retraction.

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

Thus Chapters I-VIII concern the properties in the small and in the large of given continuous mappings, and Chapters IX-X the implications of the in­ troduction of F-equivalence. The reader will observe that the direct proof of the equalities L = V = P stems from the introduction of the concept of contour (for any continuous mapping) in Chapter VI by using Caratheodory!s prime end theory. The concept of generalized Jacobian is introduced only in Chapter VIII where really needed, particularly in test and proof of the second theorem; thus the first theorem and the proof of the equalities L = V = P, dis­ cussed in Chapters V, VII do not depend upon the concept of Jacobian. Some new theorems are obtained which assure that the basic metric concepts (area, total variations, multiplicity functions), introduced by means of the operators lim inf and Sup, can also be obtained as ordinary limits of regular processes. Two appendices have been added, Appendix A and Appendix B. Appendix A gives a direct proof of theorem (1 7 -^* ii); Appendix B, a short account of the Weierstrass integral on a surface of finite area, used recently by various authors in questions of analysis and calculus of variations. The book is self-contained inasmuch as surface area is concerned. For the basic concepts of real variable theory and topology references are made only to well known books as S. Saks, Theory of the Integral; L. M. Graves, The Theory of Functions of Real Variables; E. J. McShane, Integration; H. Hahn and A. Rosenthal, Set Functions (in §25); M. H. Newman, Elements of Topology of Plane Sets of Points; G. T. Whyburn, Analytic Topology; R. L. Moore, Foundation of Point Set Theory (in Chapter X); B. v. Kerekjarto, Topologie (in Chapter VI).

§2 : SOME BASIC DEFINITIONS FOR CURVES 2 .1 .

Basic Notations

We denote by p = (x1, x2, ..., x^) any point (vector) of the real Euclidean space E^ and by |p - q| = [(x1 -y1 )2 + ... + (x^-y^ ) 2 ] 2 the distance between any two points P = (x^ xN ), q = (y1, ..., yN ) of If p(u) = [x (u), r = 1 , ..., N], ucl = [a < u < b], denotes any real vector function of the real variable u, then (T, I): p = p(u), ucl, is said to be a MAPPING from I into E^, or a CURVE C: p = p(u), uel, in E^, or a PATH CURVE, or a PARAMETRIC CURVE, and also the short notation C: (T, I)

§2 . SOME BASIC DEFINITIONS FOR CURVES (2 .3 ) shall be used. By [C], or T(I), or GRAPH of C, is meant the set of all points peE^ covered by C. If p(u) is con­ tinuous in I then the mapping (T, I) and the curve C are said to be continuous. If p(u) is continuous in I and linear in each part of a finite subdivision of I into subintervals (i.e., each component of p(u) has this property), then (T, I) is said to be a QUASI LINEAR MAPPING (briefly q.l. ) and C a POLYGONAL LINE. If each point pe[C] is image of only one point, orone segment of I, then the mapping (T, I) is said to be MONOTONE and the curve C to be an ARC or a SIMPLE CURVE, Following the common use, by SUBCURVE C T of any curve C, or ARC C 1 OF A CURVE C is denoted a curve C T: p= p(u), uel1, where I’ is any subinterval of I. If C is a simple curve, its graph [C] shall be denoted simply by C. These definitions should not give rise to misunderstandings. 2.2.

Jordan Length

Given any curve C: p = p(u), uel, continuous or not, by length in the sense of C. Jordan [I] and L. Scheeffer [1] (briefly Jordan length) is meant the Supremum of the ordinary lengths of the polygonal lines inscribed in C, i.e., n 1

(C) = Sup

Y,

lp(uj_) -

i=1 for all finite sets of points a = uQ ’ Ob­ viously, 1(C) coincides with the elementary length for poly­ gonal lines. For N = 1, C: x 1 = f(u), a < u < b, and 1 (C) is simply the total variation V[f] of the real-valued func­ tion f(u), uel. As is well known, f is said to be of bound ed variation (BV)if V[f] < + 00, and to be absolutely con­ tinuous (AC) if, given e > 0, there is a d > 0 such that for any finite system (o^, p^), i = 1, 2, ..., m, of non­ overlapping subintervals of I we have z|f(^i ) - f(a^)| < e whenever £(Pj_ - a^) < 5 .

2 .3

- Lower Semicontinuity of Jordan Length

The Jordan length 1(C) is a lower semicontinuous functional in the class of all curves C. This property can be expressed in the following simple form:

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

(i) If C: p = p(u), uel, Cn : p = pR (u), uel, n = 1,2, is any sequence of curves and p(u) = lim Pn (u) as n— ► for all uel, then 1(C) < 11m 1 (C ) as n — ►- oo.

. »

PROOF: Given e > 0 there exists, by definition of 1 (C), a finite system of points a = uQ < u1 < ... < um = b such that Z|p(ui ) - P(^i_1)l > KG) - 2”1e if 1 (C) < + oo; > e”1 if 1(C) = + oo. Then because of Pn (u) — ^p(u) at each of them + 1 points u = uQ, u1, ..., um, there is an n such that |p(ui ) - Pn (ui)l < for all n > n,

m"1 e

i = o, 1, ..., m.

I p ( u j l ) - P ( u j__1 )| < I p ^ u ^

Hence

- Pn ( u l _ 1 )| + 2 _1 m"1

i = 1, 2, ..

m

e,

•

Finally 1^Cn^ - zIPn (ui) “ Pn ^ui-i ^! > zIP(ui) - P(ui_-))l - 2“! e, i.e.

1

(Cn ) > 1(C) 1

(C) < +

- e for oo;

all

n > n, If

l(Cn ) > e“1 - 2~1 e

if 1(C) = + oo. Thus (i) is proved. The present proof holds also if n Is a continuous parameter. The following example shows that the statement above cannot be improved, in general, even if the pointwise convergence Pn (u) --► p(u) for all uel is replaced by the uniform con­ vergence Pn (u) p(u) in I. Indeed, let C: x = u, y = o, 0 < u < 2 it; Cn : x = u, y = n-1 sin n2u, o < u < 2*, n = 1, 2, ... . Then prP* 1 (C) = 2it, l(cn ) > n, l(Cn ) — ► + oo as n — oo.

*NOTE:

Whenever

1(C) < + oo, p^ — ► p,

1

(CR ) — ► 1(C),

§2 . SOME BASIC DEFINITIONS FOR CURVES (2 .5 ) then the set of the tangents to the curve Cn converges asymptoti cally (in the sense of P. Riesz [1]) toward the set of the tangents to the curve C [L. Tonelli, I]. Other properties of curves of finite length have been given by L. Tonelli [I]. For necessary and sufficient conditions in order that l(Cn )-- **1(C) and for further properties on convergence in length of continuous curve see C. R. Adams and J. A. Clarkson [1], C. R. Adams and H. Levy [1], M. C. Ayer [2], M. C. Ayer and T. Rado [1], G. Ya. Areskin [1].

*2 .4 . Alternate Definitions of Jordan Length If C: p = p(u), uel, is any continuous curve and for any sub­ division a = uQ < u1 < ... i = 1, 2, ..., m, [diameter of a set is the Sup of chords lengths]. Then we have 1 (C) = lim 1 (P) as a — ► 0. A proof of this fact is given in L. M. Graves [I]. Thus for every con­ tinuous curve C the Jordan length is a limit. Another definition of Jordan length for continuous curves is the following one. Let [®] be the collection o f all sequences * = tPn :p = pn (u),uel,

n = 1, 2,

...]

of quasi linear mappings [polygonal lines] such that pn (u) -- ► p(u) as n -- ► oo for all uel. Then we have 1(C) = Inf

lim 1 (P ) n — ► oo n

for every continuous curve. The proof is analogous to the one given in (2.3) for lower semicontinuity and is omitted. Here, as in (2.3), the substitution of pn T P ? o v pn -- ► p is irrelevant.

*2.5.

Further Properties of Jordan Length

The Theorems (1 1 ), (1 2 ), (I3) given in (1.1 ) hold also in E^ and are certainly the main properties of Jordan length. The first part of (1-, ) is an obvious consequence of the elementary inequalities

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

|xr - yr l < IP - Q.I < |x1 - y 1 | + ... + |xN - yN | r = 1, •-•, N, for Euclidean distance (2.1). Besides the original papers of C. Jordan [I], and L. Tonelli [I; 1, 2], very simple proofs of (1 1) and ( 1 ) are given by S. Saks [I, p. 123], L. M. Graves [I, p. 213], and C. Goffman [I, p. 2 4 6 ]. If we denote by s(u), a < u < b, the length of the curve image of the partial interval [a, u] of I = [a, b], then s(a) = 0, s (b) = 1 (C), and the further statement holds: (i) For every continuous curve C: p = p(u), uel, of finite Jordan length, s(u) is a continuous monotone non­ decreasing function and 3(u) = +

ip *2 x 1 + x2 + ..

a.e. in [a, b] [L. Tonelli, I, pp- 71-105; L- M. Graves, I, p. 2 13; S. Saks, I, p. 123]. It will also be pointed out that the Theorem (1 2 ) of (1 .1 ) con­ tains for N = 1 the following well known statements: If f(u), a < u < b, is BV in [a, b], then the derivative f !(u) exists a.e. in [a, b], is an L-integrable function in [a, b], and

'a The equality sign holds in this relation if and only if is AC in [a, b].

2.6.

f(u)

Frechet Equivalence and Frechet Distance for Curves

A function u = h(v), vel^ uel, is said to be a homeomorphism from a closed Interval I1 onto a closed interval I provided h(v) is a real strictly increasing or strictly decreasing con­ tinuous function and u = h(v) maps I1 onto I. According to M. Frechet [1] any mapping, or curve C: p = p(u), uel, is said to be (Frechet) equivalent (briefly F-equivalent) to another mapping C1: p = q(v), vel1, (briefly C ~ C] ) if, for every e > 0 the following statement holds: ( a ) there exists a homeo­ morphism u = h e(v) from I1 onto I such that

§2 . SOME BASIC DEFINITIONS FOR CURVES (2 .7 ) |p [h€(v)] - q(v)| < e for all vcl1. Obviously (i) C ~ C; (ii) C ~ C1 implies C1 ~ C; (iii) C ~ C ^ C^ ~ C2 implies C ~ C2- The class of all mappings which are F-equivalent (to a given mapping C : p = p(u), uel, and therefore to one another) is said to be a FRECHET CURVE C, and each mapping of the class is a repre sentation of C. For the sake of brevity it is often said that they are representations of C. The Jordan length 1(C) is Frechet invariant, i.e., C ~ C1 implies 1 (C) = 1 (0 ^). More generally, given any two curves C, C1 as above, let 5 = | |C, C1|| > 0 be the infimum of all numbers e > 0 for which property ( a ) holds. The number 5 is said to be the FRECHET DISTANCE (F-distance) of C and C ^ The following properties are easily proved: (i) ||C, C11| = 0 if and only if C - C1; (ii) ||C, C1|| = ||C1, C||; (iii) ||C, C1|| < ||C, C2|| + ||c2, C, II; Civ) C ~ C', C1 ~ Cj Implies ||C, C, || = ||C1, ||.

NOTE: The definition of F-equivalence given by L. M. Graves [I] is only a different form of the one given above. The identifica­ tion can be obtained by using known theorems on the limit of monotone functions (see also 3 7 * 1 )• We shall denote by {M, N) the usual distance of two point sets M, N, that is, {M, N) = Inf |p — q| for all peM, qeN. It is easy to prove that for every two curves C, C1 we have {[C], [C1]) < ||C, C^ ||, that is, the distance of the graphs [C], [C^ of two curves is < the Frechet distance of the two curves.

2 .7

* Open, Closed, Oriented Curves

In all the definitions above the curves C, C1 are essentially considered as OPEN CURVES, as we shall see, even if p(a) = p(b). Indeed a curve C is said to be a CLOSED CURVE if not only p(a) = p(b) but the end joints a and b of I are logically identified. In such a way the natural ordering on I is trans­ formed into a cyclic ordering and the segment I becomes topo­ logical equivalent to the boundary of a circle. Accordingly in the definition of F-equivalence for closed curves as well as in the definition of F-distance between two closed curves,all homeomorphisms u = h(v), vel^ uel, from I1 onto I must be considered as homeomorphisms between circumferences. The same modification must be introduced in the definition of simple closed curve (Jordan curve). It is often convenient to define as positive one of the two

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

natural senses on I. Then the curve C: p = p(u), uel, is said to be ORIENTED and in the definitions of F-equivalence and F-distance we have to restrict ourselves to only those homeo­ morphisms which map the positive sense on I into the positive sense on I1. For instance, the two curves C: x = u, y = o, o < u < 1; C ' : x = 1 - u , y = o, o < u < 1, are F-equiva­ lent as non oriented curves, are not F-equivalent as oriented curves (the sense of the increasing u is taken as the positive in both). If C: p = p(u), a < u 2 it rh) depends upon oc and can be made equal to any arbitrary number M > 2* rh. If n = m^, m — ► «>, then a(P) — ► + 00. Thus it is possible to define sequences [Pn ] of polyhedral surfaces approaching to, and inscribed in a right

CHAPTER I.

INTRODUCTORY CONSIDERATIONS

cylinder S, and whose areas approach any number elementary area, or + oo.

M >

the

NOTE: The example above was published first in the second mimeo­ graphed edition of the "Cours" of C. Hermite [I, pp. 3 5 -3 6 , 1882-83], from a letter of H. A. Schwarz, and then reproduced in the collected papers of H. A. Schwarz [I, 2nd vol., pp. 3 09—311^ 1890]. The letters of H. A. Schwarz to A. Genocchi have been published recently by U. Cassina [l].

CHAPTER II.

LEBESGUE AREA

**************************************************************************

55

. THE LEBESGUE AREA L 5

-1 * Admissible Sets

A set A of the w-plane E is said to be admissible in each of the follow­ ing cases: (a) A is a CLOSED SIMPLE JORDAN REGION J, i.e., the bounded complementary region of a simple closed continuous curve in E plus the curve itself; (b) Ais a CLOSED JORDAN REGION J of order of connectivity v, o Fmrr> = as L < L m m Pwv,mn - ► T as n --► oo we have pm

as

C A -

L = + oo].

C t

Fm n c p m , n * , '

Since

A°

n -oo^ ^rnn " ^^mn' T' Fmn^ n *-oo,

and it is not

restrictive to suppose

Fmn ^ A°*Let n = n (1 ) di n (-| ) < U let n = n(m)

any integer such that be any integer such that

as

V x K m ) < ra"^ L' + € if L 1 1 = + 00. By (L.j) (for the number L ’ 1 ) there exists a compact set K C A° and a a > o such that, for all q.l. mappings (P,F)with K C F C A,d(T, P, F) < a, we have a(P, F) > L 1T - € [or > e”1]; hence, in either case, a(P, F) > L ! + e. By (L2 ) (for the number L 1) we deduce that for some of the previous (P, F) we have a(P, F ) < L ! +e, a contradiction. This proves the unicity of L. (d) We have proved that a number L satisfying (L1) and (L2 ) exists [namely that defined in (5*8)] and is unique. Thus each number L satisfying (L^ ) and (L2 )

§5-

THE LEBESGUE AREA L (5-9)

coincides with the Lebesgue area. proved.

Thus (iv) is completely

(v) For any c. mapping from an admissible set A into E^ let (P, F) denote any q.l. mapping, F C A. Then we have L(A, T) = lim a(P, P) as P° \ A0 and d(P, T, F) — ► o. This statement is a consequence of (iv).

(see also

6 -7 >

Note).

(vi) If the vector function p(w) = p(u, v) is constant with respect to v, or to u, in each component of A, then L(A, T) = o. PROOF: Indeed it is possible to approach (T, A) by means of q.l. mappings (P , Fn ): p = pn(w), W€Fn> F° | A°, where pn (w) is constant with respect to v (or u) in each component R of F . Then a(Pn> F ) = o, L(A, T) = o. Thereby (vi) is proved.

NOTE 2: The statement (vi) implies that a surface S degenerated into a curve has area zero. This fact agrees with intuition because such a sur­ face S, reduced to a path curve C, canbe approached as close as we want by polygonal lines, thus of elementary area zero. This holds even in the case where the curve C isan Osgood curve as above, or a Peano curve filling a square Q, i.e., [S] = [C] = Q, or filling a cube K in E^; i.e., [3 ] = [C] = K There is no contradiction in all this be­ cause curves and surfaces are here path-curves and path-surfaces, and C and its graph [C], as well as S and its graph[S], are different concepts. On the other hand, the unit square Q as the mapping nu, S: x = u, y = v, (u, v)eQ, has area1; the mapping S2: x = sin 2 it y = v, (u, v)eQ, "covering Q 2n times", has area 2n; the degenerate surface : x = cp(u), y = t(u), (u, v)eQ, where C: x = cp(u), y = \jr(u), o < u < 1, is the Peano curve filling Q, has area zero.Nevertheless, CS1] = [S2] = [S3] = Q.

Another statement of the same nature as (vi) is the following: For any c. mapping (T, A) whose graph T(A) is contained in a finite system Z of straight lines 1 we have L(A, T) = o. For the proof a sequence (PR, Fn )> ^ T, has to be modified in such a way that P^----- ► T, Pn/Fn^ ^ Then a^^, F^) = o, L(A, T) = o. The details can be left to the reader. Finally, we may conclude with the following statement: For any c. mapping (T, A)from an admissible set A into E^ we have

CHAPTER II. LEBESGUE AREA L(A, T) = 0 provided the graph T(A) is contained in an arc r, or in a simple closed curve rT, or in the sum of a finite system of arcs r and simple curves r! in E^. Let oc denote any plane in E^ and the orthogonal projection of E^ on oc. Then given any c. mapping (T, A) from an admissible set A C E2 into E.-, let us consider theplane mapping (tT, A) from A into oc. (vii) Forevery plane a C E^ we have L(A, particular, L(A, Tp ) < L(A, T), r = 1 , 2 , 3.

T) < L(A,

T).

PROOF: If (PR, Fn ) Is any sequence of q.l mappings such that PR -- ► T, a(Pn, Fn )-- ^I«(A, T) as in (i), a(Ta Pn, Fn ) < a(Pn, Fn ) we have also Pn T and (5 *7 )* Thus by( 5 •8 ) we have L(A,

T) < lim a(xa PR, Fn ) < L(A, T)

Statement (vii) is often denoted as the projection prin­ ciple for Lebesgue area.

5.10.

Lower Semicontinuity of the Lebesgue Area

(i) If (T, A) is any c. mapping, and (Tn, An ), n = 1 , 2, ..., is any sequence of c. mappings such that Tn — ► T, then L(A, T) < lim L(An, Tn ) as n -PROOF:

The hypothesis

Tn -- *-T

implies (5 *3 )

*n c V r c A- < t A0, d(T„, T, An ) — * 0 as n -- Set 1 = lim L(An, Tn ) as n — ► » and let us consider any arbitrary sequence of figures fR C A0, f^ f A0 . By (5 *1 , ii), given any compact set C A° we can determine an integer n = n(Kffi) as large as we want such that Km C A^ C A0, TR ) < 1 + m”1, d(T, Tn, An ) < m ” 1 . In addition, having so fixed n = n(K^, we can determine (5 -9 , iv) a q.l. mapping

n = 1, 2, ... is any sequence of q.l. mappings with Pn ---► T, a(Pn, Fn )— ► L(A, T) (5 -9 ), and (P^, Fn ) is the corresponding sequence with P^ = t Pn, then Pi—

T',

Pn ) - a(Pn, Fn )

(5 -7 ), and, hence, L(A, T l) < lim a(P^, Fr ) =

11 m

a(Pn, Fn )

= L(A, T)

Thus L(A, T ’) < L(A, T) and analogously L(A, T) < L(A, T !); hence (i) is proved.

5»12.

An Axiomatic Characterization of Lebesgue Area

A great deal of research has been dedicated to the problem of axiomatic definition of area. We shall consider the area as a functional (T, A) defined in the class L of all c. mappings (T, A) from an admissible set A into E^. We mention and prove here the following two statements (see also 7-5 )• (i) For any functional Fn ) -- T By (b) and (a), we have *(T, A) < lim (Pn, Fn ) = lim a(Pn, Fn ) = L(A, T) Thus (i) is proved. (ii) For any functional $(T, A) as above satisfying the conditions (a), (b) and (c) there exists a sequence of q.l. mappings (Pn, F ) such that Pn — ►T and a(Pn, Fn ) -- ► $(T, A); we have (T, A) = L(A, T). PROOF: If (PR, F )denotes any sequence as in (c), then by (5 •10 ) we have L(A, T) < lim a(Pn, Fn ) = lim a(PR, Fn ) = (T, A) Thus L < 4> and, by (i), also is proved.

5 .1 3 *

L = 0.

Thereby (ii)

Sufficient Conditions for the Finiteness of Lebesgue Area (i) THEOREM: [C. B. Morrey, 1 , 2 , 3] If (T, A): p = p(w) weA Is a c. mapping from any admissible set A of the w-plane E2, w = (u, v),into the p-space E^, p = (x, y, z), and if (a) the functions x(u, v),y(u, v), z(u, v), components of p(w), are of bounded variation and absolutely continuous in the sense of Tonelli [i.e., BVT and ACT in A°, see (3.2)] and if (b) the partial derivativesxu, xy, ..., zy, which, by (a), exist a.e. in A0 are L2-integrable in A0 ; then L(A, T ) < + «>and

§5L(A, T) = (A° )

J

THE LEBESGUE AREA L (5-13)

J du dv = (A°) J = (A0 )

where couples

J

J2 + j| + J2 j

i [eg - F2

*+5 du dv =

du dv.

Jg, J^ are the ordinary Jacobians of the (y, z), (z, x), (x, y), (5-5)-

(ii) THEOREM [C. B. Morrey, loc. cit.]: The same con­ clusions as in (i) hold if, for the BVT, ACT functions x, y, z it is known that at least two have bounded partial derivatives (and the others are L-integrable); or that for each couple of partial derivatives, say (X u , y v ) , etc., there are two numbers p > 1, q > 1, p -1 + q~1 = 1 such that x^ is L^-integrable in A° and yy is L^-integrable in A0, etc. It is convenient to prove both statements (i) and (ii) in Chapter III (9-8). The following corollary can be deduced: (iii) The same conclusions as in (i) hold provided (a) and (b) are verified in every F°, where F is any closed figure F C A0, and J = J(w) is L-integrable in A0. PROOF:

Let

[fn l be any sequence of closed figures

C A °> fn C f n + l ' f n t A ° - T h e n f n } ------ " ( T ’ f ) (5-3) and, by (5* 10), L(A, T ) < l i m L ( f n, T). On the other hand, by fR C Ait follows L(fn, T) < L(A, T) hence, lim L(f ,T) */2~ (more precisely = 2) while the classical Integral is 1. As we have stated in (1 .4 ), we shall prove in Chapters V and VTII a nec­ essary and sufficient condition for the finiteness of the Lebesgue area and a necessary and sufficient condition in order that L(S) is given by the classical integral (statements L 1 and L2 of (1 .4 )).

NOTE 1: It is not known whether the conclusions of the Theorem (i) above hold -under the conditions that x, y, z are only ACT in A° and J(w) is L-integrable in A0.

§5 • THE LEBESGUE AREA L (5-13) NOTE 2: Given the mapping (T, A): x = x(w), y = y(w), z = z(w), weA, from any admissible set A, let us suppose that the functions x, y, z are continuous in A and the first partial derivatives xu, xy, ..., zy are continuous in A0 . Under these particular hypotheses it is easy to define, as we shall see, a sequence (P , Fn ) of q.l. mappings with Pn — ► T , a(Pn, Fn ) — -I (A, T ) , where I (A, T ) = (A°) J J dw is the area-integral. Let (Fr ) be the sequence of figures Fn C A0, Fn C Fn+1, F° f A° (defined in 5-6, Note 2). Since the functions x, y, z, xu, xv, ..., zy are continuous in the compact set F , they are also uniformly continuous —1 in Fn and, hence, there is a 6>n > o such that |T(w) - T(wf)l < (2n) ,

Ix^w) - x^(w*)|, ..., |zv(w) - Zv(w' )|< (nl^ |Fn l)~1 for all w, w ?eFn, |w - w T| < 5n> where |Fn | is the measure (area) of Fn and = max C |x |, |y|, ..., |zy |] in Fn « Let us divide Fn intoequal squares q of side-length 2~’m with m large enough in order that 2-m+1 < an^ ^ ^ t h e corresponding subdivision of Fn into equal triangles t obtained by dividing each q into two triangles t by means of one of the diagonals. If t = w w 1 w2 is any of these tri­ angles, Wj_ = (u^, v^), i = o, 1, 2, we have either u 1 = uQ + 1, v1 = vQ, u2 = u , v2 = v + 1, 1 = 2"m, or the same relations hold where each 1 is replaced by + 1 - If X(w) = Au + Bv + C is the linear functionwith X(w^) =x^ = x(u^, v^), i = o, i, 2, we have Au^ +Bv^ + C = Xj^, i = o, i, 2, and then A = (x1 - x ) I"1, B = (x2 - xQ ) 1 - 1 . Thus X^, Xy are constant in t and their constant values are *u = [x(uo \

+

V

" x(uo' vo }]l_1 = xu (uo + 91 *

v o ]>

= [x(uQ,vQ + 1) - x(uQ, vQ)]l-1 = xy(u0, vQ + e«l)

where o < e, e1 < 1. An analogous conclusion holds in the other cases. Thus we have |XU - xQ |, |XV - xy | < (niY^ |F |)“1 everywhere in t. If X(w), Y(w), Z(w), weFn, are,the q.l. functions which are linear in each teSR and take the values of x, y, z at the vertices of the triangles t, let (Pn, Fn ): x = X(w), y = Y(w), z = Z(w), weFR . Then we have |Xu - x j , - Zy| < (nlVL^ |Fn l)_1 ,

Yv - *u yv l ^ |Xu " xu l |Yv ! + |xu ! |Yv " ^v1
s. If s = + oo, then L = s = + oo and (ii) is proved. Suppose s < + oo, hence, L(A_^, T) < + oo for every i = 1, 2, ... . For each i there is a sequence (Pin> Fj_n ) — ^ (T, Ai ) as n — ► », F^n C A?, FQn| Aj, and we can suppose d(T, P. , F. )< n“1 as well as a(P. , F. ) < L(A±, T) + 2“^ . Also, Pn - F m + F2n + ... + is a figure, Fn C A , and F±n F - n =0, i + j, i, j = i,2, ..., n. Let us prove that F° j A0. In­ deed any point peA° belongs to a set A?. By Fin } Ai as n — ^ 00 ^ follows P£F?n for all n large enough; hence, P€F° for n large. This im­ plies F° | A0. Let (Pn, Fr )be the q.l. mapping defined by Pn (w) = Pj_n (w) F o r each i = 1, 2, ..., n. Then d(T, PR, FR ) < n_1 (V

Fn } — " (T'

A)>

and n

L(A, T)


L (Ai' T) + 2 1-n

" “ i= i

We have proved that have L = s.

S a(pm' pln) i=l

Fn ) =

L(A, T) < s

11m

(s + 2-n) =

n and, by (i), we

NOTE: The conditions of (ii) are obviously satisfied if A Is an open set and A^ are the components of A, or if A is a finite sum of closed disjoint Jordan regions A^. Further additivity theorems shall be proved in (2 1 .4 ). The main properties of L(A, T) as a set function will be con­ tained in §25.

(iii) If (T, A) is an c. mapping from an admissible set, if is a closed pol. reg. jt C A, if S denotes any finite sub­ division of I into closed non-overlapping pol. regions * *, then L(*, T) > z L(*1, T).

n

§5 • THE LEBESGUE AREA L (5*15) PROOF: We have i t° ) Z * ! ° where the sets a ’0 are open and disjoint. Therefore, by (i) and by (5 -9 * ii)> also L(*, T) = L(*°, T) > Z L(*l°, T) = Z L(it!, T). (iv) If (T, A) is any c. mapping from an admissible set A, and (AQ ) is a sequence of admissible sets A^ C A with An c A ^ , a£ t A0, then lim L(An, T) = L(A, T) as n — ► oo. PROOF: By An C A it follows that L(A^ T) < L(A, T) and, hence, L(A, T) > lim ^(A^, T). On the other hand, (T, An ) — ► (T, A) and, by (5-10), also L(A, T) < Thus

11 m

L(An , T) < lim L(AQ, T) < L(A, T) .

L(A, T) = lim L(AQ, T)

5 »15

as

n — ► oo.

» A Triangular Inequality

(i) For any c. mapping (T, A), let (Tp, A), r = 1, 2, 3, be the three c. mappings (5 -4 ). Then r _

— L2(A, Tr )] < L(A, T) .

PROOF: Let (Pn, Fn ), n = 1, 2, be any sequence of q.l. mappings with (Pn, Fn ) — ► (T, A), a(Pn, Fn ) — L(A, T) (5 -9 , 1 ). Let (Ppn,Fj, r = 1, 2, 3, be the corresponding plane mappings (5 . 4 )• Since for any weA we have “ Tr(w )l < |Pn (w) - T (w)I, then d(Pm , Tp, Fn ) < d(Pn, T, Fn ), r = 1, 2, 3. Therefore, L(A, Tr ) <
. Let Sn be atypical subdivision of Fn for Pn - We can suppose that Sn is regular, that each triangle A = Pn (t), is non-degenerate, has a right angle, and diameter < n”1. For each teSn let

t = w,

and let

A

v2 w 3,A = p 1 P2 P3, P1 = Pn (w1 ), 1 =

have a right angleat

1, 2,

p 1. Let

qi = T^i^ = TPn^wi^ 1 = 1' 2' 3' A * = ql q2 q3' Let (P^, Fn ) be the q.l. mapping from Fn into E^ defined as the mapping which is linear on each teS and maps t onto A 1.First let us observe that we have kj_

- q.j I = |t (p ± ) -

i ^ j*

t (p j )i

i) j = U

< |p± 2; 3

- Pj I ,

■

Therefore area. A'
o, there is a 5 = 6 (e, ) such that any two points -X* w, w 1eJ , with |w - w 1| 1 , let em be a number such that |T(w) - T(wf)l < m”1 for all w, w !eF, |w - w 1| < By (6.i,iv) there is a q.l. mapping w f = H^w), from F into another figure F^ C F°, such that iH^w) - w| < €m - Then F^ is a compact set, C F°. Since F° f F°, there is a smallest integer n = n(m) > m such that F^ C Fn - Finally, let (P*, F) be defined by = Pn V

Then

IT(w) - P*(v)| < IT(w) - T[Hm (w)]| + + |TtHm (w)] - P^H^w)]! < m_1 + n_1 < 2m_1 for all

weF.

On the other hand,

a(P*, F) = a(Pn, F^) < a(Pn, Fn ) < L(F, T) + m 1

61

of

CHAPTER II.

LEBESGUE AREA

/ * \ The sequence (P , F), m = 1, 2, ..., is a sequence hence, L (F, T) < L(F, T). Thus L* < L and finally * L = L.

*6.3.

Lebesgue Equivalence

Any two c. mappings (T, A), (T 1, A 1) are said to be Lebesgue equivalent if there is a homeomorphism w ! = H(w) between A and A ’ such that T(w) = T 1 H(w) for all weA. This is only a particular case of the Frechet equivalence we will introduce in Chapter IX (see §31 ), where we will need it. Of course a necessary condition for Lebesgue equivalence (as well as for Frechet equivalence) is that there are homeomorphisms between the two admissible sets A and A* and Topology gives nec­ essary and sufficient conditions for this. We need not go into this question. The following statement assures the invariance of Lebesgue ‘area for Lebesgue equivalence. (i) If (T, A), (Tf, A 1) are Lebesgueequivalent, then L(A, T) = L(AT, T f). PROOF: Let w f = H(w) be a homeomorphism between A and A 1 such that T = T 1 H. Let (P , F ), n = 1 , 2, ..., be any sequence of q.l. mappings (from F^ into EQ ) such that F„ C F„,-, F^n C A°,1 iii j 11 n+ 1 F° j A0, d(Pn, T, Fn ) < n-1 , a(Pn, Fn ) < L(A, T) + n_1. Thus H(Fn ) = ^ is a finite sum of disjoint Jordan regions J and F^ C A l°. Denote by 3> o the minimum of the following distances: (a) [F^ , A'*}, (b) the minimal mutual distance between the various components J of F^; (c) for each J the minimal mutual distance ■X" between the various components of J . Obviously F^° \ A ’0, o, as n — ► Since F^ is compact and [ F^, A* ) > 3 $n > the set Un of all points w* with (w*, F ^] < 2 an is also compact and UR C A f°. Thus T 1 Is uniformly con­ tinuous on Un and there is an tj , ° < T)n < such that |T!(w) - T !(w!)| < n”1 for all w, w feUn, |w - w 1| < Tj . For each component R of Fn there exists (6.1, ii) a q.l. mapping w f = hn (w) from R onto another pol. reg. R ’ such that |hn (w) - H(w)| < Tjn for all weR. Since nn < an,

cr ,

§6 . SOME ALTERNATE DEFINITIONS OF LEBESGUE AREA (6.4) the various regions R f are disjoint and contained in Un and, hence, in A ! . In addition the mappings w T = hR (w), one for each R C F , can be thought of as a particular homeomorphism from Fn into a figure F^ C Un C A ,q. Let (P^, F^) be the q.l. mappingde­ fined by P^ = Pn h^1. Let w 1 be any point w ' e A ' ° and let y x C A !° be a closed circle of center w 1 and radius r > o. Then the set 7 = H~1(7 !) is compact and hence there is an n such that 7 C FR for all n > n. We can take n in such a way that also r\n < r for all n > n. Then 7 C R where R is a closed pol. reg. component of F .Let J = H(R), R* = hn (R), J = JQ -(J1 + ... + j^)o, R 1 = R^ - (Rj + ... +R^)°. We have now y' C J and, hence, w ?€j°, {w!,J*] > r, and also

By (6.1, iii) wehave w l€R!° for all n > n. This proves that F*° f A l°. On the other hand, for every w ’eF^ and w = (w1),1 we have —

|T ?(wT) - P ^ w 1)! < |T *[hR (w)] - T 1[H(w)]| + + |T 1[H(w)] - T(w)|

+ |T(w) - Pn(w)|
n = 1, 2, ..., be any sequence of q.l. mappings such that F° n3 n”1' a(Pn, Fn ) < L(J, T) + n”1. For any integer m let t] > o be a number such that |T(w) - T(wf)| < m”1 for all w, w 1eJ, |w - w 1| < r\m and let be a homeomorphism from J onto a pol. reg. R^C J° such that iH^w) - w| < T)m for all wej. Then for each m there is a smallest integer n = n(m) > m such that Rm C Fn and we can define(Pm, J) by Pm =P ^ , n = n(m). It is immediate that Pm is a g.q.l. mapping and a(Pm, J) = a(Pn, Rm ) < a(PR, FR ) < L(J, T) + m”1. On the other hand,

f


Rn ) = a(pn> J )and iH^w) - w| < r\n Then for any point wej let r = (w, J*}and n be an integer such that tj < 2~1r for all n > n. By (6.1, iii) we have weRO for all n > n. Consequent­ ly R° — J°. By the same argument, given any compact set K C J°, there is an n such that K C R° for all n > n. Thus we can determine a subsequence [R ] with R0^ * 0J as m — ► ». Let [Rr] denote this

f

subsequence.

On the other hand

|Pn (v) - T(w)| < |Pn H^1(w)

-T H^ 1(v)| +

+ |T Hn-1(w) - T(w)| < n_1

+ n-1 = 2n-1 .

Thus [(Pn, Rn )> n = 1, 2, ...] is a sequence 0 and L(J, T) < L*(J, T). In short we have proved that, if L < + 00, then L* < L < + 00.Analogously, if L* < + 00, thenwe have L < L* < + 00. Therefore we conclude that either L = L* = + «>, or L = L* < + 00. Thereby (ii) is proved.

*6.5.

An Application of g.q.l. Mappings. An Additivity Theorem for L

(i) LEMMA: Given N points p^eE^, i = 1, 2, ..., N, and any € > 0, there is a a > 0, 0 < a < e , and a c. mapping t : p ’ = x(p), PeF3> from E^ onto itself, quasi linear in E^, such that (1 ) |t(p)-t(pi) | < I P “ P !| for all p, p 1€E^; (2) |x(p) -p| < e for all P €E^; (3) t is constant in the solid sphere F^ of center p^ and radius a; (b) every triangle A of E^ is mapped by t into a polyhedral surface A 1 with a(A* ) < a (a).

CHAPTER II.

68

LEBESGUE AREA

PROOF:For every pair of real numbers h > 0 , uQ, let t = t(u; uQ, h) be the q.l. mapping from - oo < u < + oo onto itself defined by v = v(u) = u + h if u < uQ - h; v = v(u) = uQ if uQ - h < u < uQ + h; v = v(u) = un - h if u > u + h. If p. = (x., y,, z.), “ PN i = 1, 2, ..., N let o = € (2- l) and let the 3N mappings t. , r = 1, 2, 3, 1 = 1, 2 , ..., N, be defined as follows. Let t^ 1 = t(x; x^, 2 2 i—2 a), and ti2> bedefined analogously where x, x^ are re­ placed by y, y^ and z, z^ respectively. Then let Ti = ti3 ti2 tii^

i =

1, 2 ,

N,

and

T = TN TN-1 ••• T2 T1 • The mapping t isobviously continuous and q.l. as the product of 3 N mappings having the same properties. Let us observe that for the mapping t defined above we have |t(u) - tCu1)! < |u - u T|, |t(u) - u| < h for all - oo < u, u 1 < + oo, and t(u) is constant in the inter­ val (uQ - h, uQ + h). Thus |T^(p)(p1 )| < |p - p*|, |Ti(p) - p| < 3 (2 2 i"^cr) for all p, p TeE3,and T\ is constant in the cube c. of E of centerp ., of side-length 2 2 i- 1 a, and hedges parallel to the axes. For all p, |T2 T,(p) hence,

p feE^ we have now IT^p) - T 1 (p1 )| < |p - p !|> T2T 1(p*)| < |T1(p) - T 1(p* )|, etc., and,

(p) - x(p')| < |Tn ... T1(p) - Tn ... T,(pi )| < |p - p 1| , and thus (1) Is proved. For each

P€E^

we

[T,Cp) - p| < 3a,

have |T2 T,(p) - T , ( p )| < 3'22a,

|T3 T2 T^p) - T2 T1(p)| < 3-24cr, ..., |T1 - 1

••• T1(p) ” Tl-2 ••• T 1(p)l ^

3

•22l-!+a,

and, hence, 2 |T±_ 1 • • - T 1 (p) - p| < 3ff (1 + 22 + 2 k + • • • +■ ~21 h' = (221'2 - 1) a ,

§6. i

SOME ALTERNATE DEFINITIONS OF LEBESGUE AREA (6 .5 } = 2 , 3 , ..., N + 1 •

In particular

|x(p) - p|

69

=

|Tn ••• T^p) - p| < (22^ - 1 )a = e, and thus (2 ) is proved. If

psFj_

then

|p - Pj_l < cr,

|T±_1 ... T^ (p)- p±| < |T1_1 ••• T^p) - p| +

Ip - p1 l < (221-2 - 1 )cr +

a

= 22l-2a

+

.

That is, T±_1 ... T 1(p)ec±, T± T±_1 ... T^p) = p± for every peF1 - Thus T^ ... T^p) is constant in F^ and the same occurs for t = ... T^ ... T^p). Thereby (3) is proved. Let A be any rectilinear triangle, A C E y and let A 1, A2, A^ be the parts of A with x < x1 -h, or x1 - h < x < x- + h, x 1 + h < x respectively, where h = 22i-2 a and i = 1. Then t11 maps A 1, A^ into parts Aj, A^ which are congruent with A 1, A^, and a2 into its projection A^ on the plane x = x 1. Thus a (Ag) = a(As )> s = 3, a(A^) < a(A2 ), and finally a(Aj1 ) < a(A) where A|1 = t ^ ( a ). Since t is the product of 3N transformations as we conclude that A 1 = t (a ) is a polyhedral surface with a(Al) < a(A), and (4 ) is proved. Thereby (i) is proved.

NOTE l: In the statement (i), if N = 1, i.e., [p] is a single point peE^ then we have a = e/3, and, as it can be deduced from the proof above, also t (S) = p where S is the solid sphere of center p and radius a.

(ii) An additivity theorem for L . If (T, J) is any c. mapping from the closed finitely connected Jordan region J and if [ J ^ , J2, ..., J^] Is any finite subdivision of J into non-overlapping closed finitely connected Jordan regions by means of a finite system [h1, h2, ..., h^] of simple arcs (open, or closed) having in common, or in common with J*, at most their end points and if T is constant on each arc h., then L(J, T ) = Z± L(J±, T).

CHAPTER II.

LEBESGUE AREA

NOTE 2: This statement is a particular case of the general state­ ment (21.4, i and Note). A direct proof of (ii) is sketched below where g.q.l. mappings and lemma (i) are used. Another application of the.lemma is given in (36.3). PROOF of (ii): By (5 -9 , ii) and (5 -1 4 ,i) we have (a) L(J, T) = L(J°, T) > 2 L(J°, T) = s L(J±, T). Now let [p] be the finite system of the points p = T(h .), j = 1, 2, ..., m, actually distinct, and for every n, letcrn, 0 < an < (2n)”1, be the number and rn be the q.l. mapping from E^ onto itself defined by (i) __1 ~> for e= (2n) and the system [p] . By (6 .4 ) there arek sequences [(Pnj_> Jj_)> n = 1, 2, ...], i = 1, 2, ...,k, of g.q.l. mappings with d(Pnl, T, J± ) < on, a(Pnl, J± ) < L(J±, T) + n"1, n = 1, 2 , ..., i = 1, 2, ..., k. Obviously, each mapping (Pnl, Jj_) = tn Pn± is also g.q.l. ,

a(pni' Ji^ - a^Pni' Ji^ and d(Pni' Pnj' Ji^ < (2n)_1* 'Hence, d(Pn^> ^ < 0n + (2n)_1 < n_1 • Now each arc h = h. is common boundary of exactly two regions, say J.,, J2, and if p = T(h)e[p] then lpnj_(^) - Pi < i = 1, 2, for every point weh. Therefore both

Pni^w ^ = Tn Pni^w ^ ^ = 2' ape cons'tant and equal on h. This implies that the mapping (P , J) defined by Pn (w) = pni^¥ ) for everyW€5 . Let d denote the maximum diameter of the triangles teS^. Let P& be the polyhedral surface formed by the sum of the triangular faces A whose vertices are the images on S of the vertices of the triangles t. For a fixed 5 , 0 < 5 < 2”1, let K 1, K !1 be the lim and Tim as d — ► 0 of the elementary areas of the polyhedral surfaces P&. S. Kempisty [6 ] has proved that if F(x,y) is continuous and ACT in Q, then K 1 = K !! = L(S) = I(S).

*

6 .1 0 .

Bibliographical Notes

The form of the definition of the Lebesgue area as givenin (5 -8 ) is used for instance by L. Tonelli [13] and H. Lebesgue. The form given in (6.2) is often used when A is a figure; the form given in (6 .4 ) is sometimes used by T. Rado [I]. Forms involving the concept of Frechet equivalence (see §3 1 ) are used by T. Rado [I] and C. B. Morrey [1] for the case where A is a simple closed Jordan region and are equivalent to the previous ones. The above mentioned statement of Topology (6 .1 , ii) on the approximation of plane homeo­ morphisms by means of q.l. homeomorphisms has been recently extended to spaces E^ by E. E. Moise [1, part. IV, p. 215].

* §7 . SOME CRITICAL CONSIDERATIONS ON AREA *7 -1 - The Projection Principle We have proved (5 -9 , vii) that

L(A,

T) < L(A, T)

for each plane

§7-

SOME CRITICAL CONSIDERATIONS ON AREA (7 .2 )

of E^; that is, for everysurface S = (T, A) the Lebesgue area L(Sa ) of the projection Sa of S on every plane a is < the Lebesgue area L(S) of S. With this it is often meant that the Lebesgue area satisfies the projection principle. oc

It must be explicitly stressed, here that no relation exists be­ tween the Lebesgue area L(S^) of the plane mapping (t^ T, A) and the 2-dimensional Lebesgue measure of the set [S^] = Ta T(A) C oc, i.e., the point set covered by S on the plane oc. We may have indeed L(S) |[S ] I, as the following ex­ amples show. Suppose first that S is a Peano curve S: x = cp(u), y = ^(u), z = 0, (u, v)eQ = [0 < u, v < 1], filling the square K = [0 < x, y < 1] of the xy-plane oc = E23* Then S = S^, L(S) = L(Sa ) = 0 (5 *9 , vi), and |[S ]| = |K| =1. Suppose now that S is the same square K OC 2 twice covered, e.g., S: x = Vu - 4 u , y = v, z = 0, (u, v)eQ, and then S = Sa, [S^] = K, L(Sa ) - 2, |K| = 1.

*7-2.

Areas of the Type of Peano

Peano's research on area [G. Peano, 2] preceded Lebesguefs and initiated what, in modern notation, could be said to be a settheoretical view-point on area. In order to characterize one of the concepts introduced by Peano, let S = (T, A) denote any c. mapping from an admissible setA C E2 into E^, let * C A denote any simple closed pol. reg. and a any plane in E^. Then T maps ontoa compact set T(jt) C E^ and it* onto a closed curve C CE^, and the plane mapping Ta = t^T maps it onto a compactset T (or) C oc and ** onto a closed plane curve CQ C oc. Let us denote by m(jt, oc) a sensible appraisal of the "area11 of the projection Ta (*) and let n(it) = Sup m(jt, oc) for all planes oc C E^. If now U is any finite system of non-overlapping simple closed regions jt C A, let Pm (S) = Sup X m

U

n(«)

iteU

We could denote Pm (S) as the Peano area of S relative to the function m(jt, oc). For surfaces of the class C f (5 -1 3 ) the definition m(jt, oc) = |T (jt)| seems both natural and close to the original definition of Peano. By such a definition of m, the area P^(S) coincides with the classical area-integral and thus m 0

CHAPTER II.

■with the Lebesgue area (5 -

LEBESGUE AREA

for all surfaces of the class

13)

C f.

Beyond this class such a coincidence is lost, in general, and various other definitions of m seem as plausible as the one above, for instance m = | Ta(rt°)|, or |[Ta (jr)]°|, etc. More generally we could define a "multiplicity function" u(p; a , it), pea, giving an appraisal of the multiplicity with which each point pea is covered by Ta (*) [and thus "o shall be a non-negative, Integral-valued function, zero outside Ta (ir), > oin T^it)]. Then m(a, * ) could be de­ fined as the integral of u(p) in a. Various definitions of x> have been proposed. Just recently H. Okamura [1] has proposed for t> the Brouwer index of the mapping ( T, it) in a. In §9 we shall take for -o(p) the absolute value|o(p; C )| of the topological Index (8.2) of the points pea with respect to CQ [-0=0 if pe[CQ]]. Such a definition corresponds to the one proposed by various authors for the so-called Geocze area (next no.) and leads to a Peano area P(S) which coincides with the Lebesgue area for all surfaces, aswe shall prove in ( 2 4 . 2 ) . On the Okamura definition see also S. Mizohata [1], M. Yamaguti [1], and J. Cecconi [10].

* 7 -3

- Areas of the Type of Banach

Let S = (T, A) be any c. mapping (surface) from an admissible set A C E2 into E^ and let (Tp, A), r = 1, 2, 3, be the projections of T on the coordinate planes E^, E22, E2^. For every simple pol. region it C A let us consider the image T(ir) ofit under T and the projections T (it) of T(it) on the coordinate planes, r = 1, 2, 3. Then it* is mapped under T onto a continuous closed curve C C E^ and is mapped under Tp onto acontinuous closed plane curve Cp C E2p, r = 1 , 2, 3. Forany choice of the function m(it, a ) (7-2), 2 2 2 let mp = m(it, E2p), r = 1, 2, 3, and iif(jt) = (m1 + m2 + m ^ ) 2 • If now U denotes any finite system of non-overlapping simple closed pol. regions it C A, let EL(S) = Sup X

“■

u

n'(r) •

«€tr

Then ^(S) is the Banach area of S relative to the func­ tion m(it, a). For m(it, a) = IT^ (it)| we have essentially

§7-

SOME CRITICAL CONSIDERATIONS ON AREA (7-4)

the original definition of Banach area B(S) [S. Banach, 2]. For all surfaces of class C ’, B(S) coincides with the area integral. Beyond this class a more refined definition of the basic function m(*, a ) is needed as for Peano area. In § 9 we shall take for m the same function as in (7*2), i.e., mr = (E2i>) ^

ta'(m) =

+ nig +

Cr )|dp ’

•

Such a definition leads to the area Bm (S) here denoted by V(S) and generally called Geocze area. V(S) coincides with the Lebesgue area for all c. surfaces as we shall prove in (2 4 .1 ). Such a definition of Geocze area V(S) has been en­ visaged by W. H. Young [2, 4 , 6], R. Caccioppoli [9], Andreoli and P. Nalli [1], and has been consistently studied by C. B. Morrey [1, 2, 3] and L. Cesari [7, 9 , 12]. As to the original Banach area B(S), the equality B(S) = L(S) has been proved for all surfaces S: x = x(w), y = y(w), z = z(w) weQ, where x, y, z are continuous and ACT in the closed square Q, and the partial 2 +G? derivatives x^, ..., zv are L -integrable in Q. for some a > o, while examples have been given to show that the equality B = L does not hold necessarily if L 2+Q?-integrability is replaced by L 2-integrability [L. Cesari, 6]. These results are related to the fact that each c. function f(w), weQ, ACT in Q with partial derivatives fu, fy L 2 ~¥CX -integrable in Q, for some a > o have an (ordinary) total differential a.e. in Q, [L. Cesari, 6], while there are functions f, ACT in Q, with derivatives p f , fy L -integrable in Q, having differential at no point of Q, [L. Cesari, All these facts have been extended to classes of functions somewhat more general by A. P. Calderon [1] and T. Rado and P. V. Reichelderfer [8] •

*7 -4 . Minkowski Area and the Set-Theoretical Viewpoint The area of Minkowski [H. Minkowski, 1] is better denoted as a 2-dim. measure of a set X in E^ (analogously for the corresponding concepts of length, volume, etc.), and it is simply defined as the limit (if it exists) of the quotient

CHAPTER II.

LEBESGUE AREA

nig (5 ): 26 as 5 — ► o, where nig(s) is the 3-dim. Lebesgue measure of the 5 -neighborhood X& of X. If this limit does not exist, then lim may be taken, or even lim, as W. Gross [2 ] has suggested. The ideas of Minkowski for length, area and vol­ ume opened the way to the more refined Caratheodory [1 ] and Hausdorff [1 ] definitions ofk-dim. measure. The definition of k-dim. Hausdorff measure of a set X in E^ has been mentioned in (2.13). The sets X C E^ of finite k-dimensional Hausdorff measure have important properties, especially tangential prop­ erties, which have been thoroughly investigated [A. S. Besicovitch, 1, 2, 3; J. P. Randolph, 1; H. Federer, 1, 2, 3, 4 ]. Since a surface S is given by a vector function S: p = p(w), weQ, i.e., by a mapping from Q into E^, it may happen that the surface S covers itself more than once; hence it is necessary to distinguish the (path) surface S from the set [S] of points covered by S, the area of the mapping S (as a functional defined in a class of mappings), from the 2-dim. measure of [S] . Thus a multiplicity function v>(p), o < *o(p) < + °° must, of necessity, be defined so as to be zero outside [S] and to give a sensible appraisal of the multiplicity at each point pe[S]. This naturally gives rise to the possibility of having ■o(p) = 0 at all those points where the surface is not 2-dimensional in character. If u(p), peEg, is one of these functions, let X^ denote the subset of [S] where -o(p) = n, n = 1, 2, ..., . Then a natural definition of Hausdorff area of S (with respect to the multiplicity function u(p)) is

H^ (S) = H^X.,) + 2H2(X2 ) + 3H2(X3) + ... + (00) H2 ( X j [Here 0 ») • a = o if a = o, (») • a = a> if a > o] . Recently H. Federer [8] has given definitions of t>(p) such that H2 (S) = L(S) for all c. surfaces S (c. mappings from a 2 cell). See also E. R. Reifenberg [1].

*7 -5 - Axiomatic Definition of Area Since Caratheodory [2], measure is any set-function f(X) defined for all sets X of a certain class of sets S and satisfying well known properties or axioms. For instance the k-dim. Hausdorff measure Hk (X) satisfies these axioms (2.13). All this is deeply investigated in S. Saks [I] and H. Hahn and A. Rosenthal [I]. A great deal of research has been dedicated

§7-

SOME CRITICAL CONSIDERATIONS ON AREA (7-5)

to the analogous question whether it was possible to define axiomatically length, area, volume, and so on, as functionals of(T) defined for all mappings T of certain classes of mappings [T]. We recall here that K. Menger [1-12] has observed that, by making systematic use of the methods of abstract metric-spaces, it is possible to consider large classes of integrals on a curve as generalized lengths, thus obtaining also an enlarge­ ment of the classes of curves to be used in Calculus of Variations. It is possible that analogous results hold for surfaces. (See also G. Bouligand [1], pp. 196-207.) The problem of characterizing through axioms the Jordan length among all functionals of curves, and the Lebesgue area among all functional of surfaces is a somewhat more special one. We have already seen in (5*12) an example of a set of axioms having the property above for Lebesgue area. Among the re­ quirements which could be taken into consideration for the characterization of the Lebesgue area we mention here, for instance, (a) the property of lower semi-continuity; (b) the principle of Kolmogoroff (5 -1 7 ); (c) the projection principle (5-9; vii; 7.1); (d) the coincidence with elementary area for q. 1. mappings; (e) the invariance with respect to congruences in E^; (f) the property to be subadditive; that is, for every subdivision of the surface in parts the area is > the sum of the areas of the parts (5 -1 4 ). For these questions see A. Kolmogoroff [1], M. Frechet [7], S. Kempisty [5, 6], G. Scorza Dragoni [1, 3 > 4 ], G. Zwirner [1], M. Pagni [1], J. Cecconi [5]* We will have occasion to state in (9 *1 5 ) the recent quite general result of J. Cecconi. For an original approach in abstract spaces see E. Silverman [1, 2]. The question whether the conditions (a), (b), (c) and (d) alone are enough for the characterization of Lebesgue area has attracted a great deal of attention (see M. Frechet [7], R. Caccioppoli [8]) but it is still unanswered, though no counter example has been found. In the analogous problem of a functional (volume) defined for all the c. mappings from a cube of the uvw-space into the xyz-space E^ it has been observed that there are at least two different functionals satisfying (a), (b), (c), and (d) [H. Federer, 6].

CHAPTER II.

LEBESGUE AREA

Nevertheless, as we proved directly in (5.12), the conditions (a), (c), plus the condition of limit there stated, are enough for the characterization of the Lebesgue area. The statement of (5.12) is only a particular case of the following general theorem on the unicity of the extension of lower semicontinuous functionals [M. Fr6chet, 1]. We shall need some definitions on metric spaces which are listed in §10. Let X be any metric space of elements x, let {x, x ’} be the distance in X, let Y be a subset of X everywhere dense in X, i.e., Y = X (thus for each element xeX there is at least one sequence [y ] of elements yn€Y such that yn — ► x, i.e., (yn, x) — ► 0). Let f(y) be a non-negative lower semicontinuous functional in Y having the following property (A): for every yeY there is a sequence [yn ] of elements yReY such that yn — ► y, f(yn ) — ► f(y)« Under these hypotheses, there is one and only one lower semi­ continuous functional F(x) defined in X, coinciding with f(y) in Y [i.e., F(y) = f(y) for every yeY] and having the following property (B): for every xeX there is a sequence [yn ] of elements yn€Y with yn — ► x, E(yn) — F(x) [M. Frechet, 7 ]. We may finally mention that A. S. Besicovitch [4 ] has ob­ served that a surface S (a c. mapping from a square Q, into Eg) may have finite Lebesgue area and cover a set [3 ] in Eg. (graph of S) of positive 3-dimensional Lebesgue measure. Such a surface, for instance, can be formed by a countable (tree-like) system of thin long cones, each of very small area and covering sets of zero Lebesgue measure; since S is con­ tinuous, the set [S] contains all limit points of such cones and it can easily be made so as to have positive measure in Eg. A somewhat analogous situation had been observed by L. Cesari [6] in comparing Banach and Lebesgue areas of flat surfaces, i.e., of mappings from Q into a plane. Let us mention here that for every surface S in Eg of finite Lebesgue area L(S) there is an "essential” subset M of [S], necessarily of 3-dimensional Lebesgue measure zero and whose 2-dimensional Hausdorff measure is equal to L(S) provided a convenient multiplicity is ascribed to each point of M [H. Federer, 8]. We shall encounter in §18 a subset M of S necessarily of Lebesgue measure zero. See also P. V. Reichelderfer [ k] for an approach to the concept of essential part of a surface.

§7-

SOME CRITICAL CONSIDERATIONS ON AREA (7-6)

The fact mentioned above of the existence of surfaces S with L(S) < + °° and I[S]I > 0 has been thought of by some as an objection to Lebesgue area, though the general validity of Theo­ rems L.,, , of (1 .4 ) show that the observed behavior is ir1 L0, 2 L-3 relevant for the theory. The following result of E. Silverman [3] concerning the axiomatic definition of area is revealing in this respect. Let M be the collection of all metric spaces, and C the class of all continuous mappings S = (T, Q) from a square Q, into an element M of m . Let cp(S) be a function­ al (area) defined for each element S of C satisfying the following conditions: (1 ) If S is a simple arc then cp(S) = 0; (2) If S is in E^ and |[S]| > 0 then cp(S) = + °°; (3) 9 satisfies Kolmogoroff1s principle, i.e., if S = (T, Q) S' = (T1, Q) and |T(w) - T(w»)| < |T'(w) - T !(w!)| for all w, w ’eQ, then cp(S) < cp(S 1)Then there is no area satisfying (1 ), (2), (3) and the following further axiom: (4 ) If S is in E^ and is sufficiently smooth, then cp(S) agrees with the generally accepted value of area [E. Silverman, 3 Lebesgue area obviously satisfies (1), (3), (4 ) but not (2) (see 5 .1 7 , i, ii; 5 -9 , iii; 5 .1 3 , 1 , ii; 5 -9 , Note 2). *7.6.

Extension of Lebesgue Area in Abstract Spaces and Generalized Surfaces

In spite of many difficulties E. Silverman succeeded In extend­ ing the concept and most properties of Lebesgue area to c. mappings from a 2-cell into a Banach space. In a further ex­ tension to all metric spaces, a particular space, the space m of the bounded sequences, was used consistently where the dis­ tance 5 of two points x = [xj_], y = [y-j_], x, yem, is de­ fined, as usual, by 8 = Sup|xj[ - yil . For each c. mapping (T, A): p = p(w), weA, from a 2-cell A into a metric space B, there is an isometric mapping (T, A) from A into m which has the same Lebesgue area. Further results have been obtained concerning the question of the axiomatic definition of area [E. Silverman, 1, 2, 3, 4 ]. In connection with the calculus of variations, L. C. Young [6, 8, 9, 11] has introduced the concepts of generalized curves and surfaces [See also E. J. McShane, 10, 11, 12; W. H. Fleming and L. C. Young, 1]. In the calculus of variations integrals over a surface S: p = p(w), weA, are considered H(f) = I(S; f) = (A) / f [p(w), J(w)] dw for every function f(p, t), p = p(x, y, z), t = (t-j, t2, t3 ) with f(p, ht) = hf(p, t), h > 0 [see Appendix B]. If for a given S we think of I(S; f ) as defined for all possible f, we interpret S as defining the Banach operator H(f), linear

CHAPTER II.

LEBESGUE AREA

in f. Then the generalized surfaces Ho(f) are defined as those Banach operators which are limit elements of elementary surfaces. By this process existence theorems of the calculus of variations have been proved which are widely more general than usual. The extremals may be generalized surfaces rather than ordinary ones.

*7 -7 « Further Bibliographical Notes For the problem of the area in the works of Archimedes, see T. L. Heath [I]. For an historical insight on the concepts of limit, integral, length and area at the origins of Calculus, see A. Agostini [1, 2] and A. Rosenthal [9]. For an outline of the development of the concept of area from the foundation of Calculus till the example of Schwarz and Peano, see F. Sibirani [1]. Besides the definition of area P(S) sketched in (7*2), an­ other concept of area, vectorial in character, was proposed by G. Peano in the same paper [2]. For recent developments of the same concepts, see F. Severi [1]• For general theories on the concept of area following the basic paper of S. Banach [2], see J. Schauder [1] and W. Gross [2]. See also 0 . Janzen [1], E. Cartan [1], V. Glivenko [1], A. S. Kronrod [2], G. Ya. Poplaskaya [1], Z. de Geocze [1 —1 5 ]• Following an observation due to A. Cauchy [1], a concept of area C(S), set-theoretical and integral geometric in char­ acter, has been studied by R. G. Helsel and T. Rado [2], and such an area C(S) has been proved to coincide with the Lebesgue area L(S) for all surfaces (i.e., c. mappings from a simple Jordan region). Following another observation due to J. Favard [1] another area F(S) integral-geometric in char­ acter has been studied by E. J. Mickle and T. Rado [1] and such an area F(S) has been proved to coincide with the Lebesgue area L(S) for all surfaces S as above. For studies on the area of harmonic surfaces S: p = p(w) (i.e., whose components are harmonic functions) as a func­ tion of their boundary lines, see E. Baiada [2] and K. H. Carlson and L. C. Young [1]. For intrinsic definitions of area, i.e., definitions in which a lesser use is made of the representation than usual, see M. Frechet [11] and H. Buseman [1, 2]. Besides the recent important results of H. Federer mentioned in (7 *4 ) and (7-5), we shall quote here the extensive re­ search, set theoretical in character, of the same author [H. Federer, 1-8], for the comparison of Hausdorff, Favard, Lebesgue and other areas. On the same subject see also E. J. Mickle and T. Rado [4 ]. An integral formula of the Gauss type for every bounded simply connected open set in E? is due to H. Federer [2]. [See also K. Krickeberg, 2.] On Gauss’ and Stokes1 formulas for continuous surfaces see also J. Cecconi [7, 8, 9, 10]. (Remark added during the correction of the proofs.) Ch. J. Neugebauer has orally communicated to me to have proved by examples that conditions (a), (b), (c) and (d) of (7.5) are not enough to characterize Lebesgue area.

CHAPTER III.

THE GEOCZE AREAS V AND U

AND

THE PEANO AREA P

**************************************************************************

§8.

THE TOPOLOGICAL INDEX

Concepts such as topological index, order of mappings, loop­ ing coefficients, etc. are generally discussed in topology [see P. Alexandroff and H. Hopf, I, Chapter XII; S. Lefshetz, I, Chapter IV; M. H. A* TTewman, I, Chapter VIIJ . Here only the particular case of the "order of a point peE2 ■with respect to a closed plane curve C C E2ft (topological index) is needed. This concept admits of a well known and very simple metrical approach which is given below (8.1-2) [Cf. P. Alexandroff and Hopf, I, Chapter XII, §1, No. 6, p. 4 6 2 ; T. Rado, II, Chapter II, 4 -3 4 , p. 1 4 9 ]- Such a metrical approach suffices for the present elementary ex­ position (8.3-10) of various properties of the topological index needed here, properties which are analytical in char­ acter and not discussed in topology. More detailed refer­ ences are given in (8.16). Connections with homotopy and intersection numbers are referred to in (8.13-15). For a far reaching use of topological indices in the theory of functions of one complex variable see Marston Morse [III].

8.1.

Polar Representation of a Continuous Curve

By using standard angular notations we can supposethat the p-plane E, p = (x, y), is counter-clockwise oriented, i.e., xoy = 2”1*. Let (p, co) be any system of polar coordinates of pole o = (0, 0) and polar axis ? in E [p radius vector, cd argument]. Then p = pQ(p) = |p - o| is a single-valued non-negative continuous function in E, while co = oi(p) is defined mod 2it at each point p + o. Nevertheless, if p ^ o is any point of E and 7 is the circle |p - p| < |p - o|, then the infinity of the determinations of cjo are given by single-valued continuous functions o> (p), p€7, congruent mod 2it. Let C: p = p(u), uel = [a < u < b], p(u) = [x(u), y(u)], be any con­ tinuous curve in E such that o is not in [C ]By a polar repre sent ation of C (relative to the cartesian representation p = p(u), uel, ) we mean C: p = p(u), cd = ao(u), uel, where p(u), cd(u ) are single-valued continuous functions in I and tp(u), ao(u)] are the polar coordinates (p, o>) of p(u) for every uel. Because of p

the function

p(u)

(u) = Pq [p(u)] = Ip (u) - o |

is uniquely determined and continuous in 83

I.

If

cd( u

),

84

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

cd?(u ) a re any two p o s s ib le f u n c t io n s cd(u), th e n cd(u) - o>! (u ) = o(mod 2 *) and, b ecau se o f i t s c o n t in u it y , th e d if f e r e n c e o>(u) - cd’ (u ) i s c o n s ta n t in I (a m u lt ip le o f 2it). Thus cd(u) i s d e te rm in e d i n I up to a c o n s ta n t , a m u lt ip le o f 2*. The c . f u n c t io n cd(u), u e l, can be d e te rm in e d th ro u g h th e f o llo w in g p ro c e d u re . L e t D = (o , [ C ]3 be th e d is t a n c e D > 0 from o to th e compact s e t [C] and l e t o>Q be any o f th e d e t e rm in a t io n s o f cd a t th e p o in t p = p Q = p (u ), uQ = a . Thus a co n tin u o u s f u n c t io n o>0 ( p ) i s u n iq u e ly d e te rm in e d b y th e c o n d it io n cd0 (p q ) = cjdq i n th e c i r c l e 70 : Ip “ P 0 I < 2" 1D. I f [u , u 1] i s th e m axim al s u b in t e r v a l o f I whose im age i s i n yQ, th e n th e f u n c t io n cjo(u) = cDQ[ p ( u ) ] i s s i n g l e ­ v a lu e d and co n tin u o u s i n [u Q, u 1 ] . S t a r t in g from u 1 w ith th e v a lu e o)1 = cd[p(u.j )] we can re p e a t t h i s p ro c e d u re , and so on; i n a f i n i t e number o f s te p s th e p o in t b i s re a ch e d , b e cau se o f th e u n ifo rm c o n t in u it y o f th e v e c t o r f u n c t io n p f u ) i n I.

8 .2 .

The T o p o lo g ic a l In d e x

F o r any p o la r r e p r e s e n t a t io n C: p = p ( u ) , o f a g iv e n co n tin u o u s cu rv e C, se t

0 (p ; C )

00 =

cjo( u

),

u e l = [a < u < b]

0( 0; C ) = ( 2jc) _1 [00(b) - 05(a)] Thus 0 ( 0; C ) does n ot depend upon th e p a r t i c u l a r c h o ic e o f th e c o n tin u o u s f u n c t io n co(u). Nor does 0 ( 0; C ) depend upon th e c h o ic e o f th e p o la r a x i s r a t 0 b e c a u se , b y r o t a t in g ? , a l l f u n c t io n s 00(u ) a re m o d ifie d b y an a d d it iv e c o n s ta n t . L e t C be any c lo s e d c u rv e . Then a i s I d e n t i f i e d w it h b , p ( a ) = p ( b ) , and b e cau se o f 00(b) = cd ( a ) (mod 2 k), we h ave t h a t m = 0 ( 0; C ) i s an in t e g e r ~ 0. L e t r be th e c irc u m fe re n c e o f any c i r c l e and e th e argum ent o f th e p o in t s o f r w it h r e s p e c t to any p o la r syste m w it h p o le a t th e c e n te r o f r . Then any homeomorphism H from I onto r tra n s­ form s each c a r t e s ia n o r p o la r r e p r e s e n t a t io n o f C on I in t o a c a r ­ t e s ia n o r p o la r r e p r e s e n t a t io n o f C on r , sa y C: p = p ( e ) , or C : p = p ( e ) , co = oo(e), where we can suppose t h a t p ( e ) , p ( b ), 00(0) a re a l l d e fin e d i n (-00, + 00) and a ls o t h a t p (e ), p ( e ) , 03(0) - me a re p e r io d ic o f p e r io d 2it. A c c o rd in g to ( 4 . 4 ) th e e q u a tio n s above g iv e a r e p r e s e n t a t io n o f C i n e v e ry i n t e r v a l (0 , 0 + 2it) o f le n g t h 2jt and a ls o 2itm = 2* 0 (0 ; C ) = 03(0 + .2*) - o>(e ) f o r a l l 0. F o r any p o in t p n o t i n [C] th e number 0 (p ; C ) can be d e fin e d a n a lo g o u s ly b y t a k in g p a s th e p o le o f any syste m o f p o la r c o o r d in a t e s (p , cd) i n E. F in a lly le t 0 ( p; C) = 0 i f pe[C]. Thus 0 (p ; C ) i s

§8.

THE TOPOLOGICAL INDEX (8 .3 )

85

defined, for every peE and is called the TOPOLOGICAL INDEX, or ORDER, of p with respect to C. Since we shall consider 0 (p; C) as a function of p, for the sake of brevity, we will denote the finite, integral, single-valued function 0 (p; C), peE, as the topological index of C in E. The index 0(PJ C) gives exact mathematical meaning to the intuitive notion of the "number of times C winds counter-clockwise round pM (see illustrations a and b ).

0 It follows from the previous considerations that 0 (p; C) = 0 whenever is contained in a sector of center p and opening < 2 it. Obviously 0(p_; C), peE, does not depend upon the system of xy-coordinates in

8 .3 .

Invariance of

0

(p; C)

[C] E.

With Respect to Frechet Equivalence

By C, C ! we shall mean always continuous closed oriented curves in E and by ||C, C 1|| the Frechet distance of C and C 1,( 2 . 6 ) . Thus C ~ C means that C and C 1 are Frechet equivalent and thus C ~ C 1 Is equiva­ lent to |jC, C T|| = 0. If A, B are any two sets in E, then by [A, B} = Inf |p - q| for all peA, qeB, we denote the distance between the sets A and B (2.6, Note). (I) If ||C, C f||< (P0 j [C]}, then 0(pQ; C) = 0(pQj C 1) and pQ is not on [C] + [Cf]. PROOF. Let pQ = o, a = ||C, C fL to, [C] ) = a + 2e; hence a >0, e> 0. We can suppose C, C ' be the images of the circumferences r, r1 of any two circles. If 0, s are the arguments of the points of r, rr with respect to their centers, then C, C ’ have (cartesian) representations C: p = p(e), C f: p = p'(s), where p, p l are periodic continuous vector-functions of period 2 it and 0, s may range each in any interval I, I 1 of length 2 n . Because of ||C, C 1||= a, there is a homeomorphism s =•s(e) between randr1 [s(e + 2 it) - s(e) = 2 it for all 0] such that, if p = p(0), p 1 = p ![s(0)], we have |p - p f| < a + €. On the other hand, Ip-ol >cr + 2e, and hence

86

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

|p* - o| > |p - o| - IP - p T| > cr +

2e

- a - € = €>

0

.

Thus o is not on [C1] and both C, C 1 have polar representations C: p = p(e), cd = 03(0); C1: p = p(s), oo = cd(s). For the sake of brevity, we denote 00(0), cjd1 [s(0 )] by 00 and cd? respectively. In the tri­ angle opp1 the angle pop1 is < 2 ”^. Indeed, in the contrary case, we would have a + € > |p - p !| > |p - o| >

a

+ 2e ,

a contradiction. Thus 00 - 001 = 2* k + A(e ) with k an integer, -2 ""1* < A(o) < 2 ~^it, and cd - !, cjdm at w = w2 congruent (mod 2it). We may consider the polar representations of C, C-, C9 * -)f * on Jt , Jtj, Jt2 respectively, which are defined by the functions o>(w), 0, cd(w), weQ, such that (1) C: p = p(w), co = cjo(w ), wejr , is a polar repre* sentation of C on Tt ; * (2) C. : p = p(w), cd = cd(w ), wejt., is a polar rep# resentation of C^ on jt^, i = 1, 2, ..., n. The proof is the same as for (i). Let it beany simple closed pol. reg. of the w-plane, w = (u, v), let (T, jt): p = p(w), wejt, be any c. mapping from it into the p-plane E, p = (x, y), let C be the continuous closed curve which is the image of -Xit under T, andlet p be any point peE. (ii) If p isnot in T(it°), then 0 (p; C) = 0. Hence 0 (p; C) =£ 0 implies peT(jt°). PROOF. If peT(jt*) = [C] then 0 (p; C) = 0; hence we can suppose p not in T(jt). Let D = (p, T(jt)}, hence D > 0, and let d > 0 be any number such that |p(w) - p(-w!)| < D whenever w, w fejt, |w - v 1 | < d(uniform continuity of p(w) in n ). Let it =it-j + ... + it be any finite subdivision of jt into non-overlapping simple closed regions each of diameter < d. Then each set T(jt1 ) and, there* fore, eachcurve C^: (T, jt^), is seen from p under an angle e and J = (JQ, J1, ..., J^), then (w, J^J > {w,J*} > € > ||J*, J|*IU i = 0, 1, ...,u. Hence, by (8.3, i), we have 0 (w; J.. ) = 0 (w; JJ ), i = 0, 1, ..., d, and w is not on J * * * nor on J f . In E - (J + J ! ), and because of (8.6, iv), the points wej are characterized by 0 (p; J ) = + 1, 0 (p; J.) = 0,1 = * the points weE - J are characterized by 0 (p; J j = 0, i = 0, 1, .. ., t>; * * or 0 (p; JQ ) = + 1, 0 (p; Ji ) = + 1 for some i = 1, 2, ..., d, and 0 (p; Jj) = 0 for all j = )= i, j = 1, 2, ...,13. The same holds for J !. Thus either wej0, wej’0 and hence wej° J f0, or weE - J, weE - J T and hence we(E - J)(E - J !)*

(i)

Properties of Convergence of

If ||Cn, C || -►o, then (a) 0 (p; C) = lim 0 (p; Cn ) as p not in [C];

n ►» -

8.7.

0

(p; C)

for all

1 , 2 ,...,^

92

CHAPTER III.

THE GEOCZE AREAS V AM) U AN D THE PEANO AREA P

(b ) 10 (p; C)| < lim |0 (p; Cn )| as n -- - oo for all (c) (E) / 1 0 (p; C)| < lim (E) / |0 (p; Cn )| as n -- * 00. PROOF. (a) is a consequence of (8.3, i); (b) is a trivial consequence of (a) because |0 (p; C)| = 0 for all pe[C]; (c) follows from (b) and Fatou's inequality [S. Saks, I, p. 29]. (ii) If 0

(1 )

||Cn, C|| --- ►0 and the functions (PJ C ), are L-integrable in E, and if

lim n -- oo

(E) /

0

(p; C )= (E) /

0

(p;

C ) + X,

0

(p; C),

X

| 0 ,

then (2)

lim(E) / n -- ► oo

10

(p; Cn )| > (E) /

|0 (p; C)| + |X|

.

PROOFLetAn be the difference between the two integrals in (l). The function 0 (p; C) is L-integrable in E; hence, given e > o, there ex­ ists a a > o such that, for any L-measurable set h C E, |h| < a, we have (h) j|0 (p; C)| < 2~1 e. Let [C] be the p-neighborhood of [C], the set of all peE whose distance from [C] is < p. Since [C] is compact, we have [C] C [C]p, [G ]p ---► [C] as p o +. Consequently |[C ] | -- ► |[C ]| as p -- ► o +, where by |A | we denote the Lebesgue measure of any measurable set A of the xy-plane. Consequently we can choose p > 0 in such a way that ItC]| < |[C]| + a.Let n be the smallest integer such that ||C ,C|| < pfor all n > n. By (8.3, i), for n > n and peE - [C] , we have 0 (p; Cn ) = 0 (p; C), while 0 (p; C) = 0 for all pe[C]. Consequently, for n > n, we have |An | = |(E) / =

| [ C ] p/ 0 (p;

=

IC]p/

= (E) /

(p; Cn ) - (E)

I 0

(p; C)| =

Cn ) - [ C ] p / 0(p; C)| < [C]p / |0(p; Cn )| + [C]p / I0 (p; C)| =

|0 (pj

10 (p;

0

i.e.,

Cn )| - [ C ] p I |0(p; C)| + 2[C ]p / |0 (p; C)| =

Cn )| - (E) I

10 (p;

C)| + 2 C[G]p - [C]) I |0 (p; C)| ,

§8.

THE TOPOLOGICAL INDEX (8.8)

93

where the last term is < e and |A^| -- |X| as n -- *-oo. Thus the lower limit (2) is > [X,| — e and because e Is any positive number, (2) Is proved.

8.8.

Semirectiflable Curves

A plane continuous curve C: x = f(u), y = ^(u), 0 < u < 1> is said to be semirectifiable if at least one of the two functions f(u), g(u) is BV in [0, 11 . (i) If C: x = f (u), y = g(u), 0 < u < 1, is semirectifiable, then |[C]| = 0. PROOF. ForInstance suppose g(t) Is BV and let V(u) be the total variation of g(t) in [0, u]; hence, 0 < V ( u ) < V < + o o , where V = V(1 ). The function f(u) is continuous in [0, 1] and also uniformly continuous; hence, given € > 0, there is a 5 > 0 such that |f(u) - f(uf)i < e for all u, u ! e [0, l], |u - u ’| < 5 . Let 0 = uo < u1 < ... < un =1 be any finite subdivision of [0,1] each of length < 5 , and let v i = v (u i )

- V ^u I - i

in parts

1 = 1 > *‘ ' n

Then we have |f(u) - f(u± )| < €,

|g(u) - g(u± )| < vi

for any < u < u^. Therefore, each arc c^ image of [ui-i' u^], is contained In the rec­ tangle r^ of center [f(u^), g(u^)] and sidelength 2e and 2v^. Hence [C] C z r^, |Z r^| < ke z v^ = 4 eV. Since e is any positive number, we have |[C]| = 0 and (I) is proved.

Let C: x = f(u), y = g(u), 0 < u < 1 be any continuous closed curve of the p-plane E, p = (x, y), and let s(x), - 00 < x < + o. Therefore a polarrepre­ sentation C: p = p(u), cd = (l ) - cu(o) = 2 « m- Let X X 0 be the minimum integer such that 1 jr>a = co(o). Then, X it + 2—”1 Jt - * < a X it + 2~ and the 2 m numbers x it + 2~1 it + 1 it, i = o, 1, .. ., 2 m-1, are all < b = o>(i) as it follows from Xit + 2*’1it + (2 m -1 )it < a + it + 2m.it - it= b. If u. is the minimum u such that 0 < u < 1, —1 o)(u) = Xit + 2 it + iit, then 0 < uQ < u1 < ... < u2m-1< where at mostone of the signs < may be replaced by =. ¥e have ou(u^)assuming the values + 2~1tc (mod 2i t ) , alternately as i takes successively the values 0, 1, . . . , 2 m-1, and because of g(u) = yQ + p(u) sin o>(u), p(u) = [f2 + g2]2 > 0, we have g ( u ^ ) > J Q g(u^) < yQ alternately as i = 0, 1, ..., 2 m-1. Consequently g(u2m-1) < yQ according as g(uQ ) > yQ and, since yQ * g(o) = gO ), the number j Q is an interior point either of £g(o), g(u.,)], of tg(u2m^ This implies t(y ) > 2m. The second inequality of (ii) is proved, while an analogous proof holds for the first in­ equality. Thereby (ii) is proved. (iii)

For any semirectifiable, closed, continuous, oriented curve C: x = f(u), y = g(u), 0 < u < 1 > the function 0 (p; C) is L-integrable in E.

PROOF. The set [C] is compact, hence contained in a square Q = [- M < x, y < M] C E and 0 (p; C) is

1*

§8.

THE TOPOLOGICAL INDEX (8.8)

B -m e a su ra b le i n E and ze ro o u t s id e Q. By d e f i n i t i o n a t l e a s t one o f th e f u n c t io n s f , g i s BV, hence by (2.11), a t l e a s t one o f th e f u n c t io n s s ( x ) , t ( y ) , say s ( x ) , i s L -in t e g r a b le in ( - » , + » ) and hence a ls o i n (-M, M). By ( i i ) we have 0 ( x , y ; C ) < 2"1 s f o r a l l - M < x , y < M w it h th e e x c e p tio n o f a t most one v a lu e o f x . Thus 0 (x , y ; C) i s L -in t e g r a b le in Q and hence i n E , and ( i i i ) i s p ro v e d . (iv)

If C, Cn , n = 1, 2, . . . , a re c o n tin u o u s , c lo s e d , o r ie n t e d c u rv e s o f th e p -p la n e E , p = ( x , y ) , if ||Cn , C||----- ► 0 and C i s s e m ir e c t i f ia b l e , th e n 0 (p ; C ) ------► 0 (p ; C ) a . e . i n E as n ----- ► 00. T h is a s s e r t io n i s a consequence o f ( i ) and o f (8.7, i ) (v ) If C, C , n = 1, 2, . . . , a re c o n tin u o u s , c lo s e d , r e c t i f i a b l e o r ie n t e d c u rv e s o f le n g t h s 1 , l n , i f ||Cn > C ||----- ► 0 a s n ------ ► 00 and 1 < M fo r a ll n, th e n ( a ) 1 < M; ( b ) 0 (p ; CR ) -----► 0 (p ; C ) a . e . i n E as n -----► 00; (c)

lim ( E ) J 0 (p ; C n — ► m o r t R ( y ) > m. B e cau se o f / s ( x ) < M, / t R ( y ) < M we have

■ O '

I1™ 1

M m_1-Let

W

J nm denote

96

CHAPTER III.

THE GEOCZE AREAS V AND U AMD THE PEANO AREA P

the sets of all points p = (x, y) where |0(p; C )| ! n or = m, respectively. Then I C I x 1 ^ and hence

11 ^ 1


m,

I n a d d it io n

^nm “^nm + J n,m + 1 + ' ' ' ’ ^^nm ^ ~~ ^^nm ^ + ^ n , m+ 1 ^ C o n s e q u e n tly , i f i s d e fin e d by =

dnm) / l°(P; cn )l>

+

then

^ n m 1 = m l J ranl + (m+1} |Jn,m + 1 1 + and hence ^nm1 = m ' V

+ 1J n , m + 1 ^ +

l ^ m + s 1 + • • < M2[m" 2 + 0 be any number and let m = m(e) be an Integer such that the last expression is < 2”" * €. Thus m(e) depends on e but not on n. Let t] = r](e) = 12 1m €, and h be any measurable set h C Q, |h| < Then h = (Q - Im )h + I^h = h ’ + h* 1 and now (h1 1) / 10(p; Cn )| < T ^ < 2~1 e, (hf) / |0 (p; Cn )| < m |h1| < m |h| < 2_1 e. Thus (h) / |0 (p; Cn )| < e for all |h| < t) and all n. Vitali's condition is satisfied and because of (b), also (c) holds [F. W. Hobson, Vol. II, p. 297 ]Thereby (v) Is proved. —

—

NOTE. The conclusion of statement (v) does not hold necessarily if the condition 1 R < M Is not satisfied. An example Is'given in (9-10, Note).

6.9. Further Properties of the Topological Index (I)

If Pj_ = (x^, y^), 1 = 1 , 2 , are any two points of E9, if d = x1 y2 - x2 y^, if the segment s = p 1 p0 does not contain the origin o = (0, 0), s: x = x1 + (x2 - x1 )u,y = y, + (y2 - y, )u, 0 < u < 1, if s: p = p(u), CD = co(u), 0 < u < 1, is the relative polar representation of s (8.1), then sgm [co(1 ) - cd(0) J= sgm d If d =)= 0, cjo( u ) = constant If d = 0 .

PROOF. We have

■■Hi 1.

p(u)

= | X

1 + (x2 -

X

1 )u

+ (y?

2^ 2

> 0

§8. for all

0
0 [t^ > 0], i = 1, 2, 3. Let p be any point p e A ° . It Is not restrictive to

98

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

suppose numbers

p = o = t^ > o,

(o, o). Then there are three real 1 = 1, 2, 3 with

X1 + fc2 X2 + *3 x3 = 0 ' fcl y, + t 2 y2 + t3 y 3 = 0 . Thenumbers d1 = j 2 - x2 d2 = x2 y3 - x3 y2, d3 = x3y 1 - x1 y3cannot be all zero because, otherwise, we would have D = 0; hence t, = t2 = t3 = d1 ; d2 = d3 and. t h i s th at

im p lie s

d-j,

D = d1

d 2,

+ d2

1 = 1 , 2 , °


P ) da| dU

Ih " 4

( Cf. 3 2 . 5 , i i ) . we h ave

Now, b y e le m e n ta ry t r a n s fo r m a t io n s ,

J

vn < h -

pc+h < h“

/»u+h

I

*b du

J

/*c+h

^u+h d p j^

Ix^ a ,

pb+h

p)| da

^a

dp I

|x ( a , p) | da

^a

/

du

«^a-h

/>c+h < h“ J dp • A°(p) J Ix^a, p)| da =

h

1 rc+h

+(P) dp

S in c e th e l a s t i n t e g r a l ap p ro ach e s i ( c ) as h ------>-0 and t ( c ) < + «> th e re e x i s t s some M < + 00su ch t h a t v n < M < + oo fo r a ll n la r g e enough. The same re a s o n in g h o ld s f o r y n and f o r each s id e o f I . Thu s l ( C n ) < M f o r a l l n la r g e enough. By x n ----- ^ x , y n z n j : y i t f o llo w s ||Cn , C ||------► 0 a s n ----------------and, by ( 8 . 8 , v ) , we have

as

J 0 (p ; Cn ) d p ------ ( E * ) J 0 (p;

( E2 )

/ 10 (p; Cn )| dp ► ( E g ) f 10(p;

n ----- ► » .

By ( 3 ) ,

dp , C)| dp

(*0 , ( 5 ), we co n c lu d e t h a t

(E* )

I 0 (p ; C ) dp =( I )

(E2 )

I 10 (p ; C ) | dp
•••>

) + b±

n> th e n

c . (ui - U.^ ) =

2_1 t g ^ ) - g(u1_1)] [f(u±) + f(ul_1 )] =

2-1 [xi-t

- xi ^i-i J + 2-1 [xi ^i - xi-i y n ] '

Let o = (o, closed curve the oriented Ti = (° pi_1

o) and C^ be the oriented 0 ^ = 0 Pi_1 p^; that is, boundary of the triangle p^). By (8.9, ii) we have

§8.

THE TOPOLOGICAL INDEX (8.10)

(E ) / 0 (x , j ;

C± ) = + a re a ( T j_) =

= 2 " 1 ( x . , , y ± - x ± y ±_ i ) • T h e r e fo r e ,

JU±

f dg - ( E )

f

o(x, j;

C1 :

"U i-1 + 2 - 1 [ x ± y ± - x 1_ 1 y.j__i ] > i = 1, 2,

.

n ,

and a n a lo g o u s ly

-

fUi Ju i-1

f 0 (x ,

g d f = (E )

y j C± ) -

J

~ 2_1 [ x i

“ Xi-1

'

1 - 1, 2, •••,n The sum o f th e n b r a c k e t s above ( i = 1, . . . , n) i s e q u a l to

xn (2 )

J

yn - x0y0 = °> hence

1 f dg = ~

f

i g d f = (E )

f

X n

0 ( x , y ; C± )

L e t u s c o n s id e r a s im p le co n ve x p o l . r e g . A i n th e a u x i l i a r y w -p la n e w = (s, t ) , o f v e r t i c e s w^, i = 1, 2 , . . . , n , c o n t a in in g a p o in t W € A ° . L e t M be th e s e t sum o f th e 2n s e g ­ m ents ww^, wi_iWj_, i ■= i , 2 , . . . , n, and l e t x = x ( w ) , y = y ( w ) , weM, be th e q . l . m apping w h ich maps each s e g ­ ment ww^, wi _ i wi l i n e a r l y onto op^, r e s p e c t i v e l y , i = 1, 2, . . . , n , wQ = wn « Then, i f A^ i s th e t r i a n g l e

106

CHAPTER III. THE GEOCZE AREAS V AND U AND THE PEANO AREA P Ai = (w wi_1 w^), 1 = 1 , ..., n, we have C: (T, A*), C^: (T, A?) and, by (8.6, i), also 0 (x, y; C) = z 0 (x, y; C± ) for all (x, y)eE - M 1, where M* = T(M) = [C] + z [C± ], !M!| = 0. Thus (1) follows from (2). (b)

Let now C be any curve of finite length L; hence both functions f(u), g(u) are BV in [0,1]. Let V 1, V2 be the total variations of f and g. By (8.8,i) we have I[C] | = o and, by (8.8, iii), 0(x, y; C) is L-integrable in E. If CR : x = fn (u), y = gn (u), o < u < 1, is the closed polygonal line inscribed in C whose vertices are the images on C of the points u we obtain by dividing [0, 1] into n equal parts, if ^ , V2n are the total variations of f , gn, we have fn ► f, gn - — *g -uni­ formly in [o, 1], vln-- ► V 1, V2n — ► V2 and also Ln -- ►L, where is the length of Cn « By (8.8, v) we have (E) /

0

(x, y; CR ) --- ► (E)

I 0

(x, y; C)

and by [L. M. Graves, I, p. 283, Th. 23] also I fn dgn -- f dg, I gn dfR -- ►/ g df. This proves (i) by virtue of (a).

8.11.

Projections of a Curve C

Let t be any orthogonal linear transformation in E^: | = ^x + 2y + a i3Z> ii = a21x + a 22y + E2r respectively • If C has finite length 1, then also C*

§8 . THE TOPOLOGICAL INDEX (8 .1 l) has finite length fore the functions

1,

and the curves 0 (y, z ; C1 ), ...,

107

t C1, C i have lengths < 1 . There| 0 (£, r\; C^) are L-integrable. Set

ur = (E2r^ f ° ( a > °r^ dcc d|3' ur = ^E2r^ ^ Cr^ da r = 1, 2, 3- By (8.10, i) we have u 1 = / f2 d f^ = - / f d f2, u 1 = / F2 d = - J d F2 and analogous relations hold by rotation of the indices 1, 2, 3. (i) If C is any closed curve of finite length in E~, then t 3 ur = arl U1 + ar2 u2 + ar3 u3' r = 1, 2, 3, and !2 ip !2 2 2 2 hence u 1 + ug + u^ = u ] + u2 + u^. PROOF. All functions f^, F^, i = 1, 2, 3, are continuous and BV in [0, 1] and fj_(o) = f^(l), F^(o) = F^(l). By (8.10, i) we have, for instance f 1 + a 22 f 2 + a 23 V + a 32 f 2 + a 33

a2 1

V

+ • • • + (o22 a 33 - « 23 a 3 2 ) / f 2 d

a31

where the non-written terms are obtained by rota­ tion of the indices. Because of f^(o) = f^(l), i = 1, 2, 3, the first Stieltjes integrals are zero, while a22 o c^ - oc2 ^ a ^ 2 = 1, ... . Thus u 1 = a11 u1 + cc12 u2 + and (i) is proved. For any c. closed oriented curve C in E^, let Cp, r= 1, 2,3, be the projections of C on the coordinate planes E2p. By (8A,ii) the functions |0 (p; Cp )|, peE2p, r = 1, 2, 3, are non-negative and measurable; hence the L-integrals vp = (E2p)/ 1 0 (p; Cp )|, r = 1, 2, 3, exist (finite, or +00). Set v = (v^ + v| + v|)2, 0 < v < + 00. Let and let

C, Cn, n = 1, 2, ..., be continuous closed oriented curves in E^ Cp, Cnp, r = 1, 2, 3, be their projections on the planes E2p, vp, v, vnp, vn be the functions above for the curves C and C . (ii) If ||Cn, C||--- 0 as n --- ► °°, then vr < liE vnr' n -- ► 00.

v -

vn, r = 1, 2, 3,

as

PROOF. The first part of the statement is a conse­ quence of (8.7, i (c)). Therefore, given e > 0, for each r = 1, 2, 3, there is an integer np

108

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

such that vnp > e”1 for all n > npif vp=+ v > v - e for all n > n if v < + c" for all n > n if vp = + «> for at least one r. Suppose now v1, v2, < + oo and put M = v1+ v2 + V3 < + oo, mp = max [0, vp - e ] , r = 1, 2y 3. Since vnp > 0 we have vnr - mr - 0 and mr =vr “ 5r 0 < 5 p< e. Therefore

where

v2 )2 n = z r v2nr >- z r m2r = z r v(vr - 5r' >

for all 2 v < v„ n >- 0, 8.12.

V 2 - 2 £r Vr Sr > V 2 - 2Me

Zv

n > n. In any case it is proved that 2 vn as n -► 00 and, since v > 0, also v- 0 and let us prove L > a. Let Sbe any typical subdivision of F into tri­ angles t (5.2) of linearity for P. Denote by S ’ the class (not vacuous because ofa > 0) of all triangles teS such that the triangles A = P(t) are not degenerate. For any teS! let a be the plane of E^ containing A and t 1 , t *’ any two triangles t M C (t?)°, t f C t°, so close to the triangle t such that

It| > 1 1 T | > 1 1 f 1 | > (1 - e) |t| . If

A 1,

A "

C

A 11

(A')°,

are the images of t ?, t 11, we have A' C A0 , | A| > | A ' | > | A ’ ' | > (1 - e)

| A| .

109

§8 . TEE TOPOLOGICAL INDEX (8 .1 3 ) Consequently, we have also z |AM | > ( 1 - e) z |A| = (1 - e) a(P, P) where Z ranges over all teS1. Denote by a > 0 the minimum of the distances {A1*, A*), {A,f*, A f*) for all teS1 and by U the compact set z t T, U C P°. Let (P ,F ), n = 1, 2, ..., be any sequence of q.l. mappings with PR -^P, -- *-L(F, P); hence t F°, dn = d(P, Pn, PR ) -- Consequently, U C F° for all n large enough and we can determine n1, suchthat U C Fn, dn c a for all n > n^ . For everyteS1 denote by t the projection of E^ on the plane a , hence (t P , t 1) is a q.l. mapping and a(x P , t 1) < a(Pn, t 1). Now we have |t Pn (w) - t P(w)| < |Pn (w) - P(w) | c dn < cr for all wet1,n > n1; in particular for all wet’*, and hence ||C^, A 1*|| < a where Cn :(Pn, t ’*). On the other hand, for any peAf1 we have {p, A 1*) > a > ||C , A f*||; hence by (8.3, i) and (8.6, iv) also 0 (p; Cn ) = 0 (p; A 1*) = + 1 . By virtue of (8.9, iii) we have finally |A1!| < a(x Pn, t f) and hence |A11| < a(Pn, t !). Consequently, for all n > n1, (1 - e) a(P, F) < Z|A11| < z ‘ This implies Thus a < L,

a(Pn,

t» ) < a(Pn, Fn )

(ii)

(iii)

.

a(P, F) < lim a(Pn, FR ) = L(P, a > L and finally a = L.

* 8.13. Given a C. C H, in H, (T, Q): (i)


0 < v < 1• For ( iii) le t P j ( u , v ) , P 2 (u, v ) , ( u, v ) e Q , be th e v e c t o r f u n c t io n s p e r fo rm in g th e two f r e e h om otopies C 1 — Cg , C 2 — C^ i n H, and s e t p ( u , v ) = p 1 ( u, 2v ) if 0 < u < 1, 0 < v < 2" 1; p ( u , v ) = P 2 ( u, 2v - 1) if 0 < u < 1, 2” 1 < v < 1 . Then p (u , v ) , (u , v ) e Q , p e rfo rm s th e homotopy C 1 — C^ i n H.

NOTE. F o r th e c o n c e p ts e . g . M. H. A. Newman [ I ] and " r e l a t i v e " homotopy = a re u se d i n M. H. A.

o f f r e e and r e l a t i v e hom otop ies see o r S . McLane [ I ] . The term s " fr e e " a re u se d i n S . M cLane; th e s ig n s — , Newman.

A c o n tin u o u s c lo s e d c u rv e C C H i s s a id to be FR E E LY HOMOTOPIC TO ZERO i n H, o r C ~ 0 (H), if C i s f r e e l y h om otopic to a c u rv e CQ re d u ce d to a s i n g l e p o in t ( o f H ) . An a n a lo g o u s d e f i n i t i o n h o ld s f o r r e l a t i v e homotopy to z e r o . L e t K be th e c i r c l e o f r a d iu s 1, l e t (p , 0) be p o la r c o o r d in a t e s whose p o le i s th e c e n te r o f K , and l e t C: p = p 1 ( u ) , 0 < u < 1 be any g iv e n c o n tin u o u s c lo s e d c u rv e , p - j ( o ) = P - j O ) Then C - 0 (H ) i f and o n ly i f t h e re i s a c . m apping (T , K ) : p = q(w) = q ( p , 0), weK, su ch t h a t q ( l , 2 k u ) =p 1 ( u ) 0 < u < 1, and T ( K ) C H. In d e e d , i f C^ 0 (H ) and p ( u , v ) , (u , v ) e Q , i s th e v e c t o r f u n c t io n p e r fo rm in g th e f r e e homotopy C 0 (H), th e n p (u , 0) = p 1 ( u ) , p (u , l ) =■ co n s t 0 < u < 1; h en ce th e f u n c t io n q(w) d e fin e d b y q ( p , 0) = p ( 2 " 1 it" 1 0, 1 - p ), 0 < 0 < 2jt, 0 < p < *1, i s s in g le - v a lu e d , c o n tin u o u s i n K , and h a s th e above p ro p ­ e r t i e s . Thus th e n e c e s s it y o f th e sta te m e n t i s p ro v e d , and a n a lo g o u s ly we can p ro v e th e s u f f i c i e n c y . Thu s C — 0 (H ) I s e q u iv a le n t to th e f o llo w in g sta te m e n t: T h e re i s a s u r fa c e S i n H w h ich i s g iv e n b y a co n tin u o u s m apping from a two c e l l and whose b o u n d a ry c u rv e i s C; i n o t h e r w ords C = 0 (H ) I f and o n ly i f C can be c o n t r a c te d to a p o in t b y a c o n tin u o u s d e fo rm a tio n i n H.

112

CHAPTER III. *

8 .1 4 .

THE GEOCZE AREAS V AND U AND THE PEANO AREA P Invariance of Homotopy With Respect to Lebesgue and Frechet Equivalence

(i)

If two curves C^, C2 are Lebesgue equivalent, then [C1] = [C2] and C1 = C2 in the set H = [C1] = [C2]. The first part is obvious, the second is proved in M. H. A. Newman [I, p. 1 7 9 ]. (ii) Iftwo curves C^, C2 are Frechet equivalent, i.e., C1~ C2 then [C1] = [C2] and C1 = C2 in the set H = [C1] = [C2]. Again, the first part is obvious; for a recent proof of the second part see L. Cesari [38]. (iii) If C1 ~ C*, C2 - c ’, c\ = C2 (H), then C1 ~ C2 (H). This is a consequence of (ii) and of (8.13, iii). Analogous statements hold where = is replaced by - . The last statement expresses the invariance of homotopy with respect to Frechet equivalence (and, therefore, also with respect to Lebesgue equivalence). (iv) IfC.j, C2 areoriented closed curves of the p-plane E and C1 — C2 in a compact set H, then for each point peE - [H] we have 0 (p; C^ ) = 0 (p; C2). This is a consequence of the definition of homotopy and of (8.6, iii). See 17-2, Note, for other statements. We shall need also the following simple statements. (v) If c^ = (a^ b^), i = 1, 2, ..., n, is a finite system of oriented consecutive curves, i.e., b± = aj_+1> 1 = 1 , 2, ..., n - 1 , and o ± = c± (H), i = 1, 2, ..., n, then also the curves c. are consecutive and c1 + ... + cn = c1 + ... + cn (H). PROOF. Let p^(u, v), (u, v)eQ = [o < u, v < 1], be any vector function p(u, v)eH, performing the homotopy c-j_ = c^, then we have c^: p = p^(u, 0), 0 < u < 1; c^: p = p^(u, 1), 0 < u < 1; Pj_(o, v) = a^, p^(l, v) = b1, ° < v < 1 , Then the function p(u, v), (u, v)eQ, defined by p(u, v) = p^(nu - i + 1, v) for all (i - 1)n"1 < u < in"1, 0 < v < 1, i = 1, 2, ..., n, is continuous in Q, and performs the relative homotopy of the curve z c.: p = p(u, 0), 0 o, we have v(jt’, T) > v(it, T) - N“1 g for some atT = jt^ C tt° [v (ji!,T) > e”1 if v(jt, T) = +00]. The system S ’ of the N regions at1 satisfy all the requirements above and

118

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

z v(*», T) > z v(jt, T) - e [> € 1J. Thus V* > V and finally V* = V. * " The relations Vp = V, r = 1, 2, 3, are particular cases of V* = V. What we have proved implies also that V(A°, T) = V(A, T), V(A°, Tp ) = V(A, T ), r = 1, 2, 3, for all c. mappings (T,A) from an admissible set A into E^.

NOTE 2. In the above definitions of Vp and V we could have consider­ ed finite systems S* of closed non-overlapping simple Jordan regions * J C A instead of simple pol. regions. If V , V* are the numbers de* fined as V by using systems S*, then we have V*> V, Vp > V , r = 1, 2 , 3. On the other hand, for each J we can define a sequence [*n] of pol. regions (6.1, Note 2) * c J°> ||*n> J*il — ^0 as n — ► and the same reasoning above proves that V < V*, V < V , r = 1, 2 , 3. * r - r Thus we have V* = V, Vp = V , r = 1, 2, 3. For the use of not nec­ essarily simple polygonal regions (or not simple Jordan regions) see (9.16).

NOTE 3 * V(A, T), V(A, Tp ) are invariant with respect to Lebesgue equiva­ lence (6.3). Indeed, if (T, A), (T1, A 1) are Lebesgue equivalent, then T 1 = TH where H is a homeomorphism between A and A 1. Given e > 0, there is a finite system S of non-overlapping simple pol. regions n with 2 v(*, T) > V(A, T) - e tor > e-1 if V(A, T) = + ]. Then H maps Sinto a finite system S ’ of non-overlapping simple Jordanregions J C A’ andV(Af, T f)> Z v(J, T 1) = Z v(«, T) > V(A, T) - e [or > g ”1].Hence V(A’, T T) > V(A, T). The same reasoning where T and T f are exchanged proves also that V(A, T) > V(A*, T T) and hence V(A, T) = V(A*, T 1). Analogously we have V(A, T ) = V(A!, Tp ), r = 1, 2, 3.

From the remark at the end of (8.2) it follows that for each plane mapping (Tp, A), the number V(A, Tp ) does not depend upon the system of car­ tesian coordinates in the plane E2p. The much deeper invariance of V(A, T) with respect to the system of xyz-coordinates in E^ will be proved in (22.3)*

§9-

THE GEOCZE M D PEANO AREAS V, U, P (9-2)

Finally let us point out that the following simple statement, whose proof does not offer difficulty, will merge into the deeper theorem (L1 of 1 .4 , or 18.10, i) when the equality V = L will be proved (2 4 .1 , i). (i) For every continuous mapping (T, A) from an admissible set A C E2 into E^ we have (1 )

V(A, Tp )< V(A, T)
V(A, T) - € [or > e"1 if V = + oo]. Theset K = Z it is compact and K C A 0 . Since A^ f A0, there is an n such that K C A^ for all n > n. If C, C^ are the curves C: (T, it*), Cn : (Tn, it*), we have||Cn, C|| < dn; hence ||Cn, C|| -- *-o as n — ►«>. By (8.11, ii) we have v(jt, T) < lim v(jt,T ) as n -*-00and hence V(A, T) - e < z v(jt, T) < Z lim v(jt, T )< lim z v(jt, Tn ) < lim VfA^, Tn ), and the first in­ equality in (i) is proved. The remaining ones are particular cases of the first one.

120

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P 9 -3

- V(A, T)

as a Set Function

(1 ) If A is an admissible set, if A ) z A. and the admissible sets A^, i = 1, 2, ... are dis­ joint, then V(A, T) > Z V(A^, T). The sign > may hold even if A = z A - The same relation holds for each T , r = 1, 2, 3. (ii) Under the conditions of (i), if A = I A^, A0 = Z A?, then V(A, T ) = £ V ( A ± , T) and similarly for T . (iii) If I is a closed, polygonal region, I C A, and I^ I2, ..., I is any finite subdivision of I into non-overlapping closed polygonal regions 1^, then V(I, T) > z V ( I ± , T). (iv) If (An ) is any sequence of admissible sets with ^CA, An C A n + ] , A^ T A°, then --lim V(An, T) = V(A, T) as n

NOTE. (22.k). 525-

Further properties of additivity shall be given in { 9 - 6 , ii) and in The main properties of V(A, T) as a set function are given in

PROOF of (i - iv). Let S = Z V(A±, T). Given e > 0, for every A^ there is a finite system of non­ overlapping simple pol. regions it c with Z ^ v(«, T) > V(A. , T) - 2_1+1 € if V (A., T ) < + oo > e -1 if V(Ai, T) = +oo, where zv(i) ranges over all iteS^. Since the sets A^ are disjoint, Sn = S1 + ... + Sn is a finitesystem of non-over­ lapping regions it C A and, if z 1 denotes any sum extended over all have V(A, T) >

Z

1

v (it,

T)

> Y(A1, T) + ... + V(An, T) - €

if all V(A±, T) < + oo, > €_1 if V(A±, T ) = + for at least one i = 1, 2, ..., n. As n -- ► we have V(A, T) > S - e [or > e”1]; finally V(A, T) > S, and (i) is proved. In (9 -5 , Note 2) we shall prove by an example that the sign > may hold even if A = z A Under the conditions of (1 1 ) let SQ be any finite system of non-overlapping regions it C A° with

§9-

THE GEOCZE AND PEANO AREAS V, U, P

(9-^)

v(jt, T) > V(A, T) - e if V(A, T ) < + ®,>e" if V(A, T) = + ■», where denotes any sum ranging over all *eS. For each *€S0 we have ir C A° = z and. hence each point of it belongs to some set A^. Let us prove that each it be­ longs to one and only one set A?. Indeed, suppose that there are two points w 1, w M €it, w !eA?, w m gA j ; i + j, and let h be any simple polygonal line joining w 1 and w M , h C jt. Since the sets A^ are disjoint, w 11 does not belong to A?. Let m be the closed subset of the points of h which do not belong to A? and let weh be the first point of m from w*. Then w + w., the arc w 1 w (wTincluded, w excluded) belongs to A?, and w must belong to some set A^, k + i Then an arc a = w 1 w2 of h containing w be­ longs to A^; hence the subarc w1 ? of w T w belongs to A^ and, therefore, is not in A?, a contradiction. Thus we have proved that each region * belongs to one and only one set A^. Consequently, we have V(A, T) - e
z ± V(I*, T ) = z± V(Ij_, T), and is proved. Statement (iii) can also be de­ directly from the definitions as (i).

The statement (iv) is an obvious consequence of (9-1) and (9-2, i).

9-k.

(I)

Lebesgue and Geocze Areas

V(F, P) = a(P, F) for every q.l. mapping from a figure F into •

(P, F)

122

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

PROOF. Let (P, F) be the given q.l. mapping; let SQ be any typical subdivision of F into triangles of linearity for P (5 -2 ); [A] be the system of all triangles A = P(t), teS0; [A ] the system of all triangles projections on E2p of the tri­ angles Ae[A], r = 1 , 2 , 3 (A, A^ may be degener­ ate). Then we have v(t, Pp ) = (E2p) / |0 (p; A*)| and, by (8 .6 , iv), also v(t, Pp )= lA^, r * 1 , 2 , Consequently v(t, P) = a (a ) for every teSQ. Finally V(F, P) > z v(t, P) = z a(A) = a(P, F); i.e., V > a.

3

Let S be any finite system of simple pol. reg. k C F. By superposition with SQ we have a finite system of simple pol. regions *! C Feach of which is contained in a triangle t€SQ.Each i teS is divided into a finite number of pol. reg. i t 1 and, i since the images C , Cp of **, under Tp are finite sums of segments, hence of measure zero in E2p, by (8 .6 , i) we have 0

(p; C ) =

Z

0

(p; C'),

|0 (p; C )| < Z

it*Cit

a.e. in E2p, r = 1 , tion in E2p, also v(«, p )
Z v2(«, p )]
J-ll ----------

^0

as m > 00 (6 .1 , iv, Note 1).Then T maps each into a simple Jordan region rm and we have rw rm C rm + 1.. Since * T is uniformlyJ m C t°, 7 continuous on J we have also ||rm, t*|| -----^0. By (6 .1 , iii) we deduce that r° T t° and hence |r°| t 11 °| = 111 . On the other hand, by (8 .6 , iv)> we have v(*m, T) = |r°| and finally V(A, T) > Z v(jtm, T) = Z lpm ^ where zranges over all teSn . As m -- > 00 we have first V(A, T ) > Z 111 = |Fn |. As n -- * 00 we have -

(T, A)

A

of the

124

CHAPTER III. finally

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

(or) V(A, T) > |T(A°)|.

Let (F ) be any sequence of figures invading A0, Fn C A0, Fn C Fn+1, F° t A0. Then the compact set T(Fn ) is contained in B = T(A°), and hence the minimum mutual distance between the various sets T(Q), which are the images of the components Q, of Fn and B*, is a number 5 n > o. By (6.1, ii) there is a q.l. homeomorphism (Pn, FR ) such that d(Pn, T, Fn ) < min [5 n, n_1 ]. Thus Pn ------- > T, Pn (Pn ) C T(A° }’a(Pn> Fn } = ■ W 1 < iT(A°) |, and finally L(A, T) < lim &(Pn, Fn ) < |T(A°)|; i.e., (p) L(A, T) < |T(A°)|. By ( a ) , ( 9 . k , ii) and (p ) we have |T(A°)| < V(A, T) < L(A, T) < |T(A°)| that is,

L = V = |T(A°)|

and (i) is proved.

NOTE 1. The statement (i) shows that L, as well as V, have the char­ acter of "interior areas". For instance if A is any Jordan region and T> a homeomorphism, maps A into a Jordan region B whose boundary curve B* covers a set of positive measure, |B*| = m > o (gee W- F. Osgood [1], H. Lebesgue [6] ), then we have L(A, T) = V(A, T) = |B |< |B| = |T(A)|.

NOTE 2. The following example is of some interest. Let A = [0 < u, v < 1 ] B = [o < x, y < 1], let C: x = f(t), y = g(t), ° < t < 1, be a simple curve of the xy-plane, of end points (1/2, 0), (1/2, 1), contained in the rhombus of vertices (1/2,0), (1/2, 1 ), 0 /4 , 1/2), (3 /4 , 1/2), and of measure 0 < m = |[C] | < 1 /4 . Let (T, A) be the homeomorphism between Q and B which maps A* onto B* identically and the segment s =[u = 1/2, 0 < v < 1] onto C. Then, if A^, A2 are the (closed) rectangles separated by s in A, and B1, B2 are the (closed) Jordan regions separated by C in B, we have L(A, T) = V(A,T) = |B°| = 1 , L(A±, T) = V(A±, T) = |B?|, where |B°| + |B°| + m = 1. Hence L(A, T) > h ( A } , T) + L(A2, T), V(A, T) > V(A1, T) + V(A2, T). The same happens if A^, A2 are replaced by the disjoint admissible sets H1 = H2 = A2 + s, H 1 + H2 = A. Let us also observe that if F denotes the sum of the two rectangles -1 - 1 A A^ and A2n =[2 +n < u < 1>0 < v < 1], we haveFn \ A as n — *00 while V(Fn,T) = L(Fn, T) < |B°| + |B°| = 1 - m. Thus L(A, T) > lim L(Fn,T), V(A, T) > lim V(Fn, T) as n --->». This fact is not in contradictionwith (5.10), (5 *1 4 , iv), (9*2, i) and (9 -3 ,iv)

§9-

THE GEOCZE AND PEANO AREAS V, U, P (9.6)

125

since we have here Pn t A, F° t A° - s but not F° f A0 . Since con­ tinuous homeomorphisms can be approached by q.l. homeomorphisms, this example shows that the convergence Fn T A cannot be used instead of F° t A0 in the definition of Lebesgue area (5.8), (Cf. 6.7). With the same notations above, let us consider now the c. mapping (T.., A) which maps s homeomorphically onto C, both A andA homeomorphically onto B-j, both A^ and A2 homeomorphically onto B 1. It can beproved that L(A, T 1) = V(A, ) = 2|B°|, L(A±, T., ) = V(A±, T,) = |B°|, i = 1, 2, and hence L(A, T 1 ) = L(A^ ) + L(Ag, T 1 ).

9.6.

Regular Mappings

Given a set A of the w-plane E2, w = (u, v), we have already denoted by A(u), A°(u) the linear sets which are the intersections of the sets A, A° with the straight line u = u, and by A(v), A°(v) the inter­ sections with the straight line v = v (3*2). Given a c. mapping (T, A) from the admissible set A of the w-plane E2 into the p-space E y p = (x, y, z), let (Tp, A) be the projections of (T, A) on the coordinate planes (5.4). The mapping (T, A) is said to be REGULAR pro­ vided there exist two countable sets [|], [-q] of real numbers u and v, both sets everywhere dense in (- », + 00) such that iTr [A°(|)]| = 0, |Tr [A°(t! )] | = 0, r = 1 , 2 , 3, for all |e[|], tje [^ ]5 i.e., provided the images of the intersections A°(|), A° ( - q ) , of A° with the straight lines u = £, v = tj, Se[§], r)e[rj]^ have projections of measure zero on the coordinate planes. We consider this condition trivially satisfied if A°U), A ° ( t]) are empty. In particular, if the components x, y, z of T(w) are of bounded vari­ ation on each open linear set A°(|), A ° ( t}) (2.12) [or at least two of the three components x, y, z have this property], then the projections above are countable sums of curvesof finite length[or semirectifiable] and hence (8.8, i) of measure zero. If A is the closed square Q, = [0 < u, v < 1], we can take for [£], lr\] any two countable sets of real numbers everywhere dense in [0,1]. The numbers u = 0, u = 1, v = 0, v = 1 need not belong to[£] and [r)]. In any case let = r

Z T [A0 (| )] + |€[|] r

Z T tA0 (t))] , r]e[t)] r

r =

1, 2, 3 .

Obviously Mp is a subset of measure zero of E2p. Let us denote by the countable collection of all rectangles (closed Intervals)

[I]

126

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

I = U ! < U < I11, Tj' < v < T)11], 1 1, S*1€ [I], T1,,T)M G [n ] with I C A. We shall also let [J] denote the collection containing each polygonal region J C A whose boundary J* is a finite sum of consecutive segments of straight lines u = £, v = t], |€[g], tje [-q]- Obviously each Je[J] is the finite sum of non-overlapping rectangles l€[I]• (i) Given any simple pol. reg. Je[J] and any subdivision S of J into rectangles Ie[I], we have (a) 0 (p; Cr ) = Z 0 (p; C^) a.e. in E2p, r = 1 , 2 , 3 ; (b) v(J, Tp ) < 2 v(I, Tp ), r = 1, 2, 3 ; (c)v(J, T) < z v(I, T), where z ranges over all l e S and where Cp: (Tp, J*), Cp : (Tp, I*), r = 1, 2, 3PROOF. Since Z[CJJ + [Cp] C Mp, IIV^I = 0, (a) holds in E2p - Mp by (8.6, i). Consequently I0 (p; Cp )| < Z|0 (p; C£)| a.e. in E2p and, by integration, we deduce (b). Finally by definition of v (9*1) and by (2.10, c ) i \ v(J,

T)

= [zr v2 (j, Tr )j2 oo l€Sn

V(A, T ) =

T) ,

111 I v(I, T ) , n -- > oo IeSn

r =

1, 2 , 3

•

PROOF. Given e > 0 there Is a finite system of N simple pol. reg. * C A such that

S

§9-

THE GEOCZE AM) PEANO AREAS V, U, P (9-6) V > Z v(jt, T) > V - e

if V < + oo, > €_1 if V = + oo, where V = V(A, T) and z ranges over all aeS. By (9.1 ) we may suppose it C A°. Since the set Z * C A° is compact, the mapping T is -uniformly continuous on z*. We can now repeat the reasoning of (6.1, Note 2). By using the rectangles I€Sn instead of the squares q.eZn> we can determine an n and, for each n > n, a simple pol. reg. rn C*° with ||r , **|| -----> 0 as n --> 00, where rn is a finite sum of rec­ tangles I€Sn * If Cn :(T, r*) C:(T, **)> we have IICn> C|| -- >0 as n -->00 for each JteS. By (8.11, ii) we can determine n so large that v(rn, T)> v(jt, T) - e/N if v(*,T) < + 00 v(r * T)> €_1 if v(*, T) = + 00. Thus by (i) above, for all n > n, we have v > X

IeS n

v(I, T) >

E

v(I, T) > Y .

" IsS n7 , IC 2r n

>

~ V

v then (T, Q) Is regular. In particular a mapping (T, Q) satisfying the conditions of (5.13) is certainly regular.

9 -7

(i)

(3 )

- A Lemma on L-Integration

If f (w), weF, r = 1, 2, 3, are L~integrable functions and S denotes any subdivision of a figure F into pol. regions jt of maximum diameter 5 > o, then

JL

PROOF. Let A&, B be the expression under limit and the second member of (3), respectively, and let z denote any sum ranging over all it€S. Then by (2.10, C ), we have

§9Ag

THE GEOCZE AND PEANO AREAS V, U, P (9-7)

= z

Er [ ( « )

/

f r

dv]'

s |z (5 )

4) dw =

}2 0, there is a de­ composition fp = gp + hp, r = i, 2, 3, where gp is continuous in F and (F) / |h I dw < e (E. J. McShane, I, p. 225). By (2.10, b) and (2.10, d) we have now 1 1. p [ («) / gr dw]2|2= z j z p [ (n) / fr dw - (it) / hp dw ]2 1
As > Z | z r [ (it) ; gr dw]2 j-2 - 36 _1

> >

f

B

-

3G

(g^ + g2 + g^j2 dw -

G

-

36

.

3e

- € >

>

1 30

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

That is, B > A 5 > B - 7 e (i) is proved.

9*8.

for all

5

< SQ. Thereby

Proof of Morrey Theorems (5.13, i), (5*13, ii)

Under the same conditions of (5-13, i), or (5-13, ii), let us denote by I(A, T)the area ^ntegralI(A, T) = (A0 ) / J dw, where J = (J^+ + J^)2 is the function defined in (5-5)* We shall prove the following statement which includes (5 *1 3 , i), (5 *1 3 , ii). (i)

THEOREM. Under the same conditions (5-13, i)> or (5-13, ii), we have L(A, T) = V(A, T) = I (A, T).

PROOF. (a) Let F , n = 1, 2, ..., be any sequence of figures Fn C F, Fn C A0, F° For every n we shall denote by Yn m ^ '

Znm^w ^

means of

_p ^ (w )

veFn ’

x, y, z nu+h

= h

vu weFn>

the inteSpal

of order

m,

i.e.,

pV+h

Jv

x(a ,

p) dor dp ,

h = m”1 ,

where m is any integer large enough in order that 2 m_1 < (Fn, A*), and analogous formulas for Y^. nm Z„„ nm (Cf. 8 .9 , v). The functions X^, Y^, Z ^ are continuous with their first partial derivatives in Fn (3 2 .5 , i) and Znm > z as m --- ►» uniformly in Fn . In addition, if ( T ^ j Fn ) is the c* mapping whose components are Zvw w is the function defined nm7 and Jnm as J for Tnm7 . we have (Pn ) f Jm d w --- ► (Fn ) / J dw as m *- °° (32.6). Thus we can determine tn = m(n) in such a way that 2 m -1 < [Fn, A*), I(Fn ) / Jm dw - (Fn ) / J dw| < n'1, d < n_1* By applying (5 -1 3 , Note 2) to the mapping

f A0. X^(w),

. THE GEflCZE M D PEANO AREAS V, U, P (9-8) (T^, Pn ) we cannow determine a mapping (P , F ) such that d(Pn; |(Fn ) ;

Fn ) < n

Jn dv -(Fn )

f

Jm

q.l.

9

dv| < XI-1 ,

where Jn is the function defined as Pn - Therefore

J

for

la(Pn, pn ) - V(A, r T ) , where V(A, t T ) does not depend upon the system of cartesian coordinates in the plane a as remarked in ( 9 - 1 , end of Note 3 ) - In particular P(A, T ) > V(A, T p ), r = 1, 2, 3 , where T p Is the plane mapping which is the projection of T on the coordinate plane E2r*

* 9-12.

Additional Observations on Peano and Geocze Areas and Bibliographical Notes

The area V defined in (9 *1 ) is obviously a variant of the Banach area mentioned in (7 - 3 )> where for narea" m(cr) of the projection a on aplane E of the image under T of a simple pol. region n C A, we take m(a) = (E) / |0 (p; C)|, C being the continuous closed oriented curve projection on E of the image under T of tc*. Thus also the area P defined in (9 - 1 1 ) is the corresponding variant of the Peano area (7*2). As we have mentioned in (7 -3 ) such a defi­ nition of m(tf) by making use of the topological index has been proposed by various authors (W. H. Young, R. Caccioppoli, E. Andreoli and P. Nalli). Consistent research on the area V is due to C. B. Morrey [1, 2, 3], E. J. McShane [3],

CHAPTER III.

TEE GEOCZE AREAS V AND U AND THE PEANO AREA P

L. Cesari [7 , 9, 12]; on the area P, L. Cesari [7 > 12] and J. Cecconi [4 , 5]; the area U is a useful variant which has been consistently studied by L. Cesari [1 2 ]. The proof given here (9 *8 ) of Morreyfs theorem is the one given by the same author [1]. Regular mappings have been consistently studied by L. Cesari [7]. Statement (9 -9 * i) has been proved by both T. Rado [3 5 ] and L. Cesari [22], together with the stronger statement (2^.3, i) mentioned in (9 -9 , Note). The proof is essentially the one given by T. Rado. Tangential propertiesof a surface S and the Weierstrass integral over a surface have been express­ ed in terms of the functions u and U [L. Cesari, 28, 23, 29, 30; cf. Appendix B of this book]. For the recent­ ly renewed interest in the Peano area P, see J. Cecconi (loc. cit.).

* 9.13.

The Index

0

(p; C)

and the Related Areas

V, U, P .

Given a continuous closed oriented curve C in the p-plane E, let 0 (p; C), peE, be the topologicalindex of C and let 0 (p; C), peE, be the auxiliary function defined by 0 (p; C) = +1, -1, 0, according to 0 (p; C) > 0, < 0, = 0. By (8 .4 , ii), 0 (p; C) is measurable and since it is bounded, |0 | < 1, it is also L-integrable. (i) If CR, C, n = 1, 2, ..., are continuous closed oriented curves in E, and ||Cn, C|| -- > 0 as n --> 00, then (a) 0 (p; C) = lim 0 (p; Cn ) for all P€E - [C]; (b) I0 (p; C )| < lim 1 0 (p; Cn )| for all peE; (c) (E) I 1 0 (p; C )| < lim (E) / |0 (p; Cn )| as n -- =>00. These assertions follow from (8.7, i)* (ii) Under the same conditions as (8.6, i),we have 0 (p; C) = 0 (p; C^), I0 (p; C )| < |0 (p; C± )|, |0 (p; C)| < |0 (p; Ci )| for all pe[E] - Q. Indeed, the first inequality is in (8.6, i); the second, is a trivial consequence. For the third let us observe that, if 0 = 0 , then the inequality is trivial; if |0 (p; C)| = 1,

§9-

THE GEOCZE M D PEANO AREAS V, U, P (9-14)

139

then 0 (p; C) + 0 and 0 (p; C^) + 0 for at least one i; therefore |0 (p; C^)| = 1 and |0 (p; C)| = 1 < |0 (p; C^)|. If (T, A) denotes any c. mapping from an admissible set A of thew-plane E into the p-space E^, if (Tp, A ) , r = 1 , 2 , 3 are the corresponding plane mappings from A into the coordinates planes E2p, if it C A is any simple closed pol. reg. and C, Cp the images of ** under T, V

let vr = v(fl, Tr ) = (E2r) ; |Cf(p; Cr )|, v = v(«, T)

=(v^ + v| + V3)

up = u(«, Tr )= (E2r) / u =

u (jc,

T)

r = 1, 2, 3 ,

,

(p; Cr ),r = 1, 2, *2 =(u^ + u2 + u^) • 5

In addition if ocis any plane in E^, jection of C on a , let m(*, T, or) = (or) J T) = Sup m(it, a

and

3

,

the pro­

|0 (p; Ca )| , T, a)

.

Thus all numbers vp, v, up, u, m, \x are finite. If S de­ notes any finite system of non-overlapping simple pol. reg. it C A, let V(A, T) = Sup X v(s, T) , S iteS U(A, T) = Sup Z u(*, T) , S JteS P(A, T) = Sup E n(x, T) . S neS Thus the functions V, U and and Peano type, respectively.

*

9 .1 4 .

P

are areas of the Banach

A Generalization of A Theorem of Complex Variable Theory

In addition to (8.6, ii) the following theorem generalizes a well-known statement of the complex variable theory: (i) If (T, k ) is any c. mapping from a simple

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

closed pol. reg. n of the w-plane E, w = (u, v), into the p-plane E !, p = (x, y), if C: (T, jt*), and there exists a point P0eE r such that T~1(pQ ) is a single point and I0(pQ^ C)l = m > 1, then there is a neighborhood U(pQ ) of pQ such that for each peU(po ) p ± pQ, there are at least m distinct points w^e*0 with T(w^) = p, i = 1, 2, ..., ra. This theorem was proved by T. Rado [i8]. More detailed state­ ments were given later by L. Cesari [10], M. Dolcher [1]. An indirect consequence of all these statements are the equali­ ties V = V = L, proved by L. Cesari, [11, 12] and P = P = L proved by J. Cecconi [5]. As H. Federer has observed [6], the statement (i) cannot be extended to mappings from a space En into a space E^, n > 3, and the corresponding "volumes'1 V and V, P and P, are then different. *

9.15.

An Axiomatic Definition of Lebesgue Area

As we shall prove in Chapter VII, the equality L = V = P holds for all c. mappings (T, A) from an admissible set A C E2 into E^• In addition, the further equalities have been proved L = V = V = P = P , and also L = U = U if L < + 00 [L. Cesari, 12, 13, 1°; J. Cecconi, k , 5]. J. Cecconi [5 ] has also proved the following statement concern­ ing the axiomatic definition of area: (i) If(T, A) denotes any c.^ mapping from a 2-cell A of the w-plane E2 into the p-space if cp(T) is any functional defined for all c. mappings (T, A), if and coincides with the elementary area for all q.l. mappings, If for any plane a C E y cp(T) is > the 2-dim. Lebesgue area of the set of points pea where 0(p; C^) + °, Ca being the projection on a of the curve C: (T, A*), then , o < x> < + °°, (5* 1 ), let (Tp> "bethe image of the boundary curve r . under T , i = o, 1, ..., v>, r = 1, 2, 3, where -L *I we suppose that r is counterclockwise oriented and * * u r1, •••, r^ are clockwise oriented. Then the functions cp(p; R, (Tr ), peE2r, defined by cp(p, R, Tp ) = 0 (p; Crl) if peEg - Tr (R*), cp(p; R, Tp ) = 0 if peTr(R*), is B-measurable in E2p.Consequently the L-integrals v*r = v*(R, Tr ) = (E2r) / |cp(p; R, Tr )|, r = 1, 2, 3 , exist (finite, or +00). Let v*(R, T) denote the number v*(R, T) = (v*^ + v*2 + v*^)2, 0 < v* < + °o. Let S de­ note any finite system of pol. regions R C A (simple, or not), and let us put

X

V*(A, T) = Sup v*(R, T) , S ReS

X

V*(A, T ) = Sup v*(R, T ) , 1 S ReS 1 r = 1,

2, 3

•

The function V*(A, T) is another variant of the Geocze area V. It is easy to prove that v*, V* have the same properties of lower semicontinuity and additivity already proved for v*, V* (8.7, i, (c); 8.11, ii; 9.2;9 -3 )- Obviously we have V < V* and, by (5*12, i) also V* < L. The areas U*, P* canbe defined analogously. NOTE. We shall prove in (2 4 .1 , i) that V = L. Hence, by V < V* 0, _i ~> 0< e< 2 diam C, there is a d = d(e, C) > 0 such that any two points p, p feC with |p - p f|< d are the end points of one and only one arc c C C with diam c < e. Indeed every subcontinuum K of C containing p and p l is an arc of C containing one of the two arcs ppl, p Tp in which p and p 1 divide C. Thus of the two arcs pp1, p ?p, at least one, say ppf, must have diameter < e because of the theorem of Hahn and Mazurkiewicz, while the other arc p*p must have a diameter > e because otherwise we would have diam C < 2 e. If K denotes any set of a metric space Q, then K is said to be well chained if for any pair p, p ! of points of K and any e > 0 there is a finitesystem [p^, i = o, 1, ..., n] of points Pj_eK with pQ = p, Pn = P ’> IPi_! " Pi I < €> 1 = 1> 2 , n.Then every connected set is well chained; every compact well chained set is connected, and hence a continuum [G. T. "Whyburn, p. 13]. Finally let us note that if J is a closed Jordan region, J C E2, and J is considered as a space, then the sets M C J which are "open" (as subsets of the space J) are those sets which are denoted as "open in j" when E2 is the total space.

NOTE 1. In the following, for the sake of brevity, by a top­ ological space we shall mean a perfectly separable, regular, topological space.

NOTE 2. We shall say that a set M is countable provided M is either empty, or finite, or denumerable. The components of an open set G form a countable collection. Let us mention here that an open set G C E2 is said to be simply connected provided we have J C G for every simple Jordan region J with J* C G.

§10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (1 0 .2 )

A point p of a topological space Q is said to be a point of condensation of a set M C Q provided e v e r y neighborhood U of p contains an uncountable collection of points of M. Let us denote by M the set of all points of condensation of a set M C Q. The following statementsare well known: (i) I )I, and M - M Is countable; (ii) if M is an uncountable set, then M is non­ empty and perfect; (iii) M = M if and onlyif M is perfect (and then M = M = M); (Iv) each closed set is the sum of a countable set and of a perfect set.

* 1 0 .2 . Semicontinuous Collections Given any sequence [A^] of sets of any topological space Q we denote by 1 = lim inf A ^ L = lim sup An [G. T. Whyburn, I, p. 14], the setsof all points peQ such that each neighborhood U of p contains points of all but finitely many sets A^, or of infinitely many sets A^, respectively. Thus 1 C L C Q and both 1 and L are closed. Obviously 1 and L are also compact if Q is compact. (i) (Zoretti’s Theorem). If [AQ] is a sequence of continua of a compact topological space and 11m inf ^ 0 , then L = lim sup An is a con­ tinuum [Cf. ,G. T. ‘Whyburn, I, p. 14]. A collection r of disjoint continuua g of a compact top­ ological space Q is said to be upper semicontinuous in Q provided that for each ger and any neighborhood U of g (i.e., any open set containing g), there exists a neighbor­ hood V of g,g C V C U such that, if h is any con­ tinuum of r intersecting V, then h C U [loc. cit.]. (II) A collection r of disjoint continua g of the compact topological space Q is upper semicon­ tinuous if and only if for any ger and for any sequence [g] of continua gn€r with gl + 0 , 1 = lim inf gn, we have 1 C L C g, L = lim sup gn [loc. cit., p. 1 2 2 ]. (iii) The collection r of the components of a closed subset of Q Is upper semicontinuous. Any collection r which is the sum of an upper semi­ continuous collection rQ covering a closed set

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

H q C Q, and of a collection of single points of Q - H , is upper semicontinuous. Every subcollection r of a given upper semicontinuous collection rQ, is upper semicontinuous.

The first part of (iii) is given in [G. T. Whyburn, I, p. 122, 1-11]; the second and third parts are consequences of the defi­ nitions . A collection r of disjoint continua g C Q is called a de­ composition of Q if r covers Q, i.e., z g = Q where z ranges over all ger. (iv) Let r be an upper semicontinuous decomposition of Q into continua g. For any closed set H C Q, the set H q sum of all ger with gH + 0 is a closed set and H C HQ C Q. For any open set G C Q, the set GQ sum of all ger with g C G is open (in Q) and GQ C G (GQ may be vacuous). The proof of the second part is given in [G. T. Whyburn, I, p. 123]. The first part can be immediately proved by considering the comple­ mentary sets in Q. Given the decomposition r as in (iv) of the compact topological space Q, let us denote by M the collection of all subsets M of Q; by G and H the subcollections of all those sub­ sets of Q which are open or closed; by m q the collection of all subsets of Q, which are sums of continua ger, i.e., M = Z g; by g q, h q the collections of all sets MeMQ which are open, or closed. Hence G Q = U0G> HQ = MQ H* A set M eMQ if and only if Me M and has the property that g M + 0 implies g C M. It is now natural to considerr as a space itself,say r, whose "points”, say g, are the continua g and whose Mneighborhoods,f and "open sets", say G, are the open sets GeG0, G = Z g, thought of as collections of elements g. Then "closed sets", say H,are the sets HeHQ thought of as collections of elements g. If M is any set MeAf and hence M = z g, let M denote the same set thoughtof as a collection of elements g and thusa subset of r. Let M f G } H be the collections ofall subsets of r, or of all open, or closed subsets of r, i.e., the same collections

§10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (10.2)

G Q, Hq when each set MeA^, Gq, Hq is thought of as a sum of continua g. (v) f is a perfectly separable, regular, compact, topological space; hence, by (1 0 . 1 , i), r is a metric space [G. T. Whyburn, I, p. 123],*

M0 ,

r

is said to be the hyperspace of the decomposition r (loc. cit. (vi) Any set M eMQ isopen, closed, compact, connect­ ed, a continuum, according as M has the same properties in r. M separates Q, or any set G€Gq in Q, according as M separates r, or the corresponding set G in r.

The statement (vi) is a trivial consequence of the various definitions listed in (10.1) and (1 0.2). Let us recall here also that by Borel collection B in the space Q (or collection of all Borel sets M C Q) ismeant the smallest class of subsets M of Q, containing all open and closed sets and such that (1 ) for any two sets A, B e B also A + B, A - B, AB belong to B ; (2) for any countable system [A^] of sets A ^ e B the sets z A^ and n A^ also belong to B . Thus b contains the empty set as well as the whole space Q (because Q = z is the sum of the basic sets Ri of 10.1) [Cf. H. Hahn and A. Rosenthal, I, p. 3; S. Saks, I, p. 7 and p. i k ] . Let us mention here that a collection of sets having properties (1) and (2) above is said to be a a-field. The statement (vi) can be completed as follows: M€MQ is a Borel set in Q if M is a Borel set in r and vice versa. As usual [H. Hahn and A. Rosenthal, I, p. 5; S. Saks, I, p. b ] ] we shall denote by P-sets the closed sets, by G-sets the open sets, by Fa[G6] those sets which are neither closed nor open but are countable sums [intersections] of closed [open] sets, etc.

NOTE. The statement (vi) does not hold for perfectsets be­ cause, for instance, while a non-degenerate continuum ger is a perfect set in Q, g is a single point in r and thus is not a perfect set. An elementger shall be denoted as an ELEMENT OP CONDENSATION of a set MeM provided for each

148

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

neighborhood U of g there is an uncountable collection of elements g_C M with g C U. (Hence g is point of con­ densation of M in r. ) The statement (vi) can be completed now as follows: M is perfect if and only if MeM0 is per­ fect and each geM is element of condensation of M.

*10.3.

Properties of Separation of Elements ger

We have already mentioned (10.1) that, given a topological space Q, and three disjoint subsets A, T, B of a connect­ ed set M of Q,, the set T is said to separate A and B in M if M - T is not connected and there is a de­ composition M - T =+ M2of M - T into disjoint sets M 1, M2 both closed in M - T and M 1 3 A, M2 3 B. Let Q, be any compact connected topological space, r any upper semicontinuous decomposition of Q into continua g and let r be the hyperspace associated with r (10.2). Let r.L u be the subcollection of all those continua ger 0 which separate Q, (i-e., such that Q, - g is not connect­ ed (1 0.1 )). Thus r.t shall denote the collection of all points ger which separate r (i.e., such that r - g is not connected). Given anytwo elements a, ber, let E(a, b) denote the family of all ger which separate a and b in Q. Thus E(a, b) shall denote the collection of all points ger which separate the points a, ber. As we know from (10.2), properties expressed in terms of the elements with tilde (i.e., elements of the space r of points g) can be expressed in terms of the same elements without tilde (elements of the collection r of continua g). We shall often prefer the latter if more errpressive. (i) Each family E(a, b) is naturally ordered in such a way that, if geE(a, b), and we denote by e”(g), e+(g) the collections of all ele­ ments g !eE(a, b) preceding, or following g, then a + b + E - g = e ~ + e , and g separates each element g fee_ from any element g M ee+ in Q, [G. T. Whyburn, I, p. 41 ]. (ii) For each family E(a, b) of points ger the set a + E(a, b) + b is compact and hence the sum of a countable set D ’, and of a set Ep which is either empty, or perfect [G* T. Whyburn, I, p. 50, 4 .1 2 ].

CONTINUOUS MAPPINGS AND SEMI CONTINUOUS COLLECTIONS (10.3)

§10

NOTE

1.

(ii) in terms

of continua ger,

of

compact subset of

Q.

states: for each family a +

collection

either empty or uncountable.

a perfect subset of

E(a, b) + b

E(a, b)

covers a

The collection a + E(a, b) + b

the sum of a countable Ep

r

the collection

E^,

if non-empty, covers

Q, and each continuum

an element of condensation of

Ep-

g

of

Ep

is

The illustration shows

two cases of families Q

is

D 1 and of a collection

E(a, b)

where

is a closed simple Jordan region

in the w-plane ments

ger

Eg . The other ele­

are all single points of

Q, not in the continua families

g

of the two

E.

(iii)

For each family For E(a,each b) family at most a countable subfamily D 1T

of elements

geE

not condensation elements of both e (g) e~(g) e+ (g) [Gr. T. Whyburn, I, p. kk, 2.1]. (iv)

is the sum of countably many families E(a, b)

NOTE 22..

[G. T. Whyburn, I, end p. 52].

The previous definitions and statements may be given

in a stronger form.

Indeed let

be the subcollection of

all those continua

ger

connected open set tion of all points

^ €G0 . * ^ Thus G C Q, GeGQ us rt ger g€? which separate either

connected open set

G C r.

let a

E(a, b) and

GeGQ . G€Gq ger

are

and. and

b Thus

which separate either

r,

collec­ c°llec_ or some

Given any two elements

a, ber,

denote the family of all

in

r

Q, or some

ger

which separate

or in In some connected open set

E(a, b)

G C Q,

Is the collection of all points is

which whichseparate separate aa and and bb either either in in

r, r,

or insome some

connectedopen open set GG CC r. r. The The statements statements (I) (i) -- (iv) (iv) still hold

[G. T. Whyburn, I, p. 61, §9]. The illustration shows three cases E± (a± (ai, b± Ei , b 1 ), families

i =

1, 2 2,, 3 , 3 ,

E(a, b)

where

ofof Q

is aa

closed Jordan region of order ■0=1 of the w-plane* The The continua geE1, g £E2,separate separate the continua

geE 0

Q;Q;

do not separate

Q nor separate a^, b^ inin Q, but but they do separate the open connected

E(a, b)

at

1 50

CHAPTER III. set and

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

G between the two broken lines and they do separate b^ in G.

a^

*1 o.k . Continuous Mappings Any single-valued mapping (T, Q): p = T(w), weQ, from a topo­ logical space Q into a topological space E issaid to be con­ tinuous at a point weQ, if given any neighborhood V of the point p = T(w)eE there exists a neighborhood U of the point weQ such that T(U) C V. By continuity of T in Q is meant the continuity of T at each point of Q. If both Q and E are metric spaces and for their topology e-neighborhoods are used, then the concept of uniform continuity can be introduced as usual. Any real single-valued function can be thought of as a mapping from Q into the space E 1 of real numbers. We re­ call here that for any set A C Q, the distance {A, p] is a non-negative continuous function of p in Q and that {A, p) = if and only if peA. The distance function p(p, q) is a con­ tinuous function of p and q in Q, [Cf. G. T. Whyburn, loc. cit.]. Let (T, Q) be any continuous (c.) mapping from a top­ ological space Q into a topological space E (1 0 .1 , Note 1 ). (i) For any compact set A C Q, the set T(A) C E is compact. The set T(Q) itself is compact if Q is compact. (ii) For any peT(Q) the set T~1 (p) C Q of all points weQ, such that T(w) = p is closed. If both spaces Q and E are metric and Q is compact the following statements hold: (iii) T is uniformly continuous IG. T. Whyburn, I, p. 25]. (iv) For any peT(Q) and e > 0 there is a 5 = s(p, e) > 0 such that for every other point qeT(Q) with |q - p| < 5 we have {T~1 (p), T” 1 (q)) < e. In particular, if (T~1, Q ’), Q T = T(Q), is single-valued, then (T~1, Q f) is a continuous mapping from Q 1 onto Q. PROOF of (iv). Suppose the statement is not true. Then there is a peT(Q), an e > 0, and a se­ quence [pn ] of points PneT(Q), pR -- ► p, such that {T~ 1 (Pn )> T~1 (p)} > e. Let wR be any point wn€T"1 (Pn ^ n = * Since Q is compact there is at least one point of accumulation

0

§10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (10.4) wQ of [wn], wQeQ, and also a subsequence [w„ ] with w„ -- ► w as k -- ► a k ° nlc ° contradiction. Thereby (iv) is proved. By ex­ amples it couldbe shown that if T~1 is not single-valued, then it may happen that 5 cannot be chosen independently of p (for instance if T: x = u, y = u v, 0 < u, v < 1 ).

Let both pact .

Q

and

E

be topological spaces and let

Q

be com­

For any peT(Q), the set T_1(p) C Q, by virtue of (ii), is closed, and hence compact, and its components are disjoint continua g C Q. Denote by r = r(T, Q) the collection of all continua g C Q, which are components of at least one set T^1(p), peT(Q). Obviously the continua ger are all disjoint, the collection r covers Q, i.e., r is a de­ composition of Q, and finally r is the collection of all maximal continua g C Q of constancy for T in Q. (v) The collection r = r(T, Q) is an upper semicontinuous decomposition of Q. PROOF. Let g be any continuum ger and [g^] any sequence, Sn€r> with Ig + 0, 1 = lim inf gn - Set L = lim sup gn, so that, by (10.2, i), L is a continuum, 1 C L C Q. Let u, v be any pair of points uegl, veL, and let U, V be two neighborhoods of u and v, respect­ ively. Then, by the definitions of 1 and L (10.2), there are points un€Sn> un€U F o r all but finitely many n; there are also points vn€^n^ vn€^ ^or ir^i-nitely many n. In con­ sequence, for infinitely many n, we have |T(u) - T (v) | < |T(u) - Ttu^l + |T(un )- T(vn )| + |T(v ) - T(v)|, where the second term is zero, and the first and third terms can be taken as small as we want. Therefore, |T(u) - T(v)| = 0, T(u) = T(v) for any veL. This implies that T

152

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

is constant onL, hence L C g f, g !er. other hand, gl + 0, 1 C L; therefore gg* + 0. This implies g = g 1, i.e., Thereby statement (iv) is proved. [See proof in G. T. Whyburn, I, p.

On the gL + LCg. another 1^2].

As a consequence of (v) and (10.2) the collection r = r(T, Q) of all maximal continua of constancy g of T in Q, can beconsider­ ed as a topological space f and this space can bemetrized. We shall see in (10.7, Note 1) a simple metric on r. As in (10.2) we shall denote by M, G, h the collections of all subsets M of Q, or of all open, or closed sets; by m q , G q , H q the correspond­ ing collections of those sets M = Z g which are sums of continua ger; by M, g > H the collections of all sets MeMQ, G q , h q when each set M is thought of as the sum of its continua g, and, therefore, a subset of r. (vi) Every upper semicontinuous decomposition r of a compact metric space Q into disjoint continua g is the collection r = r(T, Q) of all continua of constancy of a c. mapping (T, Q). Indeed, if we consider the hyperspace r defined in (10.2) and we define (T, Q) as the mapping p = p(w) which maps eachpoint weg into the element p = ger, then (T, Q,) is obviously a c. mapping from Q onto r and r = r(T, Q). In addition (T, Q) is monotone [Cf. G- T. Whyburn, I, p. 125].

NOTE. As a particular case of (vi), if Q, is a simple closed Jordan region of the w-plane E2 (2-cell)^ and no element ger separates J, nor E2,then also r is a 2-cell, i.e., r is homeomorphic to a closed circle c [R. L. Moore, 1] and thus r isthe collection r(T, Q) of some monotone c. mapping (T, Q) from Q, onto c.

In connection with this remark also the following statement shall be mentioned which is a particular case of various statements due to J. W. T. Youngs [7 ](vii) If (T, Q): p = p(w), weQ, is any monotone mapping from a simple closed Jordan region Q of the w-plane E2 into a Peano space S in which a metric has been chosen, if T has the property that no element g of r(T, Q) separates Q, nor E2 then, given any e > 0,

§1 0 . CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (1 0 .5 ) there exists a homeomorphism (H, Q) from Q, onto T(Q) such that d(T, H, Q) < e.

*1 0 .5 - The Countable Set With A Mapping

D(T, Q) (T, Q)

Associated

For the sake of simplicity, and because this is the case we will have occasion to apply, let Q be any closed Jordan region of finite order 0 < -o < +00 of the w-plane E2 w = (u, v), and let (T, Q); p = T(w), weQ, be any c. mapping from Q into the Euclidean p-space E^ (or any E^), p = (x, y, z). Let r = r(T, Q) be the upper semicontinuous decomposition of Q into maximal disjoint con­ tinua of constancy of T in Q. Thus we could consider Q, as a compact connected (Peano) space and we could associate to (T, Q) the hyperspace f of the decomposition rFor later use we will consider here a variant of this process. Let Qq be any closed square of the plane E2 containing in itsinterior the whole set Q, Q C r* the decomposition of Qq we obtain by adding to r all single points weQ,Q - Q. By (10.2, iii) r* is an upper semicontinuous decompo­ sition of Qq and we shall denote by r* the hyperspace of the decomposition ^Q0 p* (10.2). Let us consider now the sub* / % collection rt of all ger* (if any) which separate Q . Obviously all * ger. are continua ger. By (10.3, iv), * u r^ is the countable sum of families E(a, b) [i.e., all ger* separating in Qq any two elements a, ber*] . For each family E(a, b) we have already associated in (10.3, ii and iii) two countable subcollections, namely the subcollections D f, D !1 of all geE which either do not belong to the perfect subfamily E^, or such that g is not an element of condensation for both subcollections e~(g) and e+ (g). Thus also the subsets T(Df), T(DM ) of E^, namely the sets of all points P€E3 which are images of some continua geD1, or geD!t, are countable. In addition we have Ep = E - D J, D M CE^. For each family

E(a, b) let us consider the subcollection

153

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

EQ = Ep - D Tf of all continua g which are elements of con­ densation of continua g 1 of both e~(g) and e+(g). We say that an element gGEQ has property P+ [P“] provided g is an element of condensation of elements g'ee+(g) [g’ee~(g)] with T(g') f T(g). We denote by E1 the sub­ collection of all elements g of EQ which have both prop­ erties P+ and P~, and by D 111 the complementary collection D»11 = EQ - E1. Thus E = Ep + D 1 = EQ + D T + D !1 E 1 + D 1 + D*f + D f1!. We may say that D !11 C EQ is the collection of those elements g€EQ (if any) which are not elements of condensation of elements g ,ee”"(g) with T(g!) + T(g), or are not elements of condensation of elements g !ee+ (g) with T(g!) + T(g). Of course D !1f need not be countable because, for instance, T could have the same value on all continua geEQ and then D !11 = E . The following simple examples will be helpful. We shall de­ note by (p, e) polar coordinates of center o in the w-plane E2, and by (r, a>, z) cylindrical coordinates in the p-space E^. I. Let Q, = [p < 1 ]and (T, Q): p = T(w), weQ, be defined by the relations r = p , = 0, z = o. Then r(T, Q) is the sum of the point o and of all lines g = [p = t], 0 < t < 1. If a = 0, "b = Si = tp = then E(a, b) is the collection of all lines g = [p = t] with 0 < t < 1, and E(a, b) = Eq = E.j, while D f1 and D M 1 are empty. II. Let h be a perfect totally disconnected set of real numbers 1 < t < 2, containing 1 and 2, and let m be the countable collection of all end points of the complementary intervals of h in (_ oof + oo); hence 1 and 2 are in m, m C h C [1, 2], and each number teh- m is an element of condensation of numbers t 1 eh, t 1 < t , as well as of numbers t 1eh, t 1 > t. Let Q = [ p < 3 J> H C Q , be the perfect set of all points w = (p, e)eQ with peh, and let G = Q, - H. Finally let (T, Q,) be defined by the relations r = {w, H), a> = e, z = o, where

§10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (1 0 .5 )

{w, H) denotes the distance of w from the set H. Then T(w) = [r = 0, z = 0] for all weH, r(T, Q,) is the sum of all points weG- and of the collection [g] of all lines g = [p = t] with teh • Finally, if a = 0 and b is any point beQ* we have E(a, b) = [g] .Here D 11 is the collection of all g = [p = t] with tem, and EQ = D !11 = E - D l* while E^ is empty. III. As in II where (T, Q) is defined by the re­ lations r = {w, H),a) = e , z = p. We have T(w) = [r = 0, z = p] for all weH, and r(T, Q) is as in II. Also E(a, b) and D 11 are as in II for the same a and b. Finally D !1f is empty and E1 = EQ . IV. As in II and III where (T, Q) is defined by the relations r = (w, H); cd = e; z = p if p < t ; z = tQ if p > tQ,where tQ is a given real number tQ = h - m, 1 < tQ < 2. Then r, E, and D M are as in II andIII, D 1f f is the collection of all lines g = [p = t] with t > tQ,teh- m, and the collection of all g = [p = t] with t < tQ, teh - m. We shall now prove the following statement: (i) For each family E(a, b) the collection has a countable image T(D,fI)-

D f1’

PROOF. By (1 0 ,3 , ii) a + E + b = E p + D f = EQ + D f + D 1!. Suppose that an element g e E i s not an element of condensation of continua g fee (g), with T(gf) T(g). This implies that there is a neighborhood U of g such that the elements g T€e""(g), g C U, are uncountably many, while for all but a countable subcollection of these we have T(g’) = T(g). If g ! is an ele­ ment of this subcollection, then T(gf) 4 T(g) and g l cannot be an element of condensation of elements g fIee (g), hence g ? is not in Ep, i.e., g ^ D 1. This implies that for all g fee"(g), g* C U, g feEp, we have T(gf) = T(g). The family of all g'eE^, g !ee"(g) with T(gr) = T(g)

155

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

is obviously ordered as E and also compact (by add­ ing a if necessary) and hence has a first element g1 [Cf. G. T. Whyburn I, p. 51, 4 .4 ]. Therefore each element SeEp which has the property of not being an element of condensation of elements of e“(g) with T(g1) + T(g) is the last element of a non-empty subfamily I = Ep[E(g^ g) + g] and T is constant on the set covered by I in Q. Since Ep is ordered and compact there exists an order preserving continuous mapping from E^ (as a collection of elements ger*) into the real inter­ val [0, 1] [G-. T. Whyburn, I, p. 5 5 , 6 .4 ], and the above subfamily E^[E(g1, g) + g] is mapped into a subinterval of to, 1]. Elements S€Ep which correspond to different values T(g) of T corre­ spond to disjoint sets I and hence to disjoint sub­ intervals of [0,1]. Since disjoint intervals form a countable family, the set T(D!11 ) is also count­ able. Thereby (i) is proved. * * Since is the sum rt = z E^(a^, b^) of a countable collection of families E^(a^, b^), let D|, D£!! be the sets defined above relative to E., and from which we * exclude, in any case, those continua ger such that gQ, + 0 [at most one for each component of Q,*, thus a finite number for each family E^]. Let D(T, Q) be the countable set D = D(T, Q) = Z± [T(D!_) + T(DP ) + T(d ” ')]

NOTE 1. If we consider the cases illustrated in (10.3> Note 2) we see that only the family E- of continua ger is contained * 1 in r^ . The continua g of the families E2, E^ do not separate any large square Qq ) Q.

NOTE 2. If Q is afinite sum of disjoint Jordan regions J^, i = 1, ..., k, and (T, Q) is any c. mapping from Q into E y let D(T, Q) be the sum Z D(T, J^) of the countable sets defined above.

If A is any admissible set of the w-plane E2 (5 - 1 ) and (T, A) is any c. mapping from A into E^, then, in any

§10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (10.6)

case, A is the countable sum of connected sets a and each a is an open connected set, or a connected set open in a Jordan region J. Let [F ]be any sequence of figures Fn such that Fn C A°, Fn C F°+1, F° f A° and such that for each a there is (at most) one component Rn of Fn, Rr C A0, and thus for each a , we have R^ C Rm+1 C Rrn+2 C ••• C a > where m is an integer m = m(a). Thus for each (T, Rn ) we can define the collection rtn = r^(T, Rn ) and the countably many fam­ ilies Eni(aj_^ rtn = Enj_- For each Eni at most finitely many elements g have points in common with R* we shall denote by E ^ C Eni the subcollection of all elements g C En^ with g C R°. Since Rn C each element gert(T, Rn ), g C R°, is an element gert (T, Rn+1)Therefore we can suppose that each family E ^ is a family L,, or a part of a family E„,, ..j Thus we have also n+ i,j ii+ i, D(T, R ^ C D(T, Rn+1) and the set D(T, a ) = lim D(T, Rn ) as n — oo is a countable set. Finally the set D(T, A) = z D(T, oc) for all components a of A is also a countable set. The set D(T, A) depends solely on A and not on the sequence [F ] which we have used. For the sake of brevity we omit the proof of this statement which we shall not need.

*10.6.

The Modulus of Continuity of

(T, Q)

Let Q be a compact metric topological space (1 o.1 , Note 1 ) and (T, Q): p = T(w), weQ, be any continuous mapping from Q, into any metric space E. For any 5 > o let o>(5 ) = Sup |T(w) - T(wf)| for all pairs of points w, w JeQ such that | w - w !| < 5 . (i) o < cd( 5 ) < diam T(Q); u>(5 ) < cd( 5 1) for all 0 < 5 < 5 1; 0 < |T (w) - T (w1 )I < a) (|w - w 1|) for all w, w ’eQ; diam T(M) < co (diam M) for every set M C Q ; ao(o + ) = o. This statement is an obviousconsequence of the definition of a)(s ), of the continuity of (T, Q) and of the consequent uniform continuity of T on Q (1 0 .^, iii). (ii) If T is not constant on Q and Q is connect­ ed, then a)(5 ) > o for every 6 > o. PROOF.

Suppose

a)(s) = o for some

6 > o.

Then

157

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

for any pair of points w, w !eQ, |w - w 1I < 5 , we have |T(w) - T(w’)| = 0, i.e., T(w) = T(w'). Hence, if a, b are any two points of Q, and a = w , wi; ...., wn = b is any finite chain of points with |w^ - vi+iI _ then T(a) = T(w.j) = ... = T(b); that is, T(a) = T(b) for any pair of points a, beQ. The existence of the chain above is a consequence of the connected­ ness of Q [G. T. "Whyburn, I, p. *\b, 8.2]. The function [diam A] The function oo(5 ) is not necessarily continuous as the following example shows. Let Q, be the set Q = [ 1 < P 0, {M, N}t < {u, v)T for all ueM, veN. PROOF. Given € > 0 there are two continua c ^ , c2 with u, wec^, w, vec2, diam T(c1 ) < (u, w)T + e, diam T(c2 ) < {w, v)T + e. Therefore o-jCg + 0 and, by (10.6, iii), also {u, v)T < diam T(c1 + c2 ) < diam T(c1 ) + diam T(c2 ) < (u, w)T + {w, v)T + 2e. Since € is any positive number, the first part of (i) is proved. Now we have {M, u } ^ < {w, u)rp for every weM and hence {M, u)T < (w, v)T + {v, u)T and because we can choose w in such a way that {w, v)T is arbitrarily close to (M, v)T, we have {M, u}T < {M, v)T + {v, u)T - Thereby (I) is proved. (ii)

{u, v)T = 0 if any only if

u, veg,

ger.

PROOF. If u, veg, ger, then T(g) is a single point and hence 0 < {u, v)T < diam T(g) = 0, i.e., {u, v)T = 0. Let us suppose {u, v)T = 0. Then, for each n = 1, 2, ..., there is a continuum cn C Q, u, vecn, with diam T(cn ) < n"1. Let 1 = lim inf cn, L = lim sup cn; hence u, vel, l + o , and, by (10.2, i), L is a continuum, 1 C L, u, veL. Forany point weL and neighbor­ hood U of w there are points w l€Cn in­ finitely many c . Therefore

159

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

|T(w) - T (u) | < |T (w) - T (w1)| + |T(w>) - T(u)| where w 1, u e c n > |T(wr) - T(u)| < n”1, and. |T(w) - T(wf)| can be taken arbitrarily small. Thus |T(w) - T(u)| = 0, i.e., T(w) = T(u) and T isconstant on L. Consequently L C g, ger, and finally u, veg, ger. (iii) If u, u ’eg, v, v ’eg’, g,g'er, then (u, v)T = {u 1, v !)t; that is (u, v)T considered as a function of u, veQ, is constant on the continua ger. PROOF. First consider the points u, v, v T, ueg, v, v 1eg’. By (i) we have {u, v 1)T < {u, v)T + {v, v l)T> (u, v}T < {u, v f)T + (v!, v }rp, where, by (ii) (v, V ?)T = {v!, v)T = o. Hence [u, v f]T < (u, v)T < {u, v ?)T> i.e., (u, v 1} = {u,v)T . Consequently we have also (u, v l)T = (u!, v t}t^ and finally [u, V }rp = {U 1 , V ? ]rp.

NOTE l. Given any pair g, g f of points of r, let g, g f be the corresponding continua g,g'er and set p(g, g l) = (u, v)T for any ueg, veg!. By virtue of (iii) p(g, g f) does not de­ pend upon the choice of u in g and v in g 1. By (i) and (ii) we have j > ( g , g^) = p(g*, g) > 0; P(g, g !) = o if and only if g = g 1; p(g, g 1) < p(g, g 11) + P(g1*, g T). Thus p(g, g M ) is an example of distance in the space r.

In the following further statements we suppose that Q is locally connected besides the hypotheses already made at the beginning of this section. Thus Q is a Peano space (10.i ). (iv) For any Peano space Q, the distance {u, v)T considered as a function of u and v, is continuous in Q. For any set M CQ also the distance (M, u)T is a continuous func­ tion of u in Q. PROOF. By (1 0 .4 , iii) T is uniformly continuous in Q, hence given e > o there is a 5 > 0 such that |T(u) - T(u’)|< e for every pair of points u, u ’eQ, |u - u 1| < 6. Hence diam T(h) < e for

10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COLLECTIONS (10.7 ) every set h C Q, such that diam h < 5 . Since Q, is a Peano space, Q is locally connected and also uniformly locally connected; hence there is a a > o such that every pair of points u, u TeQ, |u - u T| < a belongs to at least one subcontinuum h of Q, with diam h < 5 . Finally (u, u f)T < diam T(h) < e for all u, u ’eQ, |u - u !| < a. Now for every four points u, u !, v, v ’eQ, |u - u ’| < cr, |v - v 1| < a we have {u, v}T < {u, u ’}T + (u!, v ’)T + (v1, v)T < {u1, v ’}rp + 2c, and, analogously, {u!, v !) < (u, v)T + 2 Thus |{u, v},p - (u1, v ’}^! < 2€. Thereby the con­ tinuity of {u, v}T in Q, is proved. Analogously for the distance function (M, u}rp.

(v)

For any Peano space Q and any two closed sets M, NeQ, there are two points ueM, veN such that {M, N)t = (u, v}T . Hence [M, N)T = min (u, v)rp for all ueM, veN.

PROOF. If a denotes a = {M, N)rp, then, for every n = 1, 2, . .., there are points u^eM, vne^with a -vn^ < a + ' Since Q is compact, also M and N are compact; hence, by two successive ex­ tractions, we can determine a sequence (n^, m = 1, 2,... of integers such that u ^ -- ► u, v ^ -- ► v as

m -- ► where ueM, veN. By (iv) we have vn)T ► {u, v)T and hence (u, v)T = a. (vi)

For any Peano space Q and for any two closed sets M, N C Q we have {M, N)T = 0 if and only if there exists ger with gM + 0, gN + o.

PROOF. By (iv), {M, N},p = {u, v),p for someueM, veN. Hence {M, N)T = o if and only if {u, v)T = 0 and, by(ii), if and only if u, veg, ger, i.e., gM + 0, gN + 0, for some ger. (vii)

For any connected set K C Q, and every two points u, veK we have {u, v}^ < diam T(K).

PROOF. Since Q, is compact, the closure is a compact set and obviously connected.

of K Thus K

K

162

CHAPTER III.

THE GEO'CZE AREAS V AND U AND THE PEANO AREA P

is a continuum containing u and {u, v]T < diam T(K) = diam T(K). (viii)

v

and, therefore,

For any Peano space Q and for any two M, N C Q and any set K C Q connected diam T(K) < {M, N)T we have either MK = 0, or NK = 0, or both.

sets with

PROOF. Let us suppose MK + 0, NK + 0, and consider two points U€MK, veNK. By the definition of (M, N)T and by (v) we have {M, N)T < {u, v)T < diam T(K), a contradiction. Thereby (viii) is proved.

NOTE 2. Neither statement (vii) nor (viii) necessarily holds if K is not connected, asthe following example shows. Let (T, Q): x = cos *u, y = sin *u, z =0 for any w = (u, v)eQ, Q = [o < u, v < 1]. Set w = (o, o), w ’ = ( 1 , o), K = (w) + (wf)- Then T(w) = (1, 0, o), T(wf) = (-1, o, 0), diam T(K) = 2, (w, w 1),p = Jt, diam T(K) < {w, w T),p-Therefore (vii) does not hold. Analogously let M = (w), N = (wr), then diam T(K) = 2, {M, N)T = [w, w f)T = it, i.e., diam T(K) < {M, N)T and MK = (w) + o, NK = (wf) + 0. There­ fore (viii) does not hold

*10.8.

Some Additional Observations and Bibliographical Notes

Let (T, Q): p = T(w), weQ, be any continuous mapping from any compact topological space Q into a metric space E. Let us re­ call here that T is said to be monotone [light] provided for every point peT(Q) the set T”1(p) C Q is connected [totally dis­ connected] . Because T is continuous and Q is compact, T~1(Q) is compact and therefore if T is monotone then T”1(Q) is a continuum. Let r = r(T, Q)be the collection of the maximal continua of constancy of T in Q and r the "hyperspace" associated with r whose points are the elements g of r. Then the open sets in f are the open sets GeG0 which are sums of continua ger. The passage from r to r can be thought of as a mapping, namely the mapping (m, Q): g = m(w), weQ, from Q into

f

§10.

CONTINUOUS MAPPINGS AND SEMICONTINUOUS COU jECTIONS (10.8)

which maps each point weg, ger, into the point ger. Then ob­ viously m is continuous in Q and monotone because for each per the set T”1(p) = g is a subcontinuum g of Q. Let us denote by (1 , r): p = 1 (g), ger, the mapping from r into E which maps each point ger into the point p = T(g)eE. Obviously 1 is continuous in r and 1 is light also; name­ ly for each peT(Q) = l(r) the set T”1 (p) is a compact sub­ set of Q whose components are continua ger and the set 1 1 m[T (p)] = 1 " (p) is a compact subset of r whose components are points ger. By definition of m and 1 we have also T = lm, i.e., T(w) = l[m(w)] for all weQ,. Hence (i) Each c. mapping (T, Q): p = T(w), weQ, from a compact topological space Q into a metric space E admits of a monotone-light factoriza­ tion T = lm, where m is a monotone mapping from Q onto the hyperspace r of the de­ composition r, and 1 is a light mapping from r into E; hence m(Q) = r, l(r) = T(Q). —

—

This statement, in the form above, is due to G* T. Whyburn [i, 2] and S. Eilenberg [5 ]* In some cases it had been observed already by B. v. KerekjartS [1]. Each time the space r is connected with this theorem, r is called the middle space of the trans­ formation (T, Q), [T. Rado, II]. As a general reference on (1 0 .1 -4 ) see [R. L. Moore, I] and [G. T. Whyburn, I]. Consistent use of the middle space r in questions of area of a surface was first made by C. B. Morrey h, 2, 3]. Note that the space r = r(T, Q) corresponds to the space used by G. T. Whyburn, for instance, at p. 1 4 2 , (the "points” of r are the single components g of the compact sets T~1(p)) and this space is consistently used in some recent research on sur­ face area. As shown for other purposes by 0 . T. Whyburn (p 126), the whole sets T~1 (p) can also be taken as points of a space rQ which of course may not be homeomorphic to r. See for more information on the same subject in (3 7 •iO* For the statement (10.5, i) and another approach to the count­ able set D(T, Q) see L. Cesari [9]. See also T. Rado [32 J. For the distance {u, v)^ of (10.7) see [G. T. Whyburn, I, p. 1 5 4 ].

16b

CHAPTER III.

*§11.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

SOME PROPERTIES OF THE EUCLIDEAN PLANE *11.1.

E2

Properties of Separation

Let E2 denote the Euclidean w-plane, w = (u, v). Thus a set M C E2 is compact if and only if M is closed and bounded. (i) If A, B C E2 are compact sets, AB = 0 if u, v are any two points aeA, beB and e is any posi­ tive number, then there is a simple closed poly­ gonal curve 1 separating a and b in E2, such that 1 (A + B) = 0 and such that every point of 1 is at a distance less than e from A. This elementary statement can be thought of as a variant of [G. T. Whyburn, I, p. 108, 3.1]. (ii) If K is a component of a compact set M C E2 7 a component of the open set E2 - K, wQ a point of 7 and e any positive number, then there exists a simple closed polygonal line 1 which separates wQ and K in Eg, with 1 M = 0, and such that each point wel as well as each point we7 separated by 1 from wQ is at a distance less than e from K- In addition, if 7 is bounded [unbounded], wQ is in the bounded [unbounded] component of E2 - 1. This statement is also a variant of [G. T. Whyburn, I, p. 109, 3.11].

*11.2. (i)

Some Junction Properties of the Plane

E2

Given any two points a, b of an open connected set G C E2, there is a simple pol. line 1 C G join­ ing a and b .

The statement is contained in [G. T. Whyburn, I, p. 3 7 , ii] where e-chains of segments are used instead of general e-chains of locally connected continua. (ii) Given any closed pol. reg. R of order v > 1 and a closed set M C R C Eg,if some two points a, b belong to different boundary curves of R and to the same component 7 of R - M, then there is a subdivision of R into simple pol. re­ gions Jt and in a new pol. reg. R 1 of order u - 1 , with ^ C 7, jtM = 0, R *M = R*M, and a and b belong tosome regions jt, and not to R 1.

§11.

SOME PROPERTIES OP THE EUCLIDEAN PLANE E2 (11.3)

The statement is an elementary one. It states simply that we can cut R along a line 1 C R, 1 M = 0, joining a and b, and that by en­ larging 1 into a simple pol. reg. it contain­ ing 1 , nM = o we divide R into one or more simple pol. regions it and a pol. region R r of order t> - i. Cf. for analogous questions, M. H. A. Newman [I]. *11.3.

Some Covering Theorems

Let (T, Q): p = T(w), weQ, be a c. mapping from the closed polygonal region Q of the w-plane E2, w = (u, v), into the p-space E^ [or any E^] and let r = r(T, Q) be the corresponding collection of the maximal continua of constancy g for T in Q (10.4). Let Sbe any finite system of simple closed non-overlapping pol. reg. q C Q° and H be any closed set, whose components are all continua ger, such that H ( z q*) = 0. Let e > o be any positive number. Then there is a finite subdivision SQ = S + S ’ + S M of Q into non-overlapping pol. regions qeS, iteS’, ReS,!, where S is the sys­ tem of all q above, S 1 is a system of simple pol. reg. it with it# = 0, S ?1 is a system of pol. reg. R (not necessarily simple) with R* H =R* Q* H and tj(it) = diam T (it) < e, r)(R) = diam T(R) < e (10.6) for all iteSf and ReSr1. Thus S + S fr is a covering of H. PROOF. Let H* = H(Q - z q) + Q*, G = Q - W . Then is compact and G is open in Q as well as in E2, and bounded, Q )z q, G C Q°. If h = HQ*, 00 = Q* - h, then h is a (linear) closed subset of Q*, and cd is the complementary, linear set, open in Q*; hence a>, if non-vacuous, is the countable sum of open disjoint arcs oc^ of Q*, i = 1, 2 , ... Let [Fn] be any sequence of closed figures Fn C Fn C G, P° f G ( 5 •1 ) , where G Is bounded. We can suppose that for each component 7 of G and integer n large enough there is only a polygonal region Rn, a component of F , with Rn C 7 (5 •1 )•

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

If for each of Q* we denote by a sequence of closed arcs such that p.^ Pim

t

ai

as

m —

oo,

th e n

m = 1> 2> ••• C p^ m+1' ^im ^ai

f ffi = [ p ,m, P2m/ . . . ,

3 ^ ] is

a finite sum of disjoint closed arcs of Q* with the f o llo w in g p r o p e r t ie s : f ffl C f°+1, fm C m , f ° f zq C F° for all n >n(m). We canalso suppose n(m) large enough so as these relations hold for all arcs 1 p of f . For every arc p let w 1, w1 be the first two points of the segment s = aw, s f = bw1 (starting from a, or b) which are on F , n > n(m), and ■x11 — hence on theboundary Rn of the same pol. reg‘. Rn, R_ n being the component of F„n contained , in7. j Thus the arc p, the segments , s1 = bw^, and an arc P ! of Rn define a simple pol. reg. *n with

*n

- °-

-K1 * We have it = p + p l + s1 + s 1 and hence ltn H s °* We have also *n 3 we shall prove that * n H = 0 for all n large enough. Indeed, in the contrary case all sets * n H would be compact and non-empty, « h) *n+lH' ^ hence also the intersection set h of all jrnH, n > n(m), would be compact and non-empty [G. T. Whyburn, I, p. 4 , 3*6]. Thus h C H, h C Jtn for all n > n(m), and since *n = *n + =0, C 7 - Rn,we should have h C 7 a contradiction, since R^ \ 7. This proves that jtn#= 0 for all n large enough. Let us fix for n an integral valuen = n T(m) > n(m) such that *nH= 0 ^or each m arcs P “ We also fix n T(m) In such a way that n !(m) > n* (m-1 ), n 1(m) > m, where n ?(0) = 1.

§11.

SOME PROPERTIES OF THE EUCLIDEAN PLANE E2 (11 .3 )

Let S* be the finite system of all closed regions R of Fm and of the m closed simple pol.regions n just now determined. It is easily seen that the closed set ^ covered by Sm verifies the relations: \

c ^+1'

of a fin it e

t*

\

c

syste m

0 S' *

811(1 Q

\

o f p o l . r e g io n s

I* * 1 R H = R Q H; while each region pletely contained in a region

ls the sum R'

w it h

is com­

Let crm = max ri cm r -> am +1 *(R1 ) for all R ^ S *m1. Then and we will prove that am -*~o as m ------ Suppose this is not true. Then there is a X > osuch that crm > X for all m and there exists alsoa sequence [R^] of closed regions with x For all m, R^ D • If K is the intersection of all the regions R^, K = R* R^ ..., then K is a continuum [G. T. Whyburn, I, p. 1 5 ]. Let us observe that for each m there is a pair of points wm, with |T(wffl) - T(w^)| > X and, since Rjh C Q, Q compact, there are certain sequences of integers m, such that wm -- ► w , w^ — where wQ, points of accumulation of [w^]. Hence wQ, w^€K, |T(wQ ) - Tfw^)) > x and hence ti(K) > X. On the other hand, let us observe that R^ C Q where F° f G as m ---hence K C Q - G = H f = fl(Q - Zq) + Q*. Also R^ Q* C Q* - f ° ,

where

f°

f

cd

as

m ------oo;

hence KQ* C Q - a> = h = HQ*. Therefore K C H and since K is connected, also K C g, where g is a component of h as well as an element of r(T, Q). Therefore t)(K) = 0 , a contradiction. This proves that am --- ► 0 as m — », and hence there exists an in such that r](R!) < € for all , m > m. Let S !f = S^1, m - m. Let cd(6 ) be the modulusof continuity of (T, Q) and, because of a)(o+) = o (1 0 .6 , (i)), let 5 q be a positive number such that 03(5 q ) < €* Let us observe that the regions qeS are all Interior to the regions R £En, n = n(m) and that q*H =0 , R* H - 0 . Thus by an arbitrary finite subdivision, we can divide each R into the regions q C R° (if any), and new simple non-overlapping closed pol. reg. Jt1 C R, o, diam ** < 5 Q. Since

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

jtH=

o for each *, also the regions * can be divided in analogous regions it1. Thusif S' is the finite system of all the regions * ’ we have * ]H = 0, < e for all jtfeS f. Thereby the required subdivision S + S ’ + S ’f of Q is ob­ tained and (i) is proved. (ii)

Under the conditions of (i) there is a finite subdivision S + S r + S 1 ' of Q into non­ overlapping pol. reg. qcS, jigS’, RgST!, where S is the given system, S T is a system of simple pol. regions *, S 11 a system of non­ simple pol. reg. R and (1 ) R* H = Q*R*H, **H = Q*jt*H, n(R) < e, ri(ir) < e for all R gS", iteS1; ( 2 ) z f1Z*r) ( p * ) < g where z* is extend­ ed over all boundary curves p* of R* and z ,f over all ReSfl.

PROOF. Each of the closed pol. reg. R g S m defined in / * (i)\ has --o+ 1 boundary curves p^, i = 0, 1, ..., d , ■ ’ j = u(R) > 0. The notations R = (p , ..., P^)* R* = (P0^ • • • > PD) (5 *1 ) will be used. Let us observe that all ReS!T with u = 0are simple pol. reg. and we can put them now in S 1. Therefore we can suppose (a) u(R) > 1 for all ReS!f. ¥e can also suppose that the following property holds: (p) any pair of points wep^, w*Gp., i + j, belong to different components we y, w '€71 of the set, open in R, R -HR, RgS!I. Indeed, if (p) is not satis­ fied, then, by(11.2, ii), there is a subdivision of R into simplepol. regions it with jt*H= 0, 11(jt) < n(R) < e, which we put in S ’, and into one region R l with R *H = R*R’HC Q*H, r\ (R’) < r\ (R) < g and R f of order -o - 1. We put R f in S f if u - 1 = 0, in S ff if -o - 1 > 0. By repeating this procedure a finite number of times weobtain a sub­ division S + S ’ + S ’1 such that both (a)and (p) are satisfied. Let N = Zt> the sum of all orders X) = 'o(R), RgS1’, 0 < N < + co. Let 03(6) be the modulus of continuity of T in Q (10.6), andlet 5 be a number such that a>(&) < cr, a = e/4 N,

§11.

E2 (1 1 .3 )

SOME PROPERTIES OF THEEUCLIDEAN PLANE

35 < {p?, pt), ReS'1.

i + j,

i, j =

1,

•••,'0,

x>

= 'o(R),

For every ReSM let us consider the compact set R H C E2 and let r* be the upper semicontinuous collection of all components of R H and of all single points of Q0 - RH, where Qq is any large square Q C (Q0 )°* Therefore (10.2, iii) r* is an upper semicontinuous decomposition of the compact set Q . For any two points aeQ0 - pQ, and bep^, let us define the family E(a, b) of the elements ger* separating a and b in Q,0 [E being ordered from a to b] . Obviously E is non-empty (because of p) and also does not depend upon the particular choice of the points aeQ0 - P0> t>€P°* On the other hand, if 70 is the component of Qq - HR which contains a, and 71 the component which contains b, then

Po ” po

H

C 7o'

P* - p I H c 7 1'

Because of (10.3, ii),

and ?o 7 1 = °-

a + E(a, b ) + b

is compact and,

since E(a, b ) C Q, a€0 o ~ P0> t>€P°j E(a, b) is also compact (and ordered). Hence [G. T. Whyburn, I, p. 31, 4 .4 ] E has a first element g0er* which separates y Q and 71 in Q0. Let y Q be the component of Q0 - gQ containing a. By (1 1 .1 , ii) (with M = HR, K = gQ, wQ = a, 7 = 7^) there is a closed polygonal line 1 * with 1 * h R = 0, and thus 1* C 7Q; separating a from gQ and such that any point wel*, as well as any point we7^ separated by 1 * from a, is at a distance < 5 from g . Let 1 be the bounded and closed simple pol. reg. whose boundary is 1 *. If 1 ) p Q, let p^ = pQ, R 1 = R. Then {w, gQ}< 5 for every wep^* because 1* already has this property and gQ C pQ C 1 ; hence T)(p* *) < 2cr . If 1 C pQ, then, because of 1 * H R = 0, we can re­ place 1* by a new polygonal line l r* interior to 1 and hence to pQ and so close to 1 that in the region between 1* and l l* there is no point of H , nor of p 1 + ... + p^. Thus gQ C 1 '° 1 ! C 1 °, and, if we set p^ = l l, we have [w, gQ] < 5 for all wep^*, ^(p^*) < 2a, as above. Then l f divides R into two regions R r, R !1 of orders -oM [R1 between l 1 and all those p^ interior to

170

CHAPTER III.

THE GEOCZE AREAS V AND U AND THE PEANO AREA P

= 1 1 including p.,; R 1 ’ between pQ, p^ and the re­ maining Pj_]• If tfT/ u fr are the orders of R f, R 1r, then f, u ff - 1 are the numbers of regions p^, I = i, ..., u, interior and exterior to I 1 * p^; hence x>' + (r>! I -i) = , o. Since R !l does not satisfy (p) we can operate as above and thus replace R 11 by a new pol. region R 1 ! of order u 1 1 - i. Thus + (t>M -i) = *o. If 1 - pQl i o, pQ - lpQ + o, then 1 *, p* have at least two distinct points in common and each bounded component of E2 - (p* + 1 *) is a simple pol. region. [This is an elementary well known fact, which could also be deduced by the non-elementary statement: G. T. Whyburn, I, p. 107, 2.5]. If 1 Q is the one region which contains g , let Po = 1o anci' as before, tw, & F o r aU wep^* = 1 *, t](po*) < 2a. Let R f be the region between p^* and those p^ contained in 1 = p^ including p 1 . Of the re­ maining simple regions (bounded components of E2 - (p* + 1 *)), those which do not contain regions p^ are simple pol. regions * which we put in S 1; those which do contain regions p^ form certain pol. reg. R !, R M , ... . If tt1, u’1, ... are the orders of the regions R ?, R 1 f, ... (finite in number), we have x>' + of* + . . . = u. Let us now repeat the procedure above for all new re­ gions R ? so determined. Since at each step the sum £uT remains the same and the orders x> do not increase, the number of the regions R f remain bound­ ed and hence, after a finite number of steps, all re­ gions R 1 will have their external boundary curve p^ verifying the inequality ^(p^*) < 2 a. We can now re­ peat the same procedure for every pair of boundary curves p|, p^, of the regions R l so obtained (i = 1 , ...,u !, j = 0, 1 , ..., instead of i = 0 , j = 1) by exchanging the use of interior and

§11. SOME PROPERTIES OP THE EUCLIDEAN PLANE E2 (1 1

.3 )

exterior boundaries. Each time we obtain a new set of regions R T with L v ' = x>, all -o1 > 1 , and the number of boundary curves p*, i = 0, ..., -c1, verifying the relation r](pj) < 2a increases each time by one. The total number of boundary curves is necessarily equal to z(u’ + 1 ) = Zd1 + z(1 ) < u + u = 2u (because of x>' > 1). Therefore, by repeating the pro­ cedure above a finite number of times, we obtain the re­ quired set of regions R ’. Thus we have Z !,Z* T](p£) < Z 1’Z*(2 a )
o any integer such that m < N(p; T, A). Then there is a finite system S of non-overlapping closed simple pol. reg. it with m < z |0 (p; C)| where we suppose 0 (p; C) + o and also that jt C A0 (8.3, i). Thus Z Tt C A0 and, since A^ j A0, there is an n such that E k C A^ for every n > n. If Cn : Tn' i1: follows ||C , C|| -- and by (8.3, i) and (8.7, i), also 0 (p; C) = 0 (p; C ) for all n large enough. Hence m < z I0 (p; C )| = Z |0 (p; CR )| < N(p; Tn, An ) for all n large enough. This proves that N(p; T, A) < lim N(p; TR, An ) as n ->-00 for every peE^. Thus the statement for Nis proved and the analogous one for ¥ follows now by FatouTs ine­ quality [S. Saks, I, p. 29]. The statement for V has already been proved in (9.2). Analogous proofs hold for V+, V”, N+, N", W+,W~. (ii)

If A^

(T, A) is any c. plane mapping (12.1 ) and any sequence of admissible sets with

A n C A ' An CiW V(An, T) -- -V(A, T),

then W(An, T) -- - W(A, T),

can

§12.

BV PLANE MAPPINGS (1 2 .5 )

and N(p; A^, T) -- «-N(p; A, T) for all as n -- ► 00. The same proof holds as for (5 .1 4 , iv) and (9 -3 ; iv).

* 1 2 .5 - The Equality

177 peE^

N* + N~ = N

Let (T, A) be any c. plane mapping from the admissible set A of the w-plane E2 into the p-plane E^. We shall need below the exceptional countable set D of points peEj, introduced in (10.5), containing all those points p which may be the images of points weA around which the topological properties of the mapping (T, A) are exceptional as we have described there. The set D was defined in (10.5) first for the case where A is a closed polygonal region and then for the case where A is any general admissible set. We recall here the definition of D for the latter case. Let (Fn ) be any sequence of figures such that Fn C A, Fn ^^ri+1' Fn I A°' Since, for every n, Fn is a finite sum ofdisjoint closed pol. reg. Q of orders o < d < + 00, we can consider the c. plane mappings (T, Q) from the compact sets Q, and the corresponding countable sets D(T, Q) defined in (10.5, Note 2). Thus by D = D(A, T) C E^ we shall denote the countable set D = Z D(T, Q), where z ranges over all components Q, of F and n = 1, 2, ... (see 1 0 .5, Note 2). (i) N(p) = N (p) + N~(p) for all points peE^ with exception at most of the points p of the count­ able set D C E ^ PROOF. For each point PeE^ and integer n < N(p) there exists a finite system S of closed non­ overlapping simple pol. regions * C A with n < z |0 (p; C)|, where C: (T, ir*) and Z ranges over all iteS. Denote by z+, z” any sum extended over all iteS with 0 (p; C) > o, or 0 (p; C) < o. Then n < z |0 (p; C)|= (z+ + z") 1 0 (p; C )| < N+ + N~. Hence N < N+ + N" for every P^E^. Let p be any point peE^ - D and n+, n~ be any two integers with o < n+ < N+ (p; T, A), 0 < n" < N”(p; T, A). Then there exists a finite sys­ tem S+ [S’] of closed simple non-overlapping pol. reg. ir+ C A [at” C A] such that n+ < Z+ 0 +(p; C), n” < z” 0 ~(p; C), where Z+, z“ are extended over all *+eS+, jt~€S”, respectively. Obviously we can suppose 0+ > o for all Tt€S+ and 0“ > o for all iteS”; hence p is not on any of the curves C. By

CHAPTER IV.

BV AND AC PLANE MAPPINGS

(8.3, i) we can suppose *+, C A° for every it+eS+, jt“eS”. Indeed we can replace each * = *+, not satisfying the above condition by another region 1 C Tt° with II*1*, II sufficiently small. Since p is not on the curves C, the polygonal lines (tf+ )*> (*")* have no point in common with the set T~1(p). Let us observe that while the regions u eS , as well as the regions Jt"’eS“, are non­ overlapping, it may happen that some region tf+eS+ is partially or completely overlapping with regions *"eS~. The closed set z+ + z“ is interior to A; hence there is an n such that z+ Jt+ + z~ C F° for all n > n, where (Fn ) the sequence of figures considered above. Consequently, if n > n, each it [it-] is interior to some region Q€^n * For each Q, containing at least one *+, or (in its interior), let us apply (11.3, iv) by taking H =(p), H = T~1(p); hence H C A, H [z+(jt+ )* + z“ (*” )*] = 0. Then there is a sub­ division SQ of each Qinto simple pol. reg. q and pol. reg. R of orders *o > 1, R = (rQ, r1,..., (a) each *+€S+,[^”eS“] is the finite sum of regions q and R; (b) for each region R, or q, contained in a region *+, or we have q* H =0, R* H

r^), such that

= 0;

(c)for any R and any boundary curve r* the curve C:(T, r*) is completely tained in a circle a not containing

of R* con­ p.

For each region jr(it = jt+eS+, or it = it eS ) let us consider now the subdivision of * mentioned in (a) and let us drop from it all parts r1, r2, ..., r^ relative to regions R = (rQ, r1, ..., r ) C it,as well as all further regions q, or R, contained in some V y } rg, ..., r^ as above.In such a way we have a sub­ division + S2jt of Tt into simple pol. reg. qeS1 and simple pol. regions r0 eS27t‘ ^ ann, and where |B^| < n^, lBonl < fop all n provided an < 1• Because of the convergence of the series ^ ( ^ + an )> we have |In | < + » for all n largeenough and as n -*°o. In addition In 3 IR+1 for |In I -- * 0 all n, and, if I = lim I , as n -> oo, |I| = 0. For every peEJ> - I there Is an integer nQ(p) such that p is not in In for any n > nQ and hence p is not in B^ + BQn for any n > n . If n^p) = max [nQ(p), n(p)], n(p) defined as in (i), then for all n > n^(p) the second alternative of (i) holds; hence 0 (p; C0) = z 0 (p; C) for all

also

andhence

CHAPTER IV.

186

n > n.j(p).

BV AND AC PLANE MAPPINGS

Thereby (iii) is proved.

NOTE 2. In the relation 0 (p; CQ ) = z 0 (p; C) of (i), as well as in the inequality |0 (p; CQ )| < z |o(p; C)| of (ii), the sum z is extended over all closed simple pol. regions Q^S^ such that q C ir. For those regions qeS^ with q C n but not q C it° we have q jt* =(= 0 and 0 (p; C) = 0 by the same reasoning utilized for all seSn2* Therefore the relations above are not affected by including or excluding these regions in the sums above.

NOTE 3 - As a consequence of Note 2we can enlarge the hypotheses of (i) and (ii) by replacing the condition it C A0 with the following weaker one: C A and there exists an integer n such thatn C Fnforalln >n. Indeed for all Q.e ^>n with q C *, n > n, either q ** =(= o and then 0 (p; C) = 0, or q C it0,hence q C A0, q* C B, [C] C B^, and the reasoning for (i) and (ii) remainsunchanged. This observation is particularly useful because, if A is a figure, we can suppose Fn = A for all n, and n any region it C A.

12.8.

The Functions v = w,

(i)

(1

V, W

as Limits and the Equalities

v+ = W +, V~ = W~, V = v+ + v~

THEOREM- For any c. plane mapping (T, A) (12.1), for any sequence (Fn ) of figures Fn C Fn+1> Fn C A, F° | A0, for any finite subdivision Sn = + S” of Fn in non-overlapping simple pol. reg. and non-simple pol. reg. w3l0Se indices dR, n^, crn — as n — ► «>, we have

)

E

111

v(q, T) = V(A, T),

n — ► «= ^e^n

(2 )

lim E (E’ n — *-°° q^S^

)

f

|0 (p; C)| = (E')

J

where C: (T, q*), BV, also (3 )

fI

f

N(p; T, A) = W(T, A),

J

In addition, if

X

H m (El) N(p; T, A) |0 (p; C)| J 1 q e S 1 n — oo ^ n

(T, A) = o.

Analogous relations hold where V, W, 0 , N are replaced by V +, W+, 0 +, N+, or V", W , 0 ", N-.

is

§ 1 2 . BV PLANE MAPPINGS (1 2 .8 ) (ii)

THEOREM. For any c. plane mapping (T, A) (1 2 .1 ), we have V(A, T) = W(A, T), V+(A, T) = W+ (A, T), V“(A, T) = IT (A, T), W+ (A, T) + W"(A, T) = W(A, T), V+(A, T) + V_(A, T) = V(A, T). PROOFS of (i) AM) (ii). We shall denote as above, by zl, z1’ any sum extended over all Q^S^, o v ReS^f. By the definition of the function N (12. l ), we know already that

(*0

z'

Io(pj c' )i< n(p ) •

Hence in (3) the difference under sign of absolute val­ ue is >0. Let € be any positive number. For any integer t > 1 and peE^, let N^(p) = t if Nt (p) > t, Nt (p) = N(p) if N(p) < t. ThenNt (p), peE^, is a non-negative, B-integrable function taking only a finite set of values. By [S. Saks, p. 22, (11.6)] we have (E') J Nt — - (E») / N = W(A, T) as t — ► Hence we can take t in such a way that (E|) / > W - e if W < + », (E£) / Nt > e " 1 if W = + 00. Let BQn be the set defined in (12.6) for the subdivision Sn and let n1 be the smallest integer such that an c 1, an < e t~1, m^ < € t”1 forall n > n^ . Then for all n > n1we have, by(12.6, ii), lBQnl < and also, by (12.6), IB^I = |T(Bn )| = ^ < € t”1, where Br = z T(q*) the sum being extended over all Q^S^ such that q C F°. Let p be any point peE£. By the definitions of N(p) (12.1) and N^(p) above, there is a finite collection SQ of non-overlapping simple pol. reg. it C A such that

(5)

£(o) |o(pj 0o )| > Ht(p)

where 0o: (T, «*) and.2^°^ denotes a sum extended over all *€S0 - By (8.3, i) we can suppose * C A0 for all *eS and also 0 (p; CQ ) + 0; hence p is not on the set M = Z^0 ^ [CQ] = z^°^ T(**). If C: (T, q*) is the curve image of any then,

CHAPTER IV.

BV AND AC PLANE MAPPINGS

by (1 2 .7 , i) there exists an n(pj which we can suppose > n1, such that, for every n > n(p), either peB^ + BQn, or 0 (p; CQ ) = E 0 (p; C) for any and where z is extended over all q C Consequently also |0 (p; C )| < Z 10 (p; C)| and, by adding all these relations and by (5), finally (or)

Nt (p) < S (0) |0(p; C0 )| < 2 (o) L 10(p; 0)|
n(p) we have either P^B^ + BQn, or Nt(p) < Z f |o(p; C)| < N(p). ¥e know that the sets B^, BQn are B-measurable and that the functions 0 (p; C) are also B-measurable. Therefore the functions

(7h (p)) ; lo(s; 0)1 di

, -

- € N_1 IQI-1 |7h (p)| • By adding all these relations and using the definition of J we have £(o)£* )7h (p)l |0 (p; C)| < s + s(o)s* I(rh z(o) (J) / |o(|j C)| dg - e . As a consequence, by using the definition of the relations above, we have V

- e < S (o) v ( * , T ) = Z (o)

SQ

and

(Ep / |0 ( g; C)| dg =

= s(o) (Q) / I0 ( C ) |

dg =

= s(o) (J) / |0 (|; C)| d| + z(o) (Q - J) / |0 (g; C)| dg < < Z ( o ) Z* |(7h ( p ) ) / 0(gj c) dg| + 2 e + (Q - J) / N< £) dg Let hQ = min h(p) for all peM; hence hQ> 0 and, for every p, 0 < p < hQ, let Kp denote the set K

(p). T h e r e fo r e K C J ° , K ----- > J ° as p = £*7, 'h-p'^' p p p -- »o +. We can determine p, o < p < h , in such a way that |J° - Kp | = |J - Kp | < cr. By using (8) we have € < z(o) z* l ( 7 h _p ( p ) ) I 0 ( 6; C) dg| +

V -

+e(°)

z* |[ r h ( p ) - 7h _p ( p ) ]

+ (Q - J) / N(g) +

26

I 0 (g;

C)d g |

+




CHAPTER IV.

BV AND AC PLANE MAPPINGS

For every n > n, let Bn, B^, BQn be the sets defined in (12.6), (12.7) relatively to Sn = S^ + S^!. Hence IB^I = < cr and, since an < 1 > also lBQ n l < (12.6). If Ln denotes the set Ln = H^ + B^ + BQn, then |H I < 2 a, |L I < 2 o + a + ko = 7 cr- Let C T: (T, q*) for *1 (1 ) any and, for each let zv denote any sum extended over all Q.6^ , q C n. If i denotes now any point SeQ - Ln, then I is not in B^ + BQn, | is not in H^, and, for every *eSQ, also it C F° and U, [C]) > (|, H ) > t ] > 3 dn * By (12.7, i) and (12.7, Note 2), where only the second alternative holds in the present case, we have 0(6; C) = Z (1 } 0(6; C'),

6eQ - 1^,

n > n .

In addition we observe that if for a certain £€Kp and s = qeS^ we have 0 (1 ; C 1) 4 0 , then £ belongs to a circle P€M, and, by (8 .6 , ii) also £eT(q). Hence 7^_p(p)T(q) 4 0 , and, because of diam T(q) < dn < p, the whole set T(q) is contained in Consequently 0 (1 ; C 1) = 0 for all £ outside ^(p)* By (9 ) and (10) we have now V -

5e




which is certainly possible because of (1 2 .8 , i). Let us observe that the functions 0 (p; C) are finite every­ where (8 .4 , ii) and that 0 < z^0 ^ |o(p; C)| < N(p; T, A), where N is L-integrable and hence N < + « almost everywhere. Let P denote the set of all points p where z^°^ |0 (p; C)| < N. Hence all points where N = + oo are in P and N - z 10 1 > 1in P and = o in E£ - P. Consequently |P| < (P) / [N - Z |0 |] = (E^) / [N - Z 10 1 ] = W - Z v = V - Z v < tr, i.e., |P| < cj.

195

196

CHAPTER IV.

BV AND AC PLANE MAPPINGS

As in the previous proof let us consider the tions 0 (g; C), we use the letter 0( (i; 5; C)

N

func­

where where C: C:(T,(T,it*), it*), Jt€SQ, and and instead of p. Each function

|

is everywhere finite and andmeasurable; measurable;hence, hence,byby

[S. Saks, I, p. 132, (10.6)], each eachfunction function o U o; U ; C)C) is is approximately continuous at almost all points peE^; i.e., there exists a measurable set density

1

at

p,

containing

e = e(p), having

p,

and

lim o(|j 0(5; C) = o(p; o(p; C)

-- ► p, p, £ee(p). as £ -----► ^ee(p). Since Since o(|; C) takes only integer values,

each function 0 OU;

C)

is constant on that part of

e(p)

in a sufficiently small neighborhood of denote this part by

contained

p.

We still

e(p).

In addition, since the intersection of a finite number of sets all dense at the same point p,

p

is dense at

then for almost almost all all point point peQ° peQ° we we can can determinedetermine

a unique set e(p) e(p) dense dense at at ppsuch such that that for all

£ee(p)

can determine 0 < h < h(p) o

and for all h(p)

0(|, 0 0(|, (|, C) C) == 0(p; 0 0(p; (p; C) C)

*eSQ *eSQ . . Consequently Consequentlywewe

in such a way that, for any

all requirements of the previous proof

hold and also there is a set

e(p) e(p; C /^(p), /^(pj,

dense at au

p and containing p such that o(s; 0(5; C) = 0 (p; C) all 3 t€S0 *€S0 *€S 0 and |ee(p), and also that |e(p)| | e (p)| > | |7h 7h (p) (p)|| (1 (1 -- aa IQI”1) IQI”1) -n

for

Let Let us us now determinedetermine

as in the previous previous proof proof and and observe observe that that for for every every

(■ey £eQ

and and nn >> n, n,

(5; (5) (5) holds holds and and also also

z ^1^^loU; loU; |0U; cC» )),where where loU; C) C)|| < z^1 , )l, over all parts qeS^, q C i ?t, and C 1: (T, q*).

For

z z^1 ^ ^ is is extended extended C:and (T, C: it*), J t * ) ,(T, Jt*),

|eQ - P - L^ we we have have also also

N(g) = 2 ^o) 10 Io(s; c)| < Z z (o) 10(gJ C')| < N(s) < + c Co O, , (6; C)| 10(1; and, as a consequence, the equality signs hold and also lo(l; lo (s ; c)| = Z (1^ (1 ' Io(?; |o(e; c')| C')|

for all

?eQ - P - Ln 5eQ

and

q€SQ . Therefore, ( 10)

o(g; 0(1; C) C) == £S (1} (1}

(1*0

0(65 o(£; C C 1) ' ) ,,

I0 (6; C)| = S (l) |0 (j; C')t , for all in

|€Q - P - Ln

Q, - P - Ln ,

q C it,

and

all functions

This implies that, 0 (|; C ?),

are zero if 0 (|; C) = 0, are > 0 are < 0 if o(i; C) < 0.

0 (|; C) > 0,

if

§12.

BV PLANE MAPPINGS (12 .9 )

197

Let us consider now, as in the previous proof, the collection of all qeS^, which are completely contain­ ed in some region it and let us recall that, if n > n, and OU; C f) + o for at least one point | of a circle then the whole set T(q) is contained in 7^(p); hence o(p; CM) is zero outside of where G 1: (T, q*). Let S ^ Sn2, be those sub­ collections of sums of all qeS^ such that (a) OU; C !) $ o for at least one point !€7^_p(p) and a point peM; (b) q C it for a *eSQ; (c) 0 (p; C) > o [< o], where C: (T, it*), C>: (T , q * ) .

Let

Sn3 =

-

Sn1 - Sn2,

be the remaining part of the collection S^. Let zn1,zn2, zn3 denote any sum extended over all Q.€Sn-j> Sn2, Sn^, respectively, and let us mention here that 2 * denote respectively sums extended over all *£S0; all qeS^ such that q C it (it one of the regions iteS0 ); all peM. By the first lines of (12.9), we have at once £nl u(q, T) < Snl v+(q, T) < V+(T, A) ,

(1 5 ) -

t n2

u(q, T) < Sn2 v“(q, T) < V~(T, A) ,

and also (16)

En1 |u(q,

T )| +

2n2|u(q,

T)| + sn3|u(q, T)|

On the other hand, for each Q.€Sn 1 , 0(5; Cr) is + 0 only in points circle /^(p); hence Znl u(q, T) = znl (E') /

0

= Sn1

(rh ( p ) ) / 0(1; C ' ) d| =

=

^ h - P e - Ln " P)

+ 2m

Crh " (rh-p e " Ln - P)]

¥e have

f

< z'

v(q, T) •

the function of one and only one

(|; C«)d^ =

0(^ C,) f

0(t;

+ C '}=X l+X l'•

CHAPTER IV.

1 9?

lXl"I < Sm

BV AND AC PLANE MAPPINGS

(7h - 7 h-P} + (7 h-p-e) + C) = z(o)(Q) / o + ( 6 ;

xi

=

c ) de - (Q - z * 7h_pe + Ln+ P) / z(o) o + ( i ;

- z(o) v+U, T) > - (Q, - z* 7h_ e + Ln + P) / N(p) dp = (Q - J) + K„ + Ln + P +

z * ( 7y, -

h

where |Q - J| < a, |K | < a, IL^I < 12 * (7^ - e)| < a. Therefore

e)

7 a,

/ N(p) dp, |P| < a,

C) dg,

§12.

BV PLANE MAPPINGS (12.9 )

X] - E v+(«, T) > v+ (1 3 ), (1 7 )

£ n1

u(q, T) >

11

199

€ and finally, using also

v+(«, T) -

10s

-

10e

> V+ -

22e

.

Analogously we can prove that (18)

-

£ n2

u(q, T) > Z(0) v-(*, T) -

From here it follows also zn2

11

e -

10€

> V" -

|u(q, T)| > V+ -

£ n1

22e

2 2 e,

> v~ “ 22e> and finaH y |£n3

u(q, T )| < Zn3 |u(q, T)| =

= (s'
n. This proves

NOTE 1 . As in (1 2 .8 , Note of all simple pol. reg.

1

66

Sn>

e ,

22e

,

(ii).

), if we denote by SnQ C the subcollection with q C then we have

Y

11m n — ► oo qeSn

Iu(q, T )| = U(A, T) •

The proof is the same as above if we only observe that in (9) the sum Z* is extended over all q of a subcollection of SnQ.

2 00

CHAPTER IV.

BV AND AC PLANE MAPPINGS

NOTE 2 . As in (1 2 .8 , Note 2 ) the limit as n — ► then then for for almost almost all all points points we we have have

peE^, peE^,

,

lim 10(p; llm XX I0(p; C)| C )| == N(p; N(p; T, T, A) A) , n --- *►- o oo o qeS^ qeS^ n

((19 1 9 a) a)

lim llm XX 0+ 0+ (p; (p; C) C) == NN+ + (p; (p; T, T, A), A), nn —— ► ► ooo o qeS^

lim Yj °~(p; C) = N"(pj T, A) . n -- ► oo qeS^ In addition, In addition, if if a.e. in a.e. in E£, , (19b) (19 h)

(T, (T, A) A)

is is

BV, BV,

then then we we have have also, also,

lim lim XX 00(p; (p; C) C) == n(p; n(p; T, T, A) A) .. nn ---► ---► oooo ^^-€ -€Sr Si ri PROOF. In PROOF. In [(12.8, [(12.8, first first part part of of the the proof proof of of (i) (i)] ] we we have pointed have pointed out out that, that, for for every every P P €E^ € E^ and and t1 t 1 >n^(p), n^(p), either either peB^++ BB0n' for all 0n' op op E|0(p; Z|0 (p; C)| C)| > > where where z z is is extended extended over over all all qeS^. An qeS^. An analogous analogous statement statement holds holds for for the the functions functions NN+, +, 0+ 0+ and and N N“ “, , 0" 0" provided provided n n> > nn 22 (p), (p), n n> > n^(p), n^(p), where n^(p), where n^(p), nn g g (p), (p), n^(p) n^(p) depend depend upon upon pp and and also also upon upon t t1 1 < < N, N, t2 t 2 ^ -- * H as n -- » oo, where H Is a subset of . In addition, since the functions N(p), N (p) are all B-measurable, all sets M, H^, H are B-measurable and [H^l > IH^I, IH^I -- > |H|. By (1 2 .4 , i) we have N(p) < lim Nn (p) for all peE^ there is an peE^ as n ---» c»; hence for every n(p) such that N (p) > N(p) - 2 ”1, and also, as a consequence, N (p) > N(p), for all n > n(p), since both N , N are integers. This implies that p does not belong to for any n > n(p) and also that p is not in H. Since p is any point of E^, the set H is vacuous and |H| = 0 . Finally iH^j -- > 0 , IH^I -- > 0 and, since N(p) is L-integrable in M, also (H^) / N -- * o as n -- > oo. Now for every point peE£ - M we have |N - N| = |Nn l = Nn = Nn - N since N = o, Nn > 0 ; for every point peM we have Nn > N,hence for every ~ ^n wehave lNn ” = Nn - N; for every peHn we have |Nn -N| = N ~ Nn < N < (Nn - N) + 2 N. Consequently (E^) / |Nn - N| = [(E^ - M) + (M - 1^) + Hj I |Nn - N| < (E‘) / (Nn - N) + 2 (1^) / N = (Wn - W) + 2 (1^) / N. Since WR - W -* 0 , (1^) J N -- * 0 as n -=» we have(E^) / |Nn - N|----- ¥ 0 . p€E2

The same reasoning holds for N+, N*, N”, N~; hence (Ep / |N+ - N"| --> 0 , (Ep / |N”- N| — > 0 . Finally, because of Ir^ - n| = |(N* - N“ ) - (N+ N")| < IN* - N| + |N~ - N| a.e. in E^, we have also (E^) / Ir^ - n| -- > 0 . Thereby (iii) is proved.

1 2 .1 2 .

(i)

Further Properties of the Functions N And n

If (T, A) is any BV plane mapping from a simple closed Jordan region A of the w-plane Eg, w = (u, v). into the p-plane E£, p = (x, y), if C: (T, A*) is the image of the (counter-clockwise) oriented boundary curve A*, then n(p; T, A) = o ( p ; C) a.e. in E ’ - [C]•

CHAPTER IV.

BV AND AC PLANE MAPPINGS

PROOF. Supposefirst that A is a simple pol. reg. A = it. Let Sn = S^ + S^ 1be a sequence of sub­ divisions of Jt into simple pol. reg. and not simple pol. reg. as in (12.7), whose indices ^ n ’ an -- * 0 as n -- > ». Let us suppose also that the series ^ ( ^ + an ) is convergent. If C 1: (T, q*) for every Q.€^ and E 1 denotes any sum extended over all then, by (12.7, iii) and (12.1 1 , ii), we have 0 (p;C) = lim E 1 0 (p; C 1), n(p; T, A) =lim E 1 0 (p; C) as n -- >00 for almost all peE^ - [C]. Thus 0 = n a.e. in E^ - [C]. Suppose now that A is any Jordanregion. Let jtn be a sequence of simple pol. reg. Jtn C A, jtR C *n+1> fl'n t A0, ||jt*, A*|| > 0 as n -- > °o. In consequence \\&n > cl l -- > 0 as n -- >00, where Cn : (T, Jt*) and also 0 (p; C ) -- > 0 (p; C) as n -- > » everywhere in E^ - [C]• Since both |0 (p; cn )l> lO(p; C)| are < the L-integrable function N(p; T, A), we have (E£ - [C]) / |0 (p; Cn ) - 0 (p; C )| -- » 0 as n -- ► 00. On the other hand, (T, it ) > (T, A) and hence ¥(A, T) < lim W(jt , T) as n -- > while ¥(it ,T) < ¥(A, T). Consequently W ( jt , T) »¥(A, T) as n -► 00 and, by (1 2 . 11 ,• iii) also (E^ - [C] ) / |n{p; T,jtn ) - n(p; T, A) | -- > 0 as n -- ► 00. Finally n(p; T, jtR ) = 0 (p; Cn ) a.e. in E2 “ * -Cn ^ where Cn : (T, jt*), n = 1 , 2 , ... . Since for each peE^ - [C] we have also peEJ> - [CR ] for all n large enough, we have n(p; T, jtn ) - 0 (p; Cn ) ----- * 0 as n -►«> for all peE^ - [C ] . Hence (E» -[C] ) / |n(p; T, *n ) - 0 (p;Cn )| ---- > 0 as n --- »00,as before. Finally (Eg - [C] ) / |n(p; T, A) -

0

(p; C)|
, p = (x , y), if H C T(A) is any given set open in T(A) and H = T”1(H), then H is open in A, H is an admissible set (5 - 1 )> N(p; T, H ) = N(p; T, A) a.e. in H, and N(p; T, H ) = 0 for all peE^ H. PROOF. Since H = T( H ) we have all peE^ - H and since H C A, N(p; T, H ) < N(p; T, A) for all N(p; T, H) =N(p; T, A) whenever (Fn ) be a sequence of figures Fn

N(p; T, H ) = 0 for also peE^. Hence N(p; T, A) = 0. Let C Fn+1> Fn C A,

Fn ^ Sn = ^n + ^n an^ sut)d.ivision of Fn into simple pol. reg. a^d non-simple pol. regions ReS^’ (1 2 -7 )> whose indices ^ crn -- >0 as n — >oo and the series+an )convergent. If C: (T, q*) for all and Z r denotes any sum extended over all Q.6^, then, by (12.11, ii), we have lim Z!|0 (p; C)| = N(p; T, A) < + oo as n -- > «> a.e. in E^. Let H f be the subset of all points peH where the equality aboveholds. Thus |Hf| = |H| . For every peH’ there is an integer n(p) such that Z 1 0 (p; C) = N(p; T, A) n(p) and we consider only those terms of z f for which 0 (p; C) * o. This implies that in each corresponding q there is at least one point weq° with T(w) = p and also that the same holds for all points of a neighborhood of p (8.6, ii) (8 .4 , i). Hence w€*° C A0, pe[T(A)]°, and, since H is open in T(A), also peH0, 2cr = (p, H*} > o. Since dn -- > 0, we can suppose n(p) large enough in order that < a for all n > n(p). Hence, if n > n(p) and is such that o(p; C) + 0, the whole set T(q)belongs to a circle y of radius c dn c a and p €7. Hence 7 C H and finally q CH. This Implies N(p; T, A) = z» |0 (p; C )| < N(p; T, h ) < N(p; T, A); that is, N(p; T, A) = N(p; T, H) for all peH1, i.e., for almost all peH. Thereby (ii) is proved.

1 2 .1 3 -

(i)

The Index

n

If (T, A) is any BV plane mapping from any ad­ missible set A of the w-plane E2 into the p-plane EJ>, if S is any finite system of non-overlapping

CHAPTER IV.

BV AND AC PLANE MAPPINGS

closed simple pol. reg. it C A, if n is the number m, = U(A, T) - Z |u(jt, T)|, 0 < |i < + oo, where z is extended over all iteS, then there exists a B-measurable set H C E^ such that (1 ) IH | < n, (2) for every peE^ - H we have N (p j T , A) = z 10 ( p j C)| < + co, N+ = Z 0 + (p; C ) , N~ = Z 0 " ( p ; C ) ,

(3)

n = Z 0 (p; C), where C: (T, it*), jteS; for every simple pol. reg. it1 C A non-overlapping any iteS and for every point peE^ - H we have 0 (p; C 1) = 0 , where C 1: (T, it1*).

PROOF. By (12.9, iii) we have U = V = W and, by (1 2 .1 ), N(p) > Z |0 (p; C )| for all peE^. Let H* be the set of all points peE^ where N(p; T, A) > Z |0 (p; C)|. By (8 .4 , ii; 1 2 .2 , i) both functions N(p) and z |0 (p; C)| are B-measurable; hence H* is B-measurable. By (8 .k , ii) 0 (p; C ) is always finite; hence z lo(p; C )|< + 00and, as a con­ sequence, all points peE^ where N(p) = + 00 are in H*. Finally, forevery peE^ - H* we have N(p) < + », N(p) - Z |0 (p; C )| =0 ; for every peH* we have N(p) - Z |0 (p; C)| > 1 . Finally |H*| = (H*) / (1 ) < (E, we have N+(p) > Z 0 +, N“(p) > Z 0 ”; hence 0 < [N+ - z 0 +] + [N“ Z 0 “] = N - Z |0 | = 0 for all peE^ - H. Thus N+ = Z 0 +, N” = Z 0 “ for all peE^ H. Consequently, also, n = N+ - N” = z 0 + - Z 0 “ = z 0 for all peEJ, - H • Let it1 be any region as above. Then, for all peE^ - H, we have N(p) < Z |0 (p; C)| + |0 (p; C !)| < < N(p) < + 00;hence 0 (p; C 1) = 0. Thereby (i) is proved.

§ 1 2 . BV PLANE MAPPINGS

(1 2 . 1

3)

Let (T, A) be any BV plane mapping from any admissible set A of the w-plane E2 into the p-plane E ^ let S = [it], S ’ = [it1] be any two finite systems of simple pol. reg. it, it1 C A. Suppose that the regions iteS are not overlapping and also that the regions it'eS1 are not over­ lapping, and let B = T(z it*), B ! =T(zTit1*), where z, s ’ are sums extended over all iteS and it’eS'. Let C: (T, it*), C T:(T, it1*), iteS, it’eST; hence B = Z [C], B ! = z1 [C1]. Let H, H T be the sets de­ fined in (i) relative to the systems S and S r; let 3 p = max diam T(it!), it'eS1, and Bp be the set of all points p at a distance < p from B. For any iteS let S' denote the subcollection of all it’eS* such that n it1 C it , let Sj be the subcollection of all it’eS1 such that it' it* 4 o for some iteS, and let S^ be the (remaining) subcollection of all it'eS’ such that it!(z it) = o. Then the subcollections S'(it) for each iteS, Sj, S^ are disjoint andZ S(it) + S' + S^ = S'. Let J be the set J = Bp + B' + H + H' . Let Z^*^ and z1 ’denote any sum extended over all it'eS(it) and it'eS* + S^ • (ii) Under the conditions above we have (1 ) 0 (p; C) = 0 (p; C ) for every iteS and for almost all peE^ - J; (2) 0 (p;C 1) = 0 for every iteS' + S,* and for almost all peE^ - J; (3 ) N(p; T, A) = z I0 (p; C ) 1 = z' lo(p; C')l for almost all peE^ - J;

(1+)

n(p; T, A) = z 0(pj C) = I' 0(pj C') peE^ - J.

for almost all

PROOF. For every it’eSj, we have it1 it* + 0; hence T(it’)B + 0 and T(it’) is contained in a circle 7 of center a point of B and radius p. Thus 0 (p; C ’) = 0 for every peEJ> - J because p is not in 7. For every itTeSJ, and peE^ - J the equality 0 (p; C 1) = 0 follows from (i). Thus (2) is proved. The set B f is closed and bounded, hence, if -n is any positive number and B^ the n-neighborhood of B 1 we have B11 -- » B 1, BT1 ► 0 as ri -- > 0 +. Let 1 |- B (Fn ) beany sequence of figures Fn C A, Fn C Fn+1> E° t A°, and let Sn = S^ + S^' beany subdivision of Fn into simple pol. regions qeS^ and non-simple pol. reg. ReS^f,whose indices dn, m^, crn — >0 as n --------- * 00 and + crn )< + 00. We can also suppose that there exists an n such that it, it’eFn for all iteS, it'e S ', n > n. By (1 2 .7, iii), if c: (T, q* )

CHAPTER IVfor each

( 21

Q.6^,

BV AND AC PLANE MAPPINGS

we have

0

(p; C)

= lim ^ n ---- > oo q c it

0

(p; C 1) =

0

(p; c),

) lim X n -- > oo q C it1

0 (p;

c),

for almost all peE^ - [c], and PeF^ - [c1] respectively and for each iteS, 7t*€Sf. Consequent­ ly both relations (2 1 ) hold a.e. in E^ - [B + B !] and thus there exists an n = n(p), P e E 2 “ [B + B f], such that, for alln > n(p), iceS, jt'eS*, 0(p; C) =

X

0(p; c) ,

q C it ( 22 )

0(p;C ') =

X

0(p;

c) ,

q C i' and also

Z U)

(2 3 )

0 (p;

C') = E U)

S

q C itf

0(p; c) •

If we suppose P£E^ - [Bp + B'],then the two sums given in (22) and (23) for 0 (p; C) and Z ^ 0 (p; CM) differ only for the terms 0 (p; c) relative to all q C it such that q it1* =j= 0 for some jcfeSf(at). If we suppose n(p) large enough in order that ^ < n, for all n > n(p), then all these terms are zero in B^. Hence o(p; C) =z^*^0 (p; c !) a.e. in E» - (B +B !). Since B f ---- > B fas ri --- » 0+,(1) 2 P 1 T) 1 is proved a.e. in E^ - (B + B f).Finally for every peE^ - (B + B ! + H +H 1 ), by (1 ) and (i), we have n(p) = Z 0 (p; C) = Z Z ^ 0 (p; C 1), N(p) = 2 |0(p; C)| < 2 L(,t) |0(p; C')| < N(p) < + » and, by (2), also n(p) = Z 0 (p; C T) N(p) = z 10 (p; C f)|. Thereby (ii) is proved. (iii)

Z

15

Under the same conditions of (ii), let u(*, T) = z ^ u(itf, T) + 5 (at) for each *eS and Z?! v(it!, T) = 5 . Then, we have

(jt)I
Z W(*, T), N(p; T, x q ) > Z N(p; T, Jt) for all pcE£.

*12.15. Let

(T, A)

The Functions

V*, W*, TJ*

be any plane mapping from any admissible set

A

of the w-plane

210

CHAPTER IV.

BV AND AC PLANE MAPPINGS

E2 Into the p-plane E^ and let R = (rQ,..., r ) denote any closed pol. reg. R C A of some order 0 < -o< + °° (5 - 1 )• Let C.,: (T, r*) be J1 the Image of the boundary curve rn-, i = o, 1 ,.. ., x>, where r* is -Xcounter-clockwise oriented and r1, • r^ are clockwise oriented. We have T(R*) = [CQ] + ... + [C^]. For any point PeEg let (p; R, T ) = 0 if peT (R*) (see 9 -1 6 ). Let cp+, cp” be the functions cp+ = 2 -1 (|cp| + cp), cp” = 2 ” 1 (|cp| - cp), peE^. Then we have 0 < cp , cp” < |cp|, cp+ + cp” = |cp|, cp+ - cp” = cp, and, by (8 .4 , ii), the functions cp, cp ,cp” areB-measurable. Therefore the L-integral v*(R, T) = (E^) / Icp(p)|, exists (finite, or + »), and the same holds for the functions v* , v” expressed in terms of cp+, cp”. Let S denote any finite collection of non-overlapping regions R C A and let V*(A, T) = Sup z v*(R, T) where z ranges over all regions ReS, and Sup is taken with respect to all S. Let us define V*, V* analogously. Finally, for each peE^, let us put N*(p;T, A) = Sup Z | c p ( R , and N*, N” be defined analogously. Obviously 0 < V*, V” < V* and 0 1 N~ < N for all peE^. The functions N*, N*, N” are lower semi­ continuous in E^ and the proof is as in (1 2 .4 ). Thus we may consider the L-integral ¥*(A, T) = (E^)/ N(p; T, A) and the analogous integrals ¥*(A, T), ¥”(A, T). Obviously we have 0 < ¥*, ¥” < ¥*, and, as in (12.3, i), also 0 < V* < ¥*, 0 < V* < ¥*, ° < V* < ¥*. If we suppose V(A, T) iv). Thus the reasoning holds for all peE^. The simple modifications of the proof can be left to the reader.

§ 1 2 . BV PLANE MAPPINGS (12.15)

(iv ) For a l l but countably many p o in ts peE^ we have N*(p; T, A) = N(pj T, A), hJ = N+, N*= N” . PROOF. Obviously we have o < N
2 . Then let us con­ sider any finite subdivision S Tof the regions jteS into simple pol. regions q C n with |q| < M " 1 a. Then we can associate all qeSl into subfamilies f, each f covering an area in A between a - M _1 a 1 _1 1 and a. We will haveless than mcr"" (1 - M )“ < 2m a 1 < M families f. Letz, zT, z^°^, z* denote any sum extended over all jteS, qeS1, qef, feS1,

2 16

CHAPTER IV.

BV AND AC PLANE MAPPINGS

respectively. Then by v < V, by (b) and (i), we have 2 v(n, T ) < 2 V(*, T) = £' V(q, T) = 2 * V(q, T) < Z* (1 ) < M; that is z v(jt, T) < M for every system S of regions iteA. Hence V(A, T) < M and, by (1 2 .8 , ii), also W(A, T) < M, i.e., (T, A) (iii) The conditions (a) and (b) are independent. This statement is proved by the following Examples A and

is

BV.

B.

(A) Example Of A Plane Mapping Satisfying (a) And Not (b)

Let A = [o < u < 1 ,o < v < 1 ] be the unit square of the w-plane Eg, w = (u, v), and B the rectangle B = [o < x < k / l , 0 < y < 1 ] of the p-plane E^, p = (x, y). Let C: x = cp(t), y = ^(t), o < t < 1 , be a simple arc of end points (o, o), (o, 1 ) covering a set of measure m > o of the rhombus r of vertices (o, o), (o, i), (1 /7 , 1/2) (~ 1 /T^ 1/2). Consequently we have [C] C r, 0 < m < |r| = 1/7* (For curves having these properties cf. W. F. Osgood [1]). Let us consider now the twin curves r 1 : u = 2/7 + (t ), v = ty(t), 0 < t < 1 ; r2: u = 5/7 + cp(t), v = \ir(t ), 0 < |t < 1, 1^, r2 C A.The curve ^divide A into three non-overlapping Jordan regions r1, r2, r^, where r 1 is between the straight line u = 0 and 1^; r2 between r1 and r2; r^ between r2 and the straight line u = 1 . Let us observe that the seg­ ment s = [u = 1/2, 0 < v < 1 ] be­ longs to r2-

cp

Let us now define a function F(u, v), (u, v)e r2, with the following prop­ erties: (a) 0 < F < 1 ; F(u, 0 ) = 0 , F(u, 1 ) = 1 for all 2/7 < u < 5 /7 ; F(l/2 , v) = v for all 0 < v < 1 ; F[2 /7 + q>(t ), ^(t)] = F[5 /7 + cp(t), t(t)] = t, 0 < t < 1 ; (p) The equation F(u, v) = t, (u, v)er2, defines one and only one continuous simple arc of r2 joining the points [2/7 + (t ), (t )], [5/7 + (t ), t(t)] and passing through (1/2, t). Here a procedure for the definition of F. Let us observe first that s divides r2 into two Jordan regions i>22 a^d that we can define a homeomorphism H between + r* 2and R* + R*, where R^, R 2 are the auxiliary rectanglesR 1 = [2/7 < u < 1/2, 0 < v < 1 ], R 2 = M / 2 < u < 5 /7 , 0 < v < 1 ].We have only to map the points [2/7 + (t), \|r(t)] [5 / 7 + cp(t ), )|r(t)], (1/2, t) into the points (2/7, t), (5 /7 , t), (1/2, t), 0 < t < 1 , and the points (u, 0 ), (u, 1 ), 2/7 < u < 5 / 7 into the same points (u, 0 ), (u, 1). By (6 .1 , i) then H can be extended as a unique homeomorphism H between r21 + r22

cp

cp

cp

§13*

and

R 1 + R2,

AC PLANE MAPPINGS (1 3 •O

coinciding with the homeomorphism

H

217 already defined between

P 21 + p 2 2 anci + R 2 ' Then the function FQ(u, v) = v, (u, v)eR1 + R2 is transfomed by H into a function F(u, v), (u, v)er2 = r2l + r22, having all the required properties (or) and (p).

Let us now define the c. mapping (T, A) as follows: Let T: x = u, y = v, if (u, vjer^ x = u - 3 /7 , y = v if (u, v)er3; x = 2/7 + cp(t), y = \|r(t) if F(u, v) =t, (u, v)er2, 0 < t < 1 . Thus T is single­ valued and continuous in A and maps A onto B. Each point (x, y)eB is the image either of one and only one point of A, or of a proper con­ tinuum (and of only one proper continuum) of A, according to (x, y)eB - [C ], or (x,y)e[CQ], where CQ is the continuous curve CQ: x = 2/7 + cp(t), y = ^(t), 0 < t < 1 , [CQ ] C B. It is easy to show that for every simple pol. reg. it C A we have Iv(jt, T)| < I* I, hence T satisfies (a). On the other hand, if (q^, q2 ) is the decomposition of A into the two non-overlapping rectangles q 1 =[0 < u < 1/2, ° < v < 1 ], q2 =[l / 2 < u < l , 0 < v < 1 ], we haveV(q1 ) + v (0-2 ) = 4 /7 - m, V(A) = 4 /7 ; hence V(q1 ) + V(q2)< V(A) and T does not satisfy (b). (B)

Example of a Plane Mapping (T, A) Satisfying (b) and not (a)

Let cp(t), 0 < t < 1 , be a continuous, singular, strictly increasing func­ tion of the real variable t, with cp(o) = 0 , cp(1 ) = 1 [for an example of these functions see, e.g., S. Saks, I. p. 101]. Hence cp(t ) is a BV, not AC function of t. Let (T, A) be the c. plane mapping from the unit square A of the w-plane, w = (u, v), into the p-plane, p = (x, y), defined as follows: T: x = cp(u), y = v, (u, v)eA. Then for every inter­ val I = [a, b; c, d], I C A, we have v(l) = V(I) = [cp(t») - 9(a)] (d - c). Since for any integer n there are families of intervals a. < t < b. _1 with z(b^ - a^) < n , 2 |cp(b^) - cp(a^)| > a where a is some positive fixed number, then (T, A) does not satisfy (a). On the other hand the formula V(I) = (d - c) [cp(b) — cp(a)] shows that V is additive as a function of interval. It is easy to show that V is additive also as a function of simple pol. region itC A. Thus (T,A) satisfies (b).

NOTE. Let us consider the example A above and let FR denote the sum of the two rectangles [0 < u < 1/2 + 1 /n, 0 < v < 1 ], [1/2 + 2/n < u < 1 , 0 < v < 1 ], n > 1 4 . Then we have L(FR, T) = |r° I + |r° | = 4 /7 - m (9 *5 , i). Therefore we have L(Fn, T) -- > 4 /7 - m, as n -- > «>, while L(A, T) = 4 / 7 and Fn -- » A, F° -- > A°, but we have not F° t A°. This shows that in the statements (5.10, i), (9 -2 , i), (5 -1 4 , iv), (9 -3 , iv) we cannot replace the convergence F° T A° by the weaker one F° -- >A°.

218

CHAPTER IV.

BV AND AC PLANE MAPPINGS

Since the mappings (T, Fn ) can be approached by means of q.l. mappings, it follows that also in the definition of Lebesgue area (5 *8 ) we cannot replace the convergence F° t A0 by F ° -- ( & - 7 ) and (6 .7 , Note).

*13.2. If f(u, v), (u, square Q, and we y = f(u, v), (u, T is BV if and say

Some Particular Cases

v)eQ,, is any real-valued continuous function in the unit consider the c. plane mapping (T, Q): x = u, v)eQ, then (T, Q) is regular ( 9 - 6 ; 12.5, Note 3 ), and only if the Tonelli total variation with respect to v, Vv[f; Q]

is finite. The image under T of any segment s C Q is semirectifiable (8 .8 ) and hence, by a theorem we shall prove In (2 1 .4 ), condition (b) is necessarily satisfied. By (9 *6 , ii) the number V(it, T), relative to any simple pol. reg. k C Q, can be obtained as a limit of sums z v(r; T) relative to finite systems of rectangles r = [a < u < b, c < v < d] Cit . Hence condition (b) is satisfied in the class of all simple pol. reg. it C Q if and only if the same condition is satisfied in the class of all rectangles r C Q._ By [S. Saks, I, p. 176] the latter condition is satisfied if and only if f (u, v) is AC in 0 < v < 1 as a function of v for almost all ue[o, 1]. Thus (T, Q) is AC if and only if the various re­ quirements for BVT and ACT definitions (3 *1 ) are satisfied with respect to v. If we consider as in (12.5, Note) the c. mapping (T, Q): x = u, y = v, z = f(u, v), (u, v)eQ, from Q, Into Eo, then we conclude that f is BVT and ACT if and only if T 1 and T 2 are BV and AC. Another particular case of some interest is the plane mapping (T, Q): x = f(u), y = g(v), (u, v)eQ, from the unit square Q, where f, and g are continuous real-valued functions in [0 , 1 ]. Then (T, Q) is regular (cf. 9 *6 , Note 3 ) and for each r = [a < u < b, c < v < d] C Q the curve C: (T, r*) Is the sum C = C 1 + C2 + c"

1

+ c^ 1

of four curves each contained in one of the straight lines x = f(a), x = f(b), y = g(c), y = g(d). Hence |C| = 0. In addition 0 (p; C ) = + 1 for all points p of the open rectangle R = [f(a), f(b); f(c), f(d)] and 0 (p; C) = 0 otherwise. Therefore v(r, T) = |f(b) - f(a)| |g(d) - g(c)| and the limit (9 -6 , ii) is then equal to V(Q, T) = V[f] • V[g], where V[f], V[g] denote the total variations of f and g in [0,1], and where the product in the right-hand member is zero whenever one of the fac­ tors is zero (even if the other factor is »). We concludethat (T, Q) Is BV if and only if either both f and g are BV in [0 , 1], or V[f] V[g] = 0, [i.e., in the latter case, either f is constant and g is any continuous function, or g is constant and f is any continuous function] . If both f and g are monotone non-decreasing then we have V(Q, T) = [f (1 ) - f(0 )] [g(i) - g( 0 )] and N(p; Q, T) = N+(p; Q, T) = n(p; Q, T) = + 1, N”(p; Q, T) = o, for all points p interior to the rectangle [f (0), f(0 ; g(o), g( 1 )], while the same functions are otherwise zero. If the mapping (T, Q) above is BV, then in anycase the image under T of any segments C Q is a semirectifiable curve C (8 .8 ) and hence ICI = 0. By (2 1 .4 , i) condition (b) is satisfied. Obviously condition (a) is satisfied if and onlyif either both f and g are AC In [o,i], or V[f] • V[g] = 0. Therefore we conclude that, if neither f nor g are constant in [0,1] then (T, Q): x = f(u), y = g(v), (u, v;eQ, is BV if and only if both f and g are BV; is AC if and only if both f and g are AC in [0,1].

§1 k. LOCAL PROPERTIES OF PLANE MAPPINGS

(1

k .1 )

219

*1 3 *3 * Bibliographical Notes The concept of AC plane mapping given in (1 3 -1 ) was introduced by [L. Cesari, 12 ] together with the examples A, B proving the independence of the conditions (a) and (b) [L. Cesari, 16] . A necessary and sufficient condition in order that a mapping is AC is given in (1 5 *2 ) of this book and due to T. Rad6 [33] and J. Cecconi [2 ]. Such a condition gives one of the forms proposed by T. Rado of the definition of the AC concept for continuous plane mappings. For other equivalent forms of the same concept see T. Rad6 [20, 32] (essential absolute continuity). J. Cecconi [3] has extended to BV and AC plane mappings the integral property given by [S. Saks, 6] for real functions of one real variable. The Banach definition of absolute continuity (ACg) [S. Banach, 2] is not equivalent to the AC definition and could be obtained by the same changes already indicated for the BVg concept (12.16). The corresponding con­ dition, say (b)B, of (b) is a consequence of condition (a)^.

*§1 4 . LOCAL PROPERTIES OF PLANE MAPPINGS *1^.1.

The Collection r for a Mapping (T, A) From any

AdmissibleSet A

Let A be any admissible set of the w-plane E2, w = (u, v). Then (5 - 1 ) A is either an open set, or a closed Jordan region A = J (of finite order u > 0), or a finite sum A = K = Z Ji of disjoint closed Jordan regions J^, or any set A C K, open in a set K. Let (T, A): p = p(w), weA, be any con­ tinuous mapping from A into the p-plane E^, p = (x, y). Let H = T(A). For any peH let T”1(p) be the subset of all weA with T(w) = p and let us consider the components g of T_1(p). Let r = r(T, A) be the collection of all components g of the set T” (p), peH. Then r is the collection of all maximal connected subsets of r on which T is constant. In (10.I1) we have already considered the collection r in the case where A is compact and then all ger are compact, hence continua g C A. In the present situation the sets g are not necessarily compact. Let us recall here that a set g is said to be "locally compactM if for every point weg there exists an open set c such that wee and the set g c is compact. Any locally compact connected set is said to be a "generalized continuum" [G. T. Whyburn, p. 16]. (i) Every element ger is a generalized continuum. PROOF. If A is compact then g is also compact and hence a continuum. If A is open, let c be an open circle of center w such that c" C A. Then by the continuity of T, T is constant not only on

220

CHAPTER IV.

BV AND AC PLANE MAPPINGS

g, and g c, but also on g ~c hence all points weg "c belong to g since g is maximal; finally g c" = g o' and g cT is closed and bounded, and hence compact. Suppose finally that A C K and A is open with respect to the compact subset K of the w-plane E2. Let c be any open circle of center w such that c" K CA. Then because of g C A C K we have also g "c= g(cK). As above, T is constant not only on g, and g(cK), but also on g(cK); hence all points weg(oK) belong to g since g is maximal. Finally g(cK) = g(cK) and g(cK) Is closed and bounded; hence g "c = g(Ac) = g(cK) is compact. The statement is proved.

*1 4 .2 . The Collections

rQ

and

r

3

rQ }

Let rQ denote the collection of all ger which are compact Hence all elements gerQare continua. Let r1 denote the collection of all gerQ which have the following property: There exists an open set U. such that g C U C A. Hence each element ger^ is interior toA; i.e., g C A°. Let Aq, A 1 be the subsets of A covered by the continua g of rQ and r1 . Hence A 3 AQ D A 1 . (i) The set AQ is open in A. PROOF. and AQ have g discuss suppose K C E2-

If A is compact, thenall g are compact = A. If A isopen, then for any g€PQ we C U = A; hence rQ = , AQ = A 1 [we shall the measurability of A 1 in(ii)]. Finally that A CK is open in a compact set Let wQ be any point wQeA0, then

¥o€go' go€ro' go comPact-For ©very wegQ we have weA, and A is open in K; hence there ex­ ists an open circle c of center w such that cK C A. By the Borel covering theorem there is a finite collection {c) of open circles c covering g0; hence, if H = z c, H is open and bounded, g0 C H, gQ C K, gQ C HK and HK is open In K. In addition HK is compact and HK C HK C A. Let F = HK(K - HK), thus F is the boundary of HK in K [G. T. "Whyburn, p. 16], F is closed, F C K, F is compact. Let us prove that F gQ = 0 . In­ deed suppose FgQ f 0 , and let w be any point

§ 14.

LOCAL PROPERTIES OP PLANE MAPPINGS (l4 . 2 )

weF gQ, then w is interior to a circle c€{c); hence there is another open circle c* of center w, radius p > 0 and c! C c. All points of c! belong to H, hence c!K C HK, and all points of K - KH are at a distance > p from w. In consequence c!P = o, w is not on F, a contradiction. This proves that gQ P = 0 ; hence {gQ, P) > 0 . Let us consider now the compact set M C A, on which T is defined, hence (T, HK) is a c. mapping. In add­ ition, since IK is obtained by finitely many sums and intersections of closed Jordan regions, EK is locally connected. Since (gQ, F} > o we have also (1 0 .7 , vi){gQ, F)t > 0 . If 2 d = (gQ, P)T and V is the open set of all W€HKQ such that {w, gQ)T < d, the set V is open in HK (10.7, iv), VF = 0 , V } g0> an& V is a sum of maximal continua of constancy 7 of T on HK. Obviously 7F = 0 for every 7 of V.In addition, because of VP = 0, V is also open in K. Each continuum 7 is contained in an element g€T; i.e., 7 C g, and g is a gen­ eralized continuum (1 4 .1 , i). Also g(HK) is compact, HK is open in K and 7 C g(HK); i.e., g(HK)+ 0 . Therefore [G. T. "Whyburn, p. 16, (10.1)] either g(HK) = g and hence g(HK) = g = 7; or g(HK) + g and every component g 1 of g(HK) intersects P. Since one of these components g ! contains 7, 7 C g ?, g*F + 0, and T Is constant on 7 as well as on g 1, we have 7 = g !, hence 7F + 0, a contradiction. This proves that the second alternative above is false; hence g = 7 for all 7 of V, V is a set open in K contained in Aq and finally wQ€V, V C AQ . Thus we have proved that Aq is open in K and hence in A. (i)

The

set

A1is open.

PROOF. Let wQ be any point wQeA1. Then wQ€g0, goer^ gQ is compact and there exists an open set U, gQ C U C A. Then for every point w€g there exists an open circle c of center w such that cT C U. There exists then a finite collection {c) of these open circles covering g; hence if H = zc, H is open and bounded, H Is compact and

221

222

CHAPTER IV.

BV AND AC PLANE MAPPINGS

gQ C H C H C U C A. Let F = H*; thus F is closed and bounded, and, as in the previous proof, gQP = o and {g, P) > 0. Then H is connected and locally connected and we can consider the c. mapping (T, H). Then {g, F)T = 2d > 0 and, if V is the set of all points w such that (g> F)t < d, then V is open, g C V C H, VH* = 0, andV is a sum of continua of con­ stancy y for T on H. As above each y is contained in an element ger, y C g, and we shall consider that component g f of gH which contains 7. Then since H is open, gff is compact,

7 C g C H, and gH + 0. By [G. T. 'Whyburn, p. 16, 10.1] we have either gH = g and gH = g = 7; or

gH + g and g !F + 0, g» = 7. In the second case we have 7H* + 0, a contradiction. Hence g = 7, for every 7 of V, V is open, V C A 1, and wQeV; i.e., A 1 is open.

*14.3.

The Collection

r2

We denote by r2 the collection of all ger., which have the following property: for any open set U, g C U C A, there exists a simple pol. reg. it suchthat g C jt°, jiC U, 0 (p; C) + 0, where p = T(g) and C: (T, **). Let r2, r2 be the subcollections of r2 (not necessarily disjoint) of all g which have the above property with 0 (p; C) > 0, or 0 (p; C) < 0. Thus r2 + r2 = r2- Let A2, A2, A2 be the sets of points weA covered by the collections r2, r2, r“. As it follows from the definitions we have r ) r Q 3 ^ )r,2 and A j AQ ) A1 ^ A 2(I) The sets Ag, A * , A“ are B-measurable. PROOF. Let us denote by r2n the subcollection of all ger9 which have the following property: In the — 1 open set U = U(g, n“ ) of all points weA such that {w, g) < n”1 there is a simple pol. reg. * such that g C *°, * C U C A, o(p; C) + 0 where C: (T, it*). Then, because of (8.3, i), the set A 2 n ^ e r e d by r2n is open and A 2 n ) A ^ n+1.. On the other hand, A z = lim (A2n A ^ n+1 A2>n+2 ... ) as n -- > 00. Therefore A2 is B-measurable and an analogous proof holds for A2 and A”. Thereby (i) is proved.

§14.

LOCAL PROPERTIES OP PLANE MAPPINGS (14.3 )

223

Let Pn be any sequence of closed figures Pn C A, Pn C Fn+1, f A0 . Then, for each c. mapping (T, F ), where Fn is compact, we have already defined a countable set Dn = D(T, Fn ) of points P^E^ (10.5). Let D = D(T, A) = D(T, F 1 ) + D(T, Fg ) + ... . Then the set D(T, A) C E^ is countable.

NOTE. The set D does not depend upon the particular sequence for the definition of D, (10.5).

(ii)

F

For each point peE£ - D and each simple pol. reg. * C A such that 0 (p; C ) + 0 [> 0, < 0] , where C: (T, jt*), there exists atleast one element g C rg, g c jt°, with T(g) = p.

PROOF. By (8.3, i) we can replace jt by a new simple pol. reg. contained In j t ° , hence in A0. Let us denote It still by j t , and let n be any integer such that jt C F° for all n > n. Let H = T~1(p), H = (p), and apply (1 1 .3 , Iv) to the pol. reg. jt with = 2“1 . If S = S1 + S !1is the subdivision of jt into simple pol. regions qeS1 and non-simple pol. reg. R = (r , r-, ..., r )€SM , — 1 —1 then we have T)(q) < 2 , r](R) < 2 H(z q* + z R*) = j t * H = 0, where j t * h = 0 since 0 (p; C) + 0 and p Is not on [c]- By (11.3,iv) we know that the set T(r*) [for every r = rQ, r^, ..., r^ of R, ReSlf] is contained in a circle c not containing p. Let us drop from S all parts completely contained in regions Pq ^R, R€Sm (11.3, iv). Then we obtain a subdivision S = S^ + S^ 1 of jt into simple pol. regions qeS^ (S’ C S') and In simple pol. regions r = rQ€S^! (rQ€R, R€S"). Ifc»: (T,q*), C tf: (T, r*) for every Q€S^ and reS^' , then each C 11 is contained in a circle c non­ containing p and 0 (p; C !t) = 0. On the other hand, jt * h = 0, (z! q* + zt! r*)H = 0, and, by (8 .6 , i), o(p; c) = z» o(p; c«) + z" o (p ; c " ) , where z!, zM are extended over all qeS1, reS^!. Finally 0 o(p; C) = z 1 o(p; C!), and there must be, therefore, at least one region q ^ S 1 such that 0 (p; C 1 ) + 0 , C,: (T, q*), and q, C «, ^(q^ < 2 _1 .

used

22k

CHAPTER IV.

BV AND AC PLANE MAPPINGS

By repeating this procedure for the region q 1 and -2 e = 2 , we obtain another simple pol. reg. qP with q 2 C q.,, 0 (p; Cg) + o where C2: (T, q2) and n (QL2 ^ < 2""2, By indefinite repetition of the same procedure we obtain a sequence [q ] of simple pol. regions qn C *, qn+1 C qn, r)(qn ) < 2 -n, 0 (p; cn )+ 0 , where Cn : (T, q ). Then the sequence [qn J has a limit continuum g and g C it, g C gn for all n. On the other hand qR h + o for all n where all sets qn, H are compact and qn+i C qR; hence gM + o. By tj(qn ) < 2 ”n we ob­ tain T](g) = 0 ; i.e., T is constant on g and therefore, T(g) = p and g is contained in a * component gQ of H . Since gn qn = o, gn C q°, and p is not in we have also gQ C g. Therefore g= g Q; i.e., g is a component of H , g is compact, ger, B £r0 and, since g C q°, also ger^. IfU is anyopen set such that g C U C A, there is an n such that g C q°, gn C U, o(p; Cn ) + o for all n > n. Hence ger2* This proof holds also for r* and r~. Thereby (ii) is proved.

*1 4 .4 . Further Properties of the Elements (i)

For every element connected.

the open set

Ser2

E 2 - g is

PROOF. Let U be any open set such that g C U C A; then there exists, by definition of r2, a simple pol. reg. it such that g C it°, jt C U. Suppose that E 2 - g is not connected. Then E2 - g is the sum of more than one component. One of these components, say 7 Q, mustcontain entirely E 2 - * as well as all points of **; another component, say 7 ^ , must be, therefore, inside it. Thus 7 1 is bounded and separated from 7 Q by g in E2* Let c be any closed circle, c C 7^9 hence, also the set V = U - c is open and g C V. By definition of r2 there must be another simple pol. reg. it* such that g C *,c>, it1 C V. Since c is not in V, c must be exterior to it* while g is interior to it1. Hence there exists a simple arc 1 , not crossing g,

§U.

LOCAL PROPERTIES OF PLANE MAPPINGS ('ik.k)

exterior to it*, joining c with some points ex­ terior to it, hence in y Q, a contradiction, since c C y ^. Thus E2 - g is connected and (i) is proved. (ii)

The elements g£r2 which satisfy both conditions: (a) p = T(g)€E» - D; (b) there exists an open set U, g C U C A, such that g ’U = 0 for all g'cr2, g' * g, T(g') = T(g) = p; have the following further properties: (c) for every simple pol. reg. it such that g C it°, it1 C U, it*M = 0, where M = T"1 (p) we have 0 (p; C ) + o where C1: (T, i^ ); (d) the value of 0(p; C^ ) does not depend upon the choice of jt1 as in (c).

PROOF. By the definition of r2, there is another simple pol. reg. it2 such that g C it°, *2 C U, 0 (p; C ) + 0 where C2: (T, it*). Then p is not in [C2 ] and hence it*M = 0. Since g C it°, g Cjt2, the set it^ + *2 is connected. We can suppose (8.3, I) that both 1^, and it2 are in A0; hence it. + * C A0 . There existsthen a figure o F, it + it2 C F , F C A , where for F we can take one of the figures Fn of (1^.3 )• Then 1^ + it2 is contained in a unique polygonal region Q of F, g C «1 + it2 C Q°. By (11 .3, iv) with H = (p), H = M = T~1(p), there is a subdivision S = S T + S !t of Q, into simple pol. reg. qeS! and not simple pol. regions R€SM , R = (rQ, r^ ..., r^), such that both it and it* are sums of regions qeST -X* and ReS’1. Let us observe that (it + it2)M = 0 , where M = T~ 1 (p) and that, by (11.3, iv) (zrq* + ZM R*)M = 0, where the sums are extended over all qeS!, ReS11, q, R C it1 + it2, respective­ ly. Again by (11 -3 > iv) each curve C M : (T, r*) (where r is any of theregions r = rQ, r^, ..., r^ relative to ReS11) is contained in a circle c not containing p; hence 0 (p; C IT) = 0. Let us consider it2 and let us drop from S all parts which are in­ terior to regions r^ ^ o relative to some ReSM and which are in it2. Weobtain a subdivision

225

226

CHAPTER IV.

BV AND AC PLANE MAPPINGS

S2 = S^ + S^ 1 of * 2 into simple pol. regions qeS^ (S£ C S 1) and simple pol. reg. r = rQ€S^ 1 (rQ€R, ReSM ). By (8 .6 , i) we have 0 \ 0 (p; c2) = E^ o(p; C f) + z^! o(p; C ff), where C!: (T, q*), C M : (T, r*) and E^' are extend­ ed over all qeS^, reS^!.Thus 0 + 0 (p; C2) = E^ 0 (p; C !) and we have 0 (p; C f) + 0 for at least one of the pol. regions qeS^. Because of (1 4 .3 , ii) and (b) this is possible only for one of the regions Q.eS^, say q = qQ, the one which contains g. Finally 0 + 0 (p; C2) = 0 (p; CQ ) where CQ :(T, ). Let us consider now and let us drop from S all parts which are interior to regions rQ relative to some ReS!! and which arein 3^. We obtain a sub­ division S 1 = Sj + Sj 1 of into simple pol. reg. qeSj (Sj C S 1) and simple pol. reg. r = rQeSj1 (rQeR, ReS* »). Let us prove that qQ cannot be inside of some region reSj r•Suppose the contrary is true. Then there exists a region r 1 relative to some ReSff and a subdivision s 0 = So + So * °^ rQ - r 1 into simple pol. regions qeS^ (S^ C S !) and simple pol. regions reS^ 1 all relative to some ReSM , while qQ is a region q = q • By (8 .6 , i) we have 0 = 0 (p; C ) = E^ 0 (p;C 1) + E^ 1 0 (p; C fl) where C: (T, r1),and where the sums are extended over allQ^S^, reS^1, respectively. In conse­ quence 0 = E^ 0 (p;C 1) where the term 0 (p; C !) = 0 (p; CQ ) is + 0 . Necessarily another term at leastIn the sum £! must be + 0 and this, because of (b) and (1 4 .3 , ii) is impossible. This proves that qQ is not inside of any region reSj1 and hence qQ is one of the regions q0 GS]. By (8 .6 , i) we have again 0 (p; C1) = £T 0 (p; Cj ) + £M 0 (p; C'), where E ! , e ! ! are now extended over all qeSj, rQeSjf, and also 0 (p; C1) = E T 0 (p; C f)‘ No term 0 (p; C]) besides 0 (p; CQ ) can be differ­ ent from zero (because of (b) and (1 4 .3 , ii)); there­ fore 0 (p; C1) = 0 (p; CQ ) and finally 0 (p; C1) = 0 (p; C ) + 0 . Thus(c) and (d) areproved. Thereby (ii) is proved. (iii)

Each point peE^ - D where N(p; T, A) < + 00 has the following properties:

§14.

LOCAL PROPERTIES OP PLANE MAPPINGS (14.4) (a)

(b)

(c)

(d)

(e)

the collection {g)p = {g1, g2, . gk ) of all g^r2 such that T(g) = p has k elements 0 < k < N(p; T, A) and k = 0 if and only if N(p; T, A) = 0 ; there exists a system of k non-over­ lapping simple pol. reg. jc^, i = 1 , 2 , ..., k, such that g^ C x ± C A; for every simple pol. reg. q such that C q°, q C q*M = 0 where M = T~ 1 (p), we have 0 (p; C) = + 0 , where C: (T, q*) and does not depend upon q C i = 1 , 2 , •••> k; |^| + |d 2 I + ... + |iDk | = N(p; T, A), z +x>^ = N+(p), = N”(p), where z + , tT are extended over all > o and < 0 , respectively; for every simple pol. reg. itC A such that gj_(A - it°) + 0, i = i, 2, ..., k, we have 0 (p; C) = 0 where C: (T, it*).

PROOF. Not knowing yet whether the collection £g)p is a finite (not even countable) set o < k < + oo. Let M be any (finite) integer, o < M < k. Let g-j* g2> •••; gj^ be any subcollection of M distinct continua of (g}p- Hence they are disjoint and also g± CA°, g±A* = 0 , i = 1 , 2 , M. Let 36 be the minimal mutual distance of the closed disjoint sets g ^ gM, A* (all but at most the last one compact); hence 5 > 0 . Let be the set of all weE2 such that {w, g^3 < 5 ; thus U^, 1 = 1 , 2 , ..., M, are open disjoint sets and g^ C C A. By the definition of rg, for every i = 1 , 2 ,..., M, there exists a simple closed pol. reg. :ri, g± C C U, 0 (p; C^) + o, where C^: (T, itj). Hence theregions are also dis­ joint. In addition, if = 0 (p; c^), also z | | < N(p; T, A) < + oo where|v>1 l > 1 for every i. In consequence M < N(p) < °° where M is any integer 0 < M < k. This implies that ° < k < N(p) < + 00, Cg^p is finite and the first part of (a) is proved.

227

228

CHAPTER IV.

BV AMD AC PLANE MAPPINGS

We can now suppose that in the previous reasoning we have M = k and then also (b) is proved, and since in each i = 1, 2, ..., k, there is one and only one element E e r 2 T ^s) = P> (ii) we have (c). By the definition of N(p) and because of N(p) < + °°, there is a finite collection CQ) of non-overlapping simple polygonal regions Q C A such that (a) N(p) = Z* |0 (p; C”)|, where C: (T, Q*)and Z* is extended over all Qe{Q). We can suppose 0 (p; £") + 0 for all Qe{Q,}, hence Q*H = 0 where H = T_1 (p) for all Q,e{Q.}. By (U.3, 1 1 ) there must be at least onege{g}p inside of each Q,and, on the other hand, by (or), no one ge(g)p can be out­ side the regions Q. Let i = 1 , 2 , ..., k, be k simple closed pol. reg. such that g^ C *|°, *i ^ *i' Q° where Q is that region Q,e{Q) containing g^, and suppose *J[*H = 0 . Then, by (1 ^.^, ii), 0 (p; C.p = + o where C|: (T, Jt|*). By (1 1 -3 , iv) there is a finite subdivision S = S ! + S 1 1 of each Q into simple pol. reg. q ’eS1 and not simple pol. reg. R = (rQ, V y ..., r^) eS11, where all regions are regions q f, satisfying the other conditions stated in (1 1 .3 , iv). By the same reasoning often used above we have N(p) = z* 1 0 (p; C)| = z± 1 0 (p; C± )| = z± |^± |, where the last sum is extended over all i = 1 , 2, ..., k. Thus (d) is proved as well as the second part of (a). The same reasoning holds for N+ and N“. Let jc be now any simple closed pol. reg. Jt C A with gi(A - it0 ) +0 , i = 1 , 2 , ..., k. If + 0 for some i, then pe[C] where C: (T, **) and hence 0 (p; C) = 0 . If g ^ * +0 , i = 1 , 2,..., k, then C A - it, i = 1 , 2 , ..., k. Let i = 1 , 2, ...,k, be k simplenon-overlapping pol. reg. such that g^ C it^,*H = 0 , C A -*, i = 1 , 2, ..., k, hence |0 (p;0 ^) 1 = + 0 , where C^: (T, *£) and Z |0 (p;C^)! = N(p),while Z 10 (p; C± )| + 10 (p; C)| < N(p). Then 0 (p; C) = 0 and (e) is proved. Thereby (iii) is proved.

§14.

LOCAL PROPERTIES OP PLANE MAPPINGS (14.5)

*14.5.

The Collections

and

Let us denote by the collection of all elements geP2 ^bich are single points. Analogous definitions hold for r*, r“ by requiring ger*, g£r“ to be simple points. Let us denote by the subcollection of all elements which have the following property: there exists an open set U, g C U C A, such that g ‘U = 0 for all g 'e r ^ , g ! 4 g, T(g!) =T(g). Analogous definitions hold for r^, r£. Let us denote by A^, A*, A”, A^, A^, k^ the sets of points covered by the elements ger3> Since all elements g€r3> •••> ri^ are single points there is no practical difference between and k y . and A^; only for the sake of analogy with the previous paragraphs we conserve the distinction. (!) The sets k^> A*, A~ are B-measurable. PROOF. Let us denote by A^n the set of all points weA which have the following property: the open circle U = U(w, n~ 1) of center w and radius n ~ 1 contains a simple pol. reg. ir suchthat we*0, j( C U C A, o(p; C) 4 0 where C: (T, it*). Ob­ viously A^n is open and since A^ = lim (A^n, A^ n+1 .••) as n -- ► «>, the set A^ is measurable. The same proof holds for A y A”. (ii) The sets

A^, A^, A^

are B-measurable.

PROOF. For every integer n let us consider the collection of all closed squares q C A, q = [a. < u < ai+1, a. < v < a. ] where aj_ = I 2“n + 3 ”n, bj = j 2 “n + 3 ~n, i, j = 0, + 1 , + 2 , ... .Let us observe first that the equality i 2 ~n + 3 ~n = i ! 2 “nt + 3 ~nI implies i = i f, n = n f. This is obvious if n = n f. If n 4 n 1, say n < n f, then the equality above implies (i1 - 2 hi) 3 n+h = (3 h - i)2 n, where h = n' - n and the prime factor 3 would be contained at least n + h times in the integer at the left and less than h times in the integer at the right, a contradiction. Thus the equality above implies n = n r, i = i f. As a consequence the numbers a^ (for equal as well as for unequal integers n) are all different, and the same happens for the numbers b.. As a further J

22 9

230

CHAPTER IV.

BV AND AC PLANE MAPPINGS

consequence, for every point weA° and n, w may belong to more than one (adjacent) square qe{q)n for at most one value of n; hence for every weA° there exists an index nQ such that w is interior to one (and only one) square qn€fq^n for all n > nQ . For every square qe{q)n let us consider the mapping (T, q) and the corresponding function t(p; T, q), peE^, defined in (12.3, Note 3)- The function \|r Is B-measurable in E^; hence the set L = L(q, n) of all points peEJ? where t(p; T, q) = 1 is B-measurable and, therefore, also the set L - L(q, n) = q° T ” 1 (L), L C q°, is B-measurable (see 16.2). Hence also the set A^ = z L(q, n) where z is extended over all qe{q)n is B-measurable, is B-measurable. and finally A* = 1 1 m A^n as n -- »00 Suppose w \[r(p; T, qn ) is the only hence weA*

be a point weA^. Then we have = 1 for all n large enough where q€{q)n withweq° and p = T(w); and A^ C A ’.

qa

Suppose w be a point w€Af. Then, for all n large enough, weq°, qn€{q}n> we i(qn, n). In each neighborhood U of w there exists a square qn, weqn and in qn there is a simple pol. reg. jt such that o(p; C) ± 0, C: (T, jt*), p = T(w), where we can suppose it C q°. In consequence of 0 (p; C ) + 0 we have **M = 0 where M = T~1 (p), hence w is not on Jt*. Let us prove that wejt0 . Suppose Indeed w outside Jt. Then there would be a neighborhood V of w (disjoint with J t ) , weV, V C q°, Vit = 0. Then there would also be a square c^, m > n, such that weq^, C V, and in (J j j a simple pol. reg. «'such that 0 (p; C 1) + o where C !: (T, j t ! * ) . Now both J t , J t fare disjoint and contained in q°,and hence \|r(p; T, qn ) > 2, a contradiction. This implies that wejt°. Thus we have proved that in each neighborhood U of w there is a simple pol. reg. k such that wejt°, n C U, 0 (p; C) 4s °> where C: (T, it*). On the other hand, if we take U = q° for some n large enough, there cannot be in qn other elements g'er^, g 1 + g, T(g1 ) = T(g) by just repeating the reasoning above. This proves that wer^ and hence A 1 C A^, while

§u.

LOCAL PROPERTIES OP PLANE MAPPINGS

A^ C A !. We have proved that A^ is B-measurable.

*1k . 6 .

A ! = A^

The Local Index

(1

k.6)

and that

j(w)

For every point weA^, by the definition of = A^, there is a neighborhood U C A such that w = g C U and such that g !U = 0 for all g fer2 with g 1 + g, T(gf) = T(g). Let p = T(w) and M = T""1(p). Let D = D(T, A) be the countable set, D C E^, defined in (10.5, Note 2) and used in 12.5, proof of (i), and 1^.3. The following statement holds: (i) For every weA^ such that p = T(w)eE£ - D and for any simple pol. reg. n such that we*0, k C U, ir*M = 0, we have 0 (p; C) + 0, C: (T, **), and the value of 0(p; C) does not depend upon the choice of n . This statement is an immediate consequence of (1^.^, ii). For every weA^ such that p = T(w)eE^ - D, let j(w) = 0 (p; C) where 0 (p; C) is defined as in (i) above. For every we (A - A^) + T“1(D) let j(w) = 0. Then the function j(w), weA, (local index), is defined everywhere in A and is every­ where finite j ^ 0. (ii) The function j(w) is B-measurable in A. PROOF. Obviously j(w) + 0 in A^ - A, where A = T"1(D), j(w) = 0 in (A - Ak) +A , where A is a countable sum of sets T” (p), peD, and hence A is B-measurable as well as A and A^ (1 4 .5 , ii). Let tQ(w) be the characteristic function of a . We will make use now of the nota­ tions of the proof of (1^.5, ii). For every square q.€(qJn set $(w; q, n) = N(p; T, q) if we L (q, n), p = T(w); $(w; q, n) = 0 if weA - L (q, n). Each of the maximal subsets of L(q, n) where N = m is constant, is B-measurable and, therefore, also T""1(K^) L (q, n) is B-measurable and there has the constant value m. Therefore all functions $ are B-measurable. Set j!(w) = [1 - tQ (w)] lim z 0, j(w) < 0. Therefore 0 < * d ,-o< ^ + 00 for every peEl and, as it is immediate, also 0 < x> < N, 0 < x>-+• < N+, ° < \T < N". (i) For every set A' C A we have o(p; T, A' ) = ■D+(p; T, A 1) + *T(p; T, A 1) for all p; for any two disjoint sets A ], A2 C A we have u(p; T, A 1 + Ag) = u(p; T, A 1) +t>(p; T, Ag) and analogous identities hold for d+ and —

This statement is an immediate consequence of the definitions.

NOTE 1. If it is a simple pol. reg. it C A and [it^, i = 1, 2, ..., n] is any finitesubdivision of * into simple pol. reg. it^, then u(p; T, it0 ) > z^ v>(p; T, it£) and analogously for u“. Indeed if K is the set K = Z T(jtj) = T(z itj), then the sign = holds for all peE^ - K, the sign > for all peK.

(ii) If(T, A) is BV and M, M ^ n = 1, 2, ..., is a sequence of sets such that M, Mn C A, Mr -- ► M as n --- then •o(p; T, MR ) ---►^(p; T, M) a.e. in E* . PROOF. Let H be the set of all peE^ where N(p; T, A)= + oo; and letD = D(T, A) be the countable set used in (14 .3 )- Then |H + D| = 0. For every point peE^ - (H + D), the set I of all points weA such that weT”1(p), weA^, is finite (i^A, iii; 1^.5). Hence also the set IM is finite and IMn -- ► IM. There exists, therefore, an in­ dex nQ such that = IM for all n > nQ. Hence u(p; T, MR ) = “ o(p; T, M) for all n > nQ. This proves that *o(p; T, Mn ) -- *-u(p; T, M)

§U.

LOCAL PROPERTIES OF PLANE MAPPINGS (1 ^-8 )

for all peE| - (H + D), by (ii) is proved. (iii)

where

|H + D| =

0

. There­

The functions u(p; T, A ) , u+(p; T, A ), ■o”(p; T, A) are B-measurable in E^.

PROOF. Let ^Q(p) "be the characteristic function of the countable set D = D(T, A), hence +0 (p) is B-measurable. Let us use the notations of (1 ^-5 , ii) and (1 ^.6 , ii). For each n and for each square qe{q)n let ¥(p; q, n) be the characteristic func­ tion of the set L(q, n) and set •of(p) =

[1

- t0 (p)] lim Z *(p;

q, n) N(p; T,

q)>

where Z is extended over all qe(q)n, q C A, and lim is taken as n ----- too.By (1 ^-5 , ii) the sets L(q, n) are B-measurable; hence all functions ¥(p; q, n) are B-measurable and finally the same holds for uT(p). As in ii) and (1 ^.6 , ii) it is immediate that of(p) = 'o(p) for all peE^. Thus u(p) is B-measurable and (iii) is proved.

NOTE 2 . Thestatement (iii) holds also for any admissible sub­ set A 1 C A; in particular for every open set A ! C A, or any open or closed pol. reg. * C A.

(iv)

If (T, A) is BV, then for every B-measurable set M C A the functions u(p; T, M), u+(p; T, M), *o“(p; T, M) are B-measurable and L-integrable.

PROOF. Because of 0 < x> < N and the analogous re­ lations for v + , xT it is enough to prove that v , t>+, t>“ are B-measurable.This measurabilityIs already proved for closed and open squares (iii and Note 2). By (i) and (ii) this measurability is proved for all B-measurable sets M C A .

*1 ^.8 . The Function

q?(M)

Suppose (T, A) is BV in A and let M C A be any Bmeasurable subset of A. Then the functions u(p; T, M), ■o+(p; T, M), xf (p; T, M) are L-integrable in E^

233

234

CHAPTER IV.

BV AND AC PLANE MAPPINGS

(l^-7 > iv) and hence the functions cp(M) = (E^) / *o(p; T, M), cp+(M) = (E^) I v+, 0 as n -- > «> (1 2 .6 ), we have (E^) / |N(p; T, A) - z N(p; T, q)| dp --- > 0 as n -- where z is extended over all

PROOF. The function N(p; T, A) is L-integrable in E^. By (1 2 .4 , ii) we know that W(Fn, T) -- >W(A, T) as n — >00 and by (i) above, T) = z V(q, T); also V(Fn, T) = W(Fn, T), V(q, T) = W(q, T) (1 2 .8 , ii). Therefore z W(q, T) --»W(A, T) as n --------- >«>,i.e., ( E p f Z N(p; T, q) -- » ( E p f N(p; T, A) as n — or (E* ) / [N(p; T, A) - z N(p; T, q)] -- > 0 as n -- > 00. Since the expression in brackets is non­ negative this result implies (ii).

* 1 4 .1 0 . Bibliographical Notes The concepts introduced in this section are due to T. Rado [2 0 ], Most of the proofs are new and based on the methods of §§1 2 ^nd 13. The proofs of (1 4 .5, ii), (1b .7, iii) are very close to Rado!s arguments. For the properties (1 4 .4 ) see [L. Cesari, 9 ] , where the same properties were obtained in another form and used for the proof of the first theorem. We shall use them in this book (§18) for the same purpose.

236

CHAPTER IV.

BV AND AC PLANE MAPPINGS

*§ 1 5 • A CHARACTERIZATION OP *15

AC

PLANE MAPPINGS

•1 • Some Lemmas

We shall give a characterization of AC plane mappings in (15.2) as a consequence of the following lemmas i - v. Let (T, A) be any c. mapping from an admissible set A of the w-plane Eg, w = (u, v), into the p-plane E^, p = (x, y), and let H C E^ be the set of all peE^ where N (p; T, A) = + °°. (i) Por every c. plane mapping (T, A) we have T(A^ - A^) C H hence, if (T, A) is BV, we have | T - A^)| = |H| = 0 . PROO^. Let wQ be any point w0 €j^3 “ \ Then wQeA^ and there is an open set U such that wQ = gQ C U C A. Let V(wQ, p ), or V(p), denote any circle of center wQ and radius p. If pQ is any number such that V(pq ) C U let V(p) be any other circle with p < pQ . Since wQeA^ - A^_, in each circle V(p) there is at least one point wegf, g fer2, T(gf) = T(g0 ) = pQ . Let 8 (p) > 0 denote the Sup of the diameters of all g Ter2, such that T(gf) = p, g f V(p) + 0. Obviously 8 (p ) > 6 (p ») for all p!< p 3 5 0 > 0 as p --- > 0 +, where is, therefore, well deter­ mined. Let us prove that 5 0 = °* Suppose indeed > 0. Then there would be a sequence [wn] of points wR and a sequence [gn] of continua Sn e r 2 such that wnegn, wn -- * wQ, gn V*(&Q ) * TCg^) = pQ. Thence there is a component continuum

0,

sn of sn V such that §n V*(8Q ) * 0, sn C V < V > wnesn ‘ In acii:iltlon> if 1 = 11m Inf L = lim sup g^, we have wQe l , 1 + 0, 1 C L where L is a continuum (1 0 .2 , i), wQeL and, if w^ is any point of g^ V*(SQ ) and w ! any point of accumulation of > then w feL. Because T is w 1 eg; constant on L, we have Leg, ger, wQeg, hence g = gQ and gQ + w , a contradiction. This proves that 5 0 = °‘ consecluerice there must be at least one continuum E , ^ vz > g-, + B> T(g-, ) = T (g0 ) = PQ> g1 C V ° ( p 0 ) and because of gQ g1 = 0 we have 2P1 = (g0, g1) > 0; hence V(p 1) g1 = 0. By repeating this procedure any finite number m oftimes, we determine a finite sequence p0 > p-j > p2 > '*'> Pm -1 >

0

§15*

237

A CHARACTERIZATION OF AC PLANE MAPPINGS (1 5 •1 )

and ra continua gj_€r2' = P0> ^ = 1 > 2 > ‘**> m> such that C V°(P-j_—-j) - V(pi ) = W^, 1 = 1 , 2 , ..., m-‘ gm C V°^pm-i^ = ¥m ; that is' the sets gi' 1= 1' **’' m are contained in m disjoint open sets WiCA. By definition of rg, for each i, ‘there is a simple pol. reg. such that gj_ C *?, C W^, 0 (po; C^) + 0 where C^: (T, n t ) and the simple pol. regions C A, i = 1 , ..., m, are disjoint. Therefore N(pQ; T, A) > Z |0 (Po> Ci^l ^ m' where z ranges overall i = 1 , 2 , ..., m. Since m is a n y finite integer we have N(pQ; T, A) = + oo, P0€H, and this proves that T(A^ - A^) CH. Thereby (i) is proved. (ii)

For every BV c. plane mapping (T,A)satisfy­ ing condition (b) of (1 3 .1 ) we have |T(A2 and N(p; T, A) = u(p; T, A) a.e. in E^.

—

)|=0

PROOF. Let [Fn ] be any sequence of figures Fn C A, Fn C Fn + 1 , F° t A0, and [Sn ] be any sequence of subdivisions Sn of Fn into simple pol. regions q as n -> ».By such that the Indices dn -- > 0 (14 .9, ii) we have(E^ ) / ^n (p) -> 0 as n -- >oo, where ^(p) denotes the difference N(p; T, A) - z N(p; T, q), and where z denotes any sum extended over all Q.€Sn * Let H be the set of all points p where N(p; T, A) = + oo, and let D = D(T, A) be the countable set used in 0 ^-3 ), D C EJ. Then |H + D| = o. For every peE^ - (H + D), by (1 ^.^,iii), the set T”1^) contains a finite number k < N < + oo of continua g1, •••, S^er2^ such that T(g^) = p and such that, if are the numbers defined in (1 ^.^, iii) we have also |*o1 | + ... + |^ |= N(p). If gl, g^, are the only g^ which are single points, and •••> % remaining ones, if a = min diam gk^' then ^or n such that dn c a, no one of the continua g^-'+l' • • • > S^ can be inside some of the squares q€{q)n (1 ^-5 , ii); hence z N(p; T, q) < | | + ... + |i>kl| = t>(p; T, A). This implies that 11m

2

n ---> oo qe{q)n

N(p; T, q) < t>(p; T) < N(p; T, A)

CHAPTER IV. a.e. In

BV AND AC PLANE MAPPINGS

E^. In consequence we have lim ^(p) > N(p) - u(p)

and, by Patou's lemma [S. Saks, I, p. 29], (EJ,)/ [N(p) - 'o(p)] < (E£) / 11m An (p) < lim (E^) /AR (p) Therefore, If K is the set where N(p) > -o(p), we have |K| = 0 . Now for every point weA2 - A^, w belongs to a continuum g € r 2 which is not a single point, hence, If p = T(w), then either peH + D, or peK. Hence T(A2 - A^ ) C H + D + K, |T(A2 - a3 )| = 0 . (iii)

If the c. plane mapping (T, A) is BV and satisfies conditions (a) and (b) of (13-1)> then for every set I C A such that |A^I| = we have |T(A^I)| = 0 .

0

PR0 0 P. Since A^ C A0, also A^I C A0 . Let e be any positive number and t)> 0 the number de­ fined by condition (a) in ( 1 3 •1 )• Let G be any set, open in A, such that A^I C G, and |G| < r\. Let [PR] be any sequence of figures Fn C G, Pn C Fn+1> t a^d observe that, since A^I C A0, we may suppose A^I C G° and Fn ^ For ©very n let Sn be a sub­ division of index dR of Fn into simple pol. regions q and suppose dR -» 0 . By (1 ^.9 , ii) we have z(E,p / N(p; T, q) -> (E') / N(p; T, G) as n -- » 00, where z is extended over all qeSn . On the other hand |G| < tj;hence z|q| < t|, Z v(q) < e and also (1 3.1 , I) Z V(q) < e, Z ¥(q) < e, i.e., z(E^) / N(p; T, q) < c. In consequence (EJj) / N(p; T, G) < €. Denote by B the set of all peE^ where N(p; T, G) > 1 . Then N = 0 for all peE^ - B and |B| < e. Let w beany point weA^I; then weA^, weA0, weG°, and in any circle c of center w, c C G , there is a simple pol. reg. it such that 0 (p; C) + 0 where p = T(w), C: (T, it*). In consequence N(p; T, G) > 1 and peB. This proves that

§15-

A CHARACTERIZATION OP AC PLANE MAPPINGS (15 •1)

T(A^I) C B; hence |T(A^I)| < |B| < e, where is any arbitrary positive number. Thus |T(Aj^I)| = o and (iii) is proved. (iv)

239 €

If the c. plane mapping (T, A) Is BV and for every set I C A with |A 2 11 = o we have also |T(A2I)| = 0, then (T, A) satisfies condition (b) of (13 •l)•

PROOP. Let H be the set of all points -p where N(p; T, A) = + oo, and let D = D(T, A) C E^ be the countable set used in (1^.3); hence |H + D| = 0. Let x be any simple pol. region jt C A and it2) be anysubdivision of n into two simple pol. regions it by means of a simple pol. line 1 C *. Then we have (12.1k ) (or) N(p; T, *) > N(p; T, ) + N(p; T, *2) for all peE2. Let B be the set of all p .where the sign > holds in (a). For every peE^ - (H + D) we denote by [g]p the finite set of continua g defined in (1 4 .4 , iii) and by -o = u(g), ge[g]p, the corresponding integers •o(g) + 0; hence N(p; T, it) = z |*01 , where z ranges over all getg]^ suchthat g C *°. Analogously for N(p; T, ir^), i = 1, 2. In consequence a point peB - (H + D) if and only if for some g€[g] , g C it0, we have gl + 0. Since T is constant on each g, all points peB - (H + D) are images of points wel which belong to at least one ge[g]^. That is B - (H + D) C T(Agl). Since |1 | = 0 we have |A21 | = 0, |T(A21 )| =0, |B - (H + D)| = 0, |B| = 0;that is, the equality sign in (or) holds a.e. in E^. By L-integration in E^ we have WU, T) = T) + W(*2, T) and also (12.8, ii) V(jt, T) = V(it1,T) + V(jt2,T). Thus the same equality can be proved for decompositions of * into any finite number of simple pol. regions and therefore condition (b) holds. Thereby (iv) is proved. (v)

If the c. plane mapping(T, A) is BVand for every set I C A with |Agl| = 0 we have |T(A2I)| = 0 then (T, A) satisfies condition (a) of (1 3 -0 -

2k0

CHAPTER IV.

BV AND AC PLANE MAPPINGS

PROOF. Suppose that the statement is not true; then there exists a number e > o and a sequence Sn of finite systems Sn of non-overlapping simple pol. regions qeSn such that Z |q| < 2~n, Z v(q, T) > e, where z ranges over all Q.GSn * The function N(p; T, A) is L-integrable, hence if H is the set of all points peE^ with N(p; T, A) = + oo we have |H| = o. Let D = D(T, A) be the countable set used in (14 .3 ). Let a > 0 be a number such that (h) / *N(p; T, A) < e for every measurable set h C E^, |h| < a. Finally let Q be a square Q C E^ large enough in order that (E| - Q) /N(p) < 2“1a. If B C E^ is the set of all points p whereN(p) + 0 [hence N(p) > 1] we have |B - Q,| < (B - Q) / N(p) < < (E| - Q) / N(p)
a. Indeed, suppose |Mn l < a then e < z v(q, T) = (Ep / z |0 (p; C )| < (Mn ) f N(p) < e, a contra­ diction. This proves that |Mn l > a and hence a < IM^I < |B| , I^QI = IMJ a - 2 _ 1 cr = 2 The sets M^,

|Mh (E’ - Q) | >

n = 1, 2 , ..., are all contained in the compact set Q, (of finite measure) and |MnQ,| > 2_1a > 0. If M = lim Mn by [H. Hahn and A. Rosenthal, I, p. 22, (3 *5 , 1 1 )] we have |M| = |lim M^l > |iim MnQ| > lim |MnQ| > 2~1 cr. In consequence M is non-vacuous. For each point peM -(H + D) the collection all elements ger2, T(g) = p, is finite and, if k is the number of its elements, we have k < N(p). On the other hand, peM^ for infinitely many n; hence at least one of the elements ge[g] must belong to a region q , Q.€Sn - Since [glp is a finite collection, there must be at leastone g€[g] Q which belongs to some q , for infinitely many n. If we set I = lim Gn, Gn = Z q°, then each point peM - (H + D) is the image of at least one element g C I, ge[g]p- This proves that

§15-

A CHARACTERIZATION OF AC PLANE MAPPINGS (1 5 *3 )

2 *H

M - (H + D) C T(A2 I) and, as a first consequence, IA2 is non-vacuous. On the other hand, |Gn l < 1=0.1 < 2 "n, |Gn + Gn+1 + ...| < 2 “n+1,

III = lim| Gn + Gn+1 + . . . | = 0,

|IA2 I = 0.

Thus we have defined a set I C A such that |IA2 I = 0 and such that |T(IA2 )| > |M - (H + D)| = |M| > 2~1a > 0 , a contradiction. Thus the con­ dition (a) holds and (v) is proved.

*15»2.

A Characterization of

AC

Plane Mappings

We conclude with the following THEOREM. A necessary and sufficient condition in order that the BY plane mapping (T, A) be AC is that for every set I C A with |IA0 | = o we have |T(IA2 )| = o. This is a consequence of the statements (i), ..., (v) of

( 1 5 -1

)-

NOTE. The condition that (T, A) is BV is essential. This is proved by the following example. Let cp(u) = u sin (u“ 1 ) if o < u < 1 , cp(0 ) = 0 , and let T: x = a Is B-measurable. We shall consider a function f(x) as a mapping y = f(x) from X into the y-space E- of all real numbers y. If B is any set of numbers y, _1 * by A = f (B) we shall mean the set of all numbers xeX with f(x)eB. (i) The single-valued real function y = f(x), xeX, Is B-measurable if and only if for every B-measurable set B C E1 also the set A = f”1(B), A C X, is B-measurable. PROOF. Suppose that the condition holds. Then for every real a the set B = [a < y < + a is B-measurable. Thence f(x) is B-measurable. Suppose that f(x) is B-measurable, and denote by P the collection of all sets B C E] such that A = f~1(B) is B-measurable. Then P is additive and also a a-field (10.2). By definition of B-measurability all sets of the form B = [ a < y < + °o] belong to P. Hence also all sets of the form B f = [a < y < b] belong to P and, since each open set is a countable sum of 2^2

§ 1 6 . AN ANALYTICAL PROPERTY OF CONTINUOUS MAPPINGS (1 6 .2 )

21*3

sets B l, P contains all open sets in E^. Thus P contains the smallest cr-field containing all open sets and hence the condition in (i) holds. Thereby (i) Is proved.

*16.2.

B-Measurable Mappings

Let X, Y be metric spaces and (T, X): y = T(x) any single-valued mapping from X into Y (not necessarily one-one). For every set B C Y we shall denote by A = T”1(B) the set of all pointsxeX with T(x)eB. The following statements are immediate consequences of the definitions: (i) If B, B 1, B2, ..., are given sets in Y and A = T-1(B), A± =T-1(B1 ), i = 1, 2, then B = B-j + B2, B =B^ - B2, B = B1 B2, B C B^ B = Z B^, B = n B^, B = lim B^ as i ---- >00, amply respectivelyA = A 1 + A2, A = A 1 - A2, A = A 1 A2, A C A = Z A±, A = n A± , A = lim A^ as i -- »». (ii) If A, A ^ A2, ..., are given sets in X and B = T(A), = T(A^), i = 1,2, ..., then A = A1 + A2, A =A 1 Ag, A C A = lim A± as i -- imply respectively B = B1 + Bg, B C B1 B2, B C B ^ B C lim B^ as i -- *00. The mapping (T, X) is said to be a "B-measurable mapping” from X into Y if for every B-measurable set B C Y the inverse set A = T~1(B) is B-measurable. The statement (i) of (16.1) assures that this definition of B-measurability Is coherent with the definition of B-measurability of real functions (Y = E 1). (iii) If (T, X) is B-measurable, if f(y), yeY, is any B-measurable function in Y and F(x), xeX, isdefined by F(x) = f[T(x)], then F(x) isB-measurable in X. PROOF. For every real a let BQ a be the set of all points yeY where f(y) > a and let Aa = f”1(Ba ). Then Ba is B-measurable, Aa Is B-measurable and AQ d is the set of all xeX where F(x) > a. Thus F(x) is B-measurable in X. (iv)

If X, Y,Z are metric spaces, if T 1:y = T^(x) is a B-measurable mapping from X into Y, and T2: Z =* T(y) is a B-measurable mapping from Y

2kk

CHAPTER V.

THE FIRST THEOREM

into Z, then T = T2 T 1 : z a B-measurable mapping from

=T2 [T^x)] X into Z.

is

PROOF. Indeed for every B-measurable set C C Z the set B = T^CC) is B-measurable in Y and the set A = T ^ 1 (B) = T " 1 [ T ” 1 ( C ) ] = T _ 1 ( C ) is B-measurable in A. This proves the B-measurability of T = T2 T 1 . (v)

If T: p = T (x) = [f -j(x), f2 (x), ..., fn (x)J, xeX, is any mapping from a metric space X into the real Euclidean p-space En, p = (y1, y2, -.•, yn ), then T is B-measurable if and only if all the single-valued real functions y = f 1 (x), . . . 9 y = f*n (x) are B-measurable.

PROOF. Necessity. Let B be any B-measurable set of the real axis - q; then wj, wj 1 -- > w 1, w», v ” -- > w2, and the corresponding sequences [gf], [g,!] have their lim sup, say L 1, L ,?, which are continua contained in g, g C n ° (10.2, 4 ). Con­ sequently we can suppose q f, q 11 close enough to q in such a way that both g 1, g 11 C * . Both w*, wj * belong to a circle of center w 1 and radius < 5 ^ < p hence their distance is < 2p and there exists an arc 1 1 C Q joining wj, wj1 of diameter < Sg. Hence 1 1 is in a circle c1 of

257

CHAPTER V.

THE FIRST THEOREM

center w 1 and radius < and c1 C n , 1 1 1 for all wel Analogously z(w) < z(w1 ) + 3 n there is an arc joining con2’ 2 ’ tained in a circle c 2 C it o and such that Since z(w) > z(w2 ) - 3"1 n ~ 1 for all welr = £(q) > n~-1 ' we have z(w2 ) - z(w1 ) = Z2 —

1

—

, In = O.

be the minimal mutual distance of g, g 1 let 2 t = min [x,! - x, x - x 1] > o, and 5 ^ be a number such that x(w) - x(w!)| < t for every ™ ^ 1 lu u*' Let w, w 1 |w w 1 o*tf C Q where Q, - min 111X11 LU^ [5 , U^y ^ J • Now W,V ^ g 1 1 is a closed pol. reg., and r\> ^ e g S w By (6 .1 , v) there are two simple pol. lines i !, I 11 C Q, respectively, and joining wj, w; and w ] 1 < w; whose points are all at a dis­ tance < 5 ^ from g ?, or g t!. Then I 1, l !f C it0 and x(w) < x for all welT, x(w) > x for all w g 1 m ; hence 1 * 1 *1 = o.

Let Cf 1 *

placing 1 ^ lg the two new arcs closed curve 7* Hence, since n 7 C it, where 7 boundary is 7*.

The four arcs do not form necessarily a simple closed curve, but by replacing 1 !, 1 11 by the minimal subarcs joining 1 ^ lg and by reby the minimal subarcs joining the 1 1, 1 11, we obtain a simple C 7t ! » + ! ,1 + I 1 + 1 . is a simple pol. reg., also is the simple pol. reg. whose

We have also 7* C Q, and we shall prove also that 7 C Q. Indeed, in the contrary case, at least one region A ., i = 1 , 2, ...., would be completely contained in 7 and hence in *°, what we have seen is not. If C: (T2, y * ) y then C is a closed curve of the \2, zx-plane E22, sum of four arcs x!, \ ^ } on such that z < zn + 3“ n z > z2 - 3 1 n”1

§ 1 6 . AN ANALYTICAL PROPERTY OF CONTINUOUS MAPPINGS (1 6 .9 ) on x < x on \1, x > x on X ' ' . If s “ _ _1 _1 _I _ is the segment s0 = [x = x, z 1 + 3 n < z < z2 - 3 n" of the plane E22, sQ has length sQ = 3"1 n~1, and 0 (x, z; C) = + 1 for all (x, z)esQ . Consequently *+°°

J

( y . z*7;* fM IH |0 0 (x, C )| dz*7. >

S^ n

^ >

If we consider now the collection

3“ 1

[yl

n

1

of the

N

disjoint simple pol. regions 7 C Q, we have +°° N(x, 00

_ z;

T2, Q) dz >

p+°° _ I I0 (x, r , J -00

2 _,

z;

_i _i C)| dz > 3” Nn“ ,

7€ [ 7 ]

where

N

is any arbitrary integer.

Hence

r> + 0 0

J

I

N(x,

z;

T2, Q) dz = + °° ,

— CO

for every x of the set i, and i has positive measure. Hence N(pj Tg, Q) is not L-integrable in E 22 and T 2 is not BV, a contradiction. This proves that |IQ Zn l = 0 for all n > 1 and, thereby, (i) is proved. (ii)

If either one of the two mappings T^, Tg is BV, then 13^ ZR | = o for all m = 1 , 2 , ..., n = 1 , 2, ... .

PROOF. Let T 2 be BV and suppose, if possible, that Ijji Zn = a > o for some m > 1 ,n > 1 . Then also Zn - Dm l = a > o sinceDm is countable and there is a closed set I C Im Zn - D' s 111 > 2~1cr such that the mappings w 1 (q), w 2 (q) are continuous in I, as in the proof of (i). Let us proceed now as in the proof of (i) and let [q] be the finite collection of points q = q± = (x, yi )el', i = 1 , 2, ..., N. Let [g] be the corresponding finite collection of distinct continua g e r ^ , g = g±, T(g± ) = q±, i = 1 , 2 , ..., N. Here geE^a^, bm ) -Dm and

260

CHAPTER V.

THE FIRST THEOREM

we willorder the points q = q^ and elements g = according to the order of the elements in V

g

Each continuumge[g] belongs to Em - T)m; hence E 2 - g is the sum of exactly two components, say M , M 1 and g Q* = o. Consequently g C Q°. Let 35 > o be the minimum mutual distance of the com­ pact disjoint sets g1, ..., g^, Q*. We can suppose for instance that for each g = g^, MQ is the un­ bounded and M 1 the bounded component of Eg - g. Since the elements a = am, b = bm belong to different components of Eg - g, let aeMQ, beM1. By (1 1 .1 , ii) there are two closed simple pol. regions *0, C Eg such that (1 ) b€jf^, ot1 C aeEg - *0, g C (2) it* [it*] separates a [b] from g in E2; (3) each point weM0 [M1 ] separated by it* [**] from a [b] in E2 is at a distance < 5 from g . If we denote by R the pol. reg. nQ of order 1, then g C R°, R* C Q. On the other hand, each point weR which does not belong to g must belong to MQ, or M 1 [otherwise g would separate E2 into more than two components]; hence each point of R is at a distance < 5 from g and, conse­ quently, R C Q and the N regions R are disjoint. Let 2 5 1 > o be the minimum distance of each g from R* and let Sg, p, 5 ^ be defined as in the proof of (i). For each qe[q] let us define q f, q M , w], w geg', wj1, w£!€g»«, g f, g f1 er^ 3 \ g 1, g 1 1 C R°, asin (i). Here we have q«, q-el', I ' CI C ^ Zn - Dm, g', g'UE^ - ^ m > 1 ; hence both g ! and g 1 1 separate Eg into only two components. It is not restrictive to suppose that a, g 1, g ,!, b are in this orderin a + Em (a, b) + b. Let a 1, p ! [o: 1 1, p M ]be the unbounded and bounded components of Eg - g ! [Eg - g ,!], respectively. Then b, **, g " C p', a, ** C a 1, b, it* C p11, a> S ! C q:m [Note that g may be either before

§ 1 6 . AN ANALYTICAL PROPERTY OF CONTINUOUS MAPPINGS (1 6 .9 ) op

after

g,

261

and T in the Let us define 5 k>

or between

ordering :11, b ]. as in the proof of (i).

By (1 1 .1 , ii) there are two simple pol. regions P r1 such that

( 1 ) beP'°, P' C p', P " C a "

P 1,

aeE2 - P " ;

(2 ) pi* [p»»*] separates b [a] from g* [g,T] in the w-plane E2; every point weP1 [P,!], as well as every (3 ) point we£T [w€al!] separated by P 1* [P,!*] from b [a], is at a distance < 5 ^ from g* [gtf]. Because of 5 ^ < 5 ^ we have first pi* pit* _ o anc^ also P !1 C P a., i t* g' «*,

b, it

C P1

c e2

Let us observe now that

W1eg, g c R°,

{g, R*} > 2S

W 1I < 6 3 < ,1 . 1,, , w1 Ware Therefore both w 1 , w 1 in a circle of center w o and radius < 5 and c1 C R hence the segment 3 11 = w ’ w]T is completely contained in '1 and therefore in R° and Q°. In addition -

z(w) < z(w1) + 3 1 n 1 = z1 + 3”1 n for all wel1 . Analogously the segment 12 = w^ w^ is completely contained in a circle c2 of center w2 and radius < 5 ^ < 5 ^, c2 C R°, and We have z(w) > z(w2 ) - 3”1 n“1 = z2 - 3 " 1 n" 1 - z1 = £(q) > n ; hence 1 1 1 2 = o. Now in P 1 hence the seg­ wi is in En PS y ment 11 intersects both P 1* and P I!*; let us denote still by 11 any minimum subsegment joining P !* and P f’*. Analogously there is a minimum subsegment of 12, _u2 let us denote It still by 1 2, joining P »* and P T1*. The two segments 1 ^, 1 , divide R into two simple is any one of these two, then pol. regions * If 7* is the sum of two arcs I 1, l ft of P f* and P M *, and of the two segments 1 ^ 1 2 . Since

51

262

CHAPTER V.

THE FIRST THEOREM

P' - P " CR°, we have also y CP' - P M C R° C Q°. Let [7] be the collection of the N disjoint simple pol. regions 7, y C Q, so obtained. If C: (T„, 7*), then C is the sum of four arcs X , , x , x 1, a,m such that z < z., + 3 - 1 n1 on X , -1 - 1 — — z > z2 - 3 n on Xg, x < x on x!, x > x on 1 exactly as in the proof of (i). The remain­ ing part of the proof is the same as for (i). There­ by (ii) is proved. (iii)

If at least one of the two mappings T^ and T2 is BV, then |Z| = o, i.e., £(q) = o for almost all points qeK^ and the components —1 of T^ (q) are elements ger(T, Q).

This statement is a consequence of (1) and of (i) and (ii).

16.10.

An Analytical Property of Continuous Mappings

We can now conclude the previous considerations with the following final (i)

THEOREM. If Q is the finite sum ofdisjoint closed, finitely connected Jordan regions of the w-plane Eg, w= (u, v); if (T, Q): p = T(w), weQ, is any c.mapping from Q into the p-space E3, p = (x, j , z ) , if (Tr, Q,), r = 1, 2, 3, are the plane mappings which are the projections of (T, Q) on the coordinate yz, zx, xy-planes E21> e 2 2 > E23, and Kp = Tp (Q), K,, C E2pj if at least two of the three plane mappings T , r = 1, 2, 3, are BV, then for almost all points qeK1, K2, K^ the components of the sets T”1(q), Tg1(q), T”1(q) are continua ger(T, Q), i.e., maximal continua of constancy for T on Q. PROOF. Suppose first that Q is a closed Jordan region. Then in order that property (iii) of (16.9) hold for almost all qeK1, Kg, K^ it is sufficient to know that for each of the following pairs of mappings (Tg, T3), (Ty T 1 ), (T1, Tg) at least one is BV. Hence if any two of T ^ Tg, T^ are BV, the property (iii) holds for K 1, Kg, K3. If Q, is any finite sum of closed disjoint (finitely connected) Jordan regions, we have only to add the

§ 1 6 . AN ANALYTICAL PROPERTY OF CONTINUOUS MAPPINGS (1 6 .1 1 )

263

exceptional sets of points qeK1, Kg, K^, and also the new sets have measure zero. Thereby the theorem above is proved. NOTE. If we denote by Fp Cthe subset of all points qeKp such that at least one component of T~1(q) is not an element ger(T, Q), then the previous theorem can be restated as follows: If at least two of .the three plane mappings Tp, r = 1, 2, 3 , are BV, then |F^ | = |F2 I = |F^|

16.11.

Examples

The theorem proved in 16.10 has wide application in the following. Some examples are convenient for a better understanding of the analytical mean­ ing of the theorem EXAMPLE 1. Let C: x = cp(u), y = t(u )> 0 < u < 1 > any representation of a Peano curve, i.e., of a continuous curve covering the square K = [o < x, y < 1], and cp(u), \|/(u) be not both constant on any subinterval of [0,1] [E. W. Hobson, I, Vol. 1, p. 4 5 5 ]- Let (T, Q) be the mapping T: x = cp(u), y = i|r(u), z = v, (u, v)eQ = [o < u, v y = o], [° c 2 belong same plane a normal to r, we can suppose that the homotopy is performed in an annular region of the same plane oe. Analogously, If C 1 and C2 are in a sphere cr whose center is in r, we can require C 1 — C2 (a).

(iii)

If R = Q - q° is any closed region of order * 0 = 1 , If C.j : p = p-j (w), weQ*, C2: p = p2 (w), weq*, are any two closed orient­ ed curves in E^ (i.e., any two continuous

§17-

SOME PROPERTIES OP HOMOTOPY FOR CONTINUOUS CURVES IN E3 (1 7 .2 ) mappings from the boundary curves Q* and q* of R), if r is any straight line in E^ with [C-| ]r = [C2]r = 0 , if 0 (pQ; C1Q) = 0 (pQ; C2Q), where pQ = a r is the intersection point of r with a plane a normal to r, and C2Q the curves projections of C ^ Cg on a, then there is a c. mapping (T, R): p = p(w), weR, with rT(R) = 0 ,p(w) = p 1 (w) on Q*, p = P2 (w) on q*. In addition if p 1 (w), p2 (w) are q.l. on Q* and q*, then we can require that p(w) is q.l. in R. PROOF. The first part is a consequence of (i) and (1 7 *1 , ii and Note 4 ). Let us now prove the second part. Let (T, R): p = p(w), weR, be the c. mapping already defined. If x(u, v), y(u, v), z(u, v) are the components of p(w) and we suppose that both p.j(w), p 2 (w) are q.l. on Q* and q*, respectively, then also x(u, v), y(u, v), z(u, v) are q.l. on Q* and q* (and continuous in R). By applying (5 *6 , iii) with e = 3"”1 {T(R), r} we can determine three new functions x f(u, v), y f(u, v), z !(u, v) q.l. on R, coinciding with x, y, z on Q* and q*. If p T(w) is the corresponding vector function, then (T 1, R): p = p !(w), weR, has all the required properties. (iv)

If Q is any simple pol. reg. and C: p = pQ(w), weQ, is any closed oriented curve in E^ (i.e., any c. mapping from the boundary Q* of Q), if r is any straight line in E^ with r[C] = 0, if °(p0; CQ ) = ° where pQ = ar is the inter­ section point of r with a plane a normal to r, and CQ the curve projection of C on a , then there is a c. mapping (T, Q): p = p(w), weQ, with rT(Q) = 0 , p(w) = pQ(w) on Q*. In addition if P0 (w) is q.l. on Q*, then we can require that p(w) is q.l. on Q.

This statement is a particular case of (iii).

See (1 7 -1 , Note

3

)*

271

272

CHAPTER V.

THE FIRST THEOREM

* 1 7 -3 - A Preliminary Statement (1 ) If C C E^ Is a continuous closed oriented curve, if pQ = ( x Q, j Q, z Q ) is any point of E3> r1, r2, r3 the three straight lines r, = [y = y 0 , z = zQ], r2 = [x = x0, z = zQ], r3 = [x= xQ, y = yQ] and R = r 1 + r2 + r3, if (a) [C]R = 0 andhence 2 S = {[C], R} > 0 ; (b) 0 (xo, j Q; CQ ) = 0 whereCQ is the projection of C on the xy-plane; (c) z > zQ - 5 for every point (x, y, z) of [C] [or z < z + 5 ], then C *= o(E3 - R). PROOF. We can suppose x 0 == zo = °' hence r-j, r2, r^ are the x,y,z-axes. Let C: p = p(u), ° < u < 1 , p( 0 ) = p(l). If (r, a>, z) is a cylindrical coordinate systemwhose axis isthe z-axis, then C has a representation C: r = r(u), cd = a)(u), z = z(u), o < u < l, where r(u), o>(u), z(u) are real single-valued continuous functions such that r(o) = r(l), z(o) = z(i), and, because of (b), also a)(0 ) = o>(l). Let us observe that for every point p = (x, y, z )e[C] we have 25 = {[C], R) < {[C], r ) < (p, r2) =(x 2 + z 2)2; hence |z| < 5 implies |x| > 5 . Analogouslywe can prove that |z| < 5 implies |y| > 5 . Let us denote by z(u, v), (u, v ) e Q = [o < u, v < 1 ], the function defined in (1 7 *1 , i) relative to the functions f 1 (u) = z(u), f2 (u) = 26 = constant, 0 < u < 1 . We shall now con­ sider the curve C 1: r = r(u), w = a)(u), z = 2 8 , o < u < 1, and the c. mapping T: p= p(u, v), (u, v)eQ, where p(u, v) = [r(u), a>(u), z(u, v)]. Obviously T performs a homotopy of C into C 1 . Each point [r(u), cd( u), z(u)]e[C] describes the seg­ ment s = [r = r(u), cd = cd(u), z = z(u, v)] as o < v < 1 , where s is parallel to the z-axis. Now either we have - 5 < z < 5 and then |x (u )| > 5 , |y (u)| > 5 and sR = 0 ; or we have z(u) > 5 and then, since f2 (u) = 2 5 , also z(u, v) > 5 for every ° < v < 1 (1 7 *1 , i, requirement 2 ) and sR = 0 . Thus T(Q)R = o and C ^ C 1 (E^ - R). By (1 7 *2 , i) we have 0 (0 , 0 ; CQ ) = o(o, 0 ; CM), where CQ, are the projections of C, CM on the xy-plane. On the Other

§17-

SOME PROPERTIES OF HOMOTOPY FOR CONTINUOUS CURVES IN E 3 (17 •3 ) 273 hand, C^ is contained in the plane a = [z = 2 6 ] and, more exactly, in an annular region A = t § < r < M ] Cor. Hence C ! -o(A) (1 7 - 1 ^ Note 2b). Since AR = o we have C 1- o(E^ - R) and finally, by (8.13, iii), C - 0(E3 - R).

NOTE. In (1 7 -3 , i) we can replace of the statement.

(ii)

sf26

to

26

in the part

(a)

Given any simple pol. reg. jt of the w-plane E2 and any c. mapping C: (TQ, j t * ) from jt* into E3 (c. closed curve), if C is contained in a sphere a of E3 then C - 0(a) and there is a c. mapping (T, jt ) such that T(jt) C a, T = TQ on j t * . If Tq is q.l. on j t * , we can require that T is q.l. on jt .

The first part of this statement is obvious, the second part is a consequence of (1 7 •1 , ii and Note 3 )• For the last part see (17 •1 , Note 2 ). (iii) Given any pol. reg. Q = (qQ, q1, ..., ) and a c. mapping (TQ, Q*) from Q* = q* + q* + ... + q* into E y let us suppose that the c. curves C|: (TQ, qt ) , i = 0, 1, ..., o, are homotop to zero In a connected open subset G of E3. Then there is a c. mapping (T, Q,): p = T(w), weQ, such that T(Q) C G, and T = TQ onQ*. If TQ is q.l. on Q* we can require that T is q.l. in Q. PROOF. Let Q T = (q^, qj, q^) be any pol. reg. interior to Q, such that q^ C (q{)°^ q^ C (q0 )°* Then Q, is divided into Q ’ and further x> + 1 pol. reg. of order 1, say rQ = (qQ, q£), r± = (qj_, q± ), i = 1, ..., *0. ¥e can define T in each of these regions r^ by using (1 7 -1 , iii and Note 3 ); i.e., so as to per­ form in r^ the homotopy of C^ into a single point P± = T(q|*), P±eG, i = 0, 1, ..., q. Thus T is constant on each boundary curve qj_*> i = 0, 1, ..., v>. Let us divide Q,' into x> pol. regions of order 1, say r^+i, ..., r^, by means of disjoint arcs joining any two points* of q^*- We shall define T on these arcs

CHAPTER V.

THE FIRST THEOREM so as to be constant there with value P - Finally let us define T on each region ’t>+i so as to perform the trivial homotopy of the point PQ = TCq^*) into the point P^ = T(q|*) along any arc 1 ^ C G (see 1 7 -1 , iii)- Then T is de­ fined in Q, and has all the re-

quired properties.

The Four Line Theorem and Further Statements Let C C E^ be any c. closed oriented curve, let p 0 = (xo' V zo> be any point of E. the straight lines „ = „D.. z r3 = [x = xD, r, j 0h •1 = [y = y 0 , z zo]' r2 = [x = xQ, z + r and R as in ( 1 7 3 , i ) • Then 2 + r2 ' X1 2 r 3 = p0 ■1 c the and ye shall suppose R[C] = o. We denote by c„, ^2, ^ plane c. closed oriented curves which are the projections of E, and C on the yz, zx, xy coordinate planes E,21 E 23' PQ1, pnp, ■029 pn^ -^03 the projections of the point pn on the coordinate t = 1, 2, 3. planes. Hence ot ■

In the present article we shall discuss further conditions in order that C — (E^ - R), i.e., in order that C can be re­ duced to a point in E~ without crossing the straight lines 19

'3 "

By (17-2, ii) the condition (or) °(P0t> ct ^ = °' t = 1, 2, 3, is a necessary and sufficient condition in order that C ~ o (E3 - rt ), t * 1, 2, 3- Hence C — o(E^ - R) implies t = 1, 2, 3, and finally (a). Thus (a) is o(Eq a necessary condition in order that C — °(^3 - R)- Neverthe­ less the same condition (a) is not sufficient as the following example shows. Let C be the closed c. oriented curve C = C1 + c2 + c^ + cv where c1 and c^ are the circumferences of the circles [x2 + y2 1], [ X 2 + J 2 = 1, z = - 1; counter-clockwise and clockwise oriented, respectively, and c2, c^ are the segments [x=\/~2 , y = n/~2 , -1 < z < 1], oriented according decreasing and increasing z, respectively, (the curve C is schema­ tized in the illustration). Obviously con­ dition ( a ) is satisfied but it is intuitively evident that C cannot be reduced to a point in E~ - R.

§17-

SOME PROPERTIES OP HOMOTOPY FOR CONTINUOUS CURVES IN E3 (17-4)

275

A first elementary sufficient condition in order that C ^ o(E3 - R) has been given in (17-3* i)> but we shall need more refined statements. Let

C, pQ, r ^ r 2 j be defined as above and let = ([C], R) > o. For the sake of simplicity we shall suppose P0 = (°> 0); hence r ^ r2, r^ are the x, y, z-axes, respectively. Let s^ s2, s^, s^ be four disjoint straight lines s1 = [y = y 1, z = z1], s2 = [x = x2, z = z Q], s3 = [x » x3, y = y3b s^ = [x = x^, y = y^b with lyj, Iz^|, |x2|, |z2l, |x3 l, |y31y |XjJ, lyrics, and such that bothstraight lines s^ s2 cross the strip between the two parallelstraight lines s^ s^. Hence z1 + z2, and either x3 < x2 < x^, or x3 > x2 > x^; either J 3 < < j k, or y3 > y 1 > yv Finally let S = s1 + s2 + s3 + s^. We shall also denote by q1, q2, q3> q^ the points q1 = s1 E21, q2 = s2 E22, Q.3 = S3 \ 3 > q^ = s^ E23. The following non-elementary statement holds: 25

(i)

THE FOUR LINES THEOREM. Under the conditions above C ^ 0 (E3 - R) if and only if C ^ 0 (E3 - S).

This theorem was given by L. Cesari by using elementary methods of topology [L. Cesari, 8]. A second and shorter proof in terms of knot theory was given later by S. Eilenberg [2].

NOTE, In order to get an intuitive understanding of statement (i) we shall observe that the system R of three lines p2 ' r~ is replaced by the system S of disjoint lines ^2 * 3 * 4 * so close to the lines r that we can move the lines s into the lines r with­ out crossing C. Statement (i) states that the system R is equivalent to S in the question of homotopy to zero of C in EQ - R (or E 3 “ S)* It is easy to see that systems S' S 1 of only three disjoint lines, or systems S f! of four lines where s^ s2 do not cross the strip between s~ and

CHAPTER V.

THE FIRST THEOREM

are not equivalent to R. Indeed, in the illustration the curve C already considered in the previous example and not homotop to zero in E^ - R, is homotop to zero in E^ - S*, as well as in E 3 - S 11. Let (T, Q): p = p(w), weQ, be any c. mapping from a simple closed pol. reg. Q, of the w-plane E2, w = (u, v), into the p-space E^, p = (x, y, z); let C: (T, Q*) be the closed oriented curve which is the image of Q* under T; let (Tp, Q), r = 1 , 2 , 3, be the plane mappings which are the projections of T on the yz, zx, xy-coordinate planes E 2 i, E22, E23; Let Fp C E2p, r = 1 , 2 , 3 , be the point sets defined in (16.10, Note) for the mapping T; let Dp C F2p, r = 1 , 2 , 3 , be the countable sets defined in (lo.ii-) and (12.5) relative tothe plane mappings (T, Q); let D C E^ be the set definedin (10.5) relative to (T, Q). Finally let pQ = (xQ, yQ, z Q ) , ry R = r 1 + rg + iy p01, P0 2^ ^03 (ii)

^efineci as above.

THEOREM. Under the conditions above, if if N(pot, Tt, Q) = 0, t = 1, 2, 3, and

R[C] = 0,

poteE2t " Dt ~ Ft' t = 1» 2, 3, if pot is a point of density of points Pe^2t Dt - Ft with N(p, Tt,Q) = 0 , t = 1 , 2, 3, then C - 0 (E3 - R).

NOTE. A simple proof of (ii) is given below where (i) is used. A direct somewhat longer proof, independent of (i), Is given in Appendix A. The statement (ii) is extended to mappings from Q, into E^, N > 3 , in [L. Cesari,^6 ].

PROOF OF (ii). Let us suppose pQ = (0, 0 , 0)and let Kj. C E2t be the set ofall points P€?2t whepe N(p; Tt, Q) = 0 and K* = - (Ft + Dt ), t = 1 , 2 , 3. For any number d > o let qt C E2t be the square of center (0, 0), side length 2 d, and sides parallel to the axes. Then Iqt K£ | : Iqt I -- » 1 as d -- » 0, t = 1 , 2 , 3 • Let 25 = {[C], R) > 0 and dQ be any number, such that 0 b ~ ] , t = 1 , 2 , 3 * Hence both squares M = [- dQ < x < 0, - dQ < y < o]> M* = [0 < x < dQ, 0 < y < dQ], M, M 1 C E23, contain points of . Let qQ3 = (x3, y3),

§17-

SOME PROPERTIES OP HOMOTOFY FOR CONTINUOUS CURVES IN E3 (17-10 q0i+ = (xu> 7^) be ^ two points qo3eM Kg, q^eM' K3- Let d, = min Ux^l, Iy3 1 , x^, y^l and let us observe that both squares M l! = [ 0 < y < d 1,0 < z < d1] C E21, M 11! = [0 < x < d ^ - d1 < z < o] C E22 contain points of K 1 and Kg, respectively. Let q0 1 = (y1, z, ), q.2 = (x2, z2 ) be any two points q0l€MM K 1, Q.o2€M1 ’!K2* The*1 we have z2 < o < z ^ , x^ < x2 < x^, J 3 < < J k, and the corresponding straight lines s^ s2, s3, s^ have the required properties. Let 3e > 0 be the minimum mutual distance of the lines s^ s2, s^, s^ and let S = s1 + s2 + s3 + • By (16.12, i) there is a finitesubdivision z f + z !! of Q into simple pol. regions qez1 and not simple pol. regions Rez1! such that (a) diam T(q), diam T(R) < e; (b) foreach ReZ’1 and for each boundary curve p* of R, the set T(p*) is completely contained in a sphere a with crS = 0; (c)H Q* = 0 where H - T~1(S), since [C]S = o. If wedenote by c the image of each curve q* under T, then c has diameter < e, since diam c = diam T(q*) < diam T(q) < c.Hence c is completely contained in a sphere aQ of radius €. Such a sphere a , either has no point in common with H, or has points in common with at most one straight line s. In the first case we have obviously c ^ 0(ao ) and hence c ^ “ ^)* In the second case aQ is contained in a sphere of radius 2e whose center is on one of the straight lines s. Suppose for example that has its center on the straight line s^ = [x = x^, y = y^] and denote by c^ the projection of c on ^23, i.e., c3: (T3, q*). Then by N(qQ3, T3, Q) = o it follows o(qQ3, c3) = o and hence, by (17-2, ii), also c =* °(E3 “ S) ^OP ©very qeS*. This implies, by (17-2, iv), that there is a c. mapping (T*, q) from q into E^ - S such that T*(w) = T(w) for every weq*. We consider now each

277

278

CHAPTER V.

THE FIRST THEOREM

region Rez,!. Since each boundary curve p* of R has an image c: (T, p*) contained in a sphere a with a S = 0 , we have c ^ 0 (a) and hence c ~ o(E^ - S). Since E^ - S is an open connected set in E^, by (1 7 -3 , iii) there is a c. mapping (T*, R) from R into E^ - S with T*(w) = T(w) for every weR*. Thus T* is defined on the whole set Q, and wehave T*(Q) C E^ - S, T*(w) = T(w) for every weQ*. Thus C — o(E^ - S) and, by (i) above, also C — °(E3 - R).

* 1 7 -5 - An Elementary Statement on Relative Homotopy (i)

If s^, i = 1 , 2 , ..., 6 , are the six closed segments joining the point pQ = (xQ, yQ, z Q ) to the points (x0 + m, yQ, zQ ), (xQ, yQ + m, z Q ) , (xQ, yQ, zQ + m), m > 0 , if 3 = 3 ^ . . . + s6, if C is any c. curve of end points p1, p 2 and [C] C S, then C = C f(S) where C f is a poly­ gonal lineof end points p^, p2-

PROOF. Let C: p = p(u), 0 < u < 1, p(u) = tx(u), y(u), z(u)]. For the sake of simplicity we can suppose xQ = yQ = zQ = o, m = 1 , and also, e.g., p 1 = (x^ 0 , ojes^ p2 = (°> j 2 ’ ° } e s 2> 0 < xi 0 < y2 < 1* Hence x(o) = x 1 > o, y(l) = y > 0 and there must be two real numbers 0 < u 1 < u2 < 1 such that (1) x(u1) = 0 ; (2 ) x(u)> o, y(u) = z(u) = o for all ° < u < u1; (3 ) y(u2) = 0 ; (4 ) y(u)> o, x(u) = z(u) = 0 for all u2 < u < 1 . Let C f: p = pQ(u), 0 < u < 1 , pQ (u) = [xQ(u), yQ(u), zQ(u)], where the c. quasilinear vector function defined by x0 (u) = x 1u 1J"1 (u1 - u) if o < u < u1; yQ (u) = y 1 (l - u2 ) ~ 1 (u - u2) if u2 < u < 1 ; xQ(u) = o, yQ (u) = o, z0 (u) = o otherwise. Let us now define x(w), y(w), z(w), w = (u, v )eR = [o < u < 1 , 0 < v < 1 ] according to

§17-

SOME PROPERTIES OF HOMOTOPY FOR CONTINUOUS CURVES IN E 3 (1 7 . 6 ) 279 (1 7 -1 , !)• Each point p(u, v) = [x(u, v), y(u, v), z(u, v)] Is on the segment s1if o ^ u < U, on s2 if u 2 < u < 1 , on the same segment s^ where p(u) lies if u 1 < u < u2. An analogous procedure holds in all other cases. Obviously C ! is the seg­ ment p ] p 2 if p ^ p 2 belong to the same segment s; the polygonal line p 1 pQ p 2 if p ^ p 2 belong to different segments s. If C is closed, i.e., P-j = P2> then, by repeating this procedure, C ! is reduced to thesingle point p 1 = p2 and then C = o(S).

*1 7 -6 . A Second Elementary Statement on Relative Homotopy (i)

(ii)

If A is any square, A C or c E^, of a plane cr in E^, if C is any c. closed oriented curve [C] C A*, then C - C l(A*) where C ! is some polygonal line with [C 1] C A*. In addition, if pQ is the center of q and o(pQ, C) = h ^ o, then C ! is A* itself counted h times and counter-clockwise oriented if h > o, counted |h|times and clockwise oriented if h < 0 , any single point of A* if h = o. If the curve C of (i) is given by a c. mapping C: P = Po 0w), weQ*, from the boundary Q* of a simple pol. reg. Q, if R is any pol. reg. R = Q - q° of order u = 1 , then we can require that the homotopy in (i) is performed by a c. mapping (T, R): p = p(w), weR, [or (T, Q): p = p(w), weQ, if h = o] such that p(w) = pQ(w) on Q*. In addition if P0 (w) is q.l. on Q* we can require that T is q.l. on R [or Q].

PROOF OF (i) M B (ii). In the statements above the plane a is supposed to be oriented, and, for the sake of simplicity, we can suppose that oc is the xy-plane, that r is the z-axis, that P0 = (°> °> °)> and A = [- 1 < x < 1 , - 1 < y < 1 , z = o]. Let q>(u) = 1 if 0 < u < 1 , and7 < u < 8 , = 2 - u if 1 < u < 3 , = - 1 if 3 < u < 5 ^ = u - 6 if 5 < u < 7 - Then cp(u), o < u < 8 , is a q.l. function

280

CHAPTER V.

THE FIRST THEOREM

with cp(o) = cp(8 ) = 1 and we can define cp(u) in (- oo, + oo) by the periodicity of period 8 . Set i|r(u) = 9 ( 2 - u) for all u. Then A*: x = cp(u), y = i|r(u), 0 < u < 8 , and C f: x = cp(hu), y = Tjr(hu), 0 < u < 8 , is a q.l. representation in [0, 8 ] of the curve C 1, [Cf] C A*. From here we can pass to a q.l. representation of C T on [0 , 1], or on q*. Since [C], [C 1 ] C A* C a, o(p; C) = 0(p; C 1) = h, by (17-2, i) we have C — C 1(E^ - r) and, by (1 7 «2 , Note 1 ), also C - C 1 [ a - (pQ )] .

Thus (i) is proved. By (17.2, iii) there is a c. mapping (T, R): p = p(w), weR, with p(w) = pQ (w) on R, rT(R) = 0, and also, as above, T(R) C a - (p0 ) * Set 5 = {pQ, T(R)} > 0 and let 7 be the open circle of center pQ and radius 5 , 7 C a. Let t bethe c. mapping from a - 7 into A* which maps each point peor - 7 into the projection p feA* of p from pQ into A*.Then the mapping T f = tT has all properties required in the first part of (ii). Suppose finally that pQ (w) is q.l. on Q*. Then, by (1 7 *2 , iii) we can suppose that (T, R) is q.l. in R and hence there is a finite regular sub­ division S of R into triangles t of linearity for T . The six straight lines p: x = + 1 , y = + 1 , y = + x, divide a into eight regions, say h, and each triangle D = T(t) into simple pol. regions. There is a new finite regular sub­ division S ! (a refinement of S) of R into triangles t 1 such that for each t ’eS, the tri­ angle D ! = T(t) has no interior points in common with the straight lines p. For each vertex w of S let p = T(w) and p 1 =t(p) = tT(w). Then if t 1 is any triangle t ? = w 1 w2 w^ of S 1, and pj^ = t(p^) = xT(Wj_), i = 1 , 2 , 3, the points p^ are on the same region h ofor, the points p£ on the same side of A*, and D ? = p| p^ p^ is a degenerate triangle A* C A*. Then the q.l. mapping (T?, R) which maps each vertex weS1 into the point pt = T(p) = xT(w)eA* hasall the properties required in (ii). Thereby (ii) is proved.

§17*

SOME PROPERTIES OF HOMOTOPY FOR CONTINUOUS CURVES IN E3 (17-7) *17.7. (i)

A Third Elementary Statement on Relative Homotopy

If s^, i = 1 , 2 , 3, b , are thefour sides of the square A of vertices (xQ, j Q, z Q ) , (xQ + m, j Q, z Q ), (xQ + m, yQ + m, z Q ), (xQ, yQ + m, z Q ), m > 0 ; if s^, i = 5 , 6 , ..., 2 0 , are the sixteen further seg­ ments joining each vertex (x^, y^, z^) of A to the four points (x|,y£, z^) not on A with x| = x^ + m, yj =

zi =

or

y| =

±

zi =

z±>

or xj_ = x±, yj_ = y±, zj_ = z ± +m,i = 1 , 2 ,3, ^ (cf. figure); if S = s., + ...+ s2Q; if C is any oriented, closed, polygonal curve with [C] C S de­ fined by a q.l. mapping C: p = pQ (w), weQ*, from the boundary Q* of a simple closed pol. reg. Q; if CQ Is the projection of Con the plane z = zQ; if 0 (pQ; C ) = h + 0 where pQ is the center of A; then there is a q.l. mapping (T, Q): p = p(w), weQ, with p(w) = pQ (w) on Q*, T(Q) C S + A, and a(T, Q) = |h| m2 . PROOF. We can suppose x = y = z= 0 , m = 1, _i _ -j U U vj pQ = (2*" , 2 ,0). Let q 1 , q2, q^ be the closed squares q^ = [- i < u, v < i], 1 = 1 , 2 , 3, and H be any q.l. homeomorphism from Q Into q^ (6.1, i). Let H ! be any q.l. mapping, as defined in (17.1, ii), mapping q^ - q° into the square r = [0 < £ < 1, 0 < t ] < 1 ] of an auxiliary £-plane, (; = (£, tj), such that q* and q* are mapped into the segments t 1 = [° < | < 1, tj = 0], t2 = [ o < | < i , n = 1]. The given q.l. representation of C on Q* is trans­ formed by H and H ! into q.l. representations of C on q* and t2, say C: p = p !(w), weq*, and C: p = p M ( 0 = P !!(£, 1 ), £et2, i.e., 0 < £ < 1 . Since p , ! ( 0 is continuous on t2, there is a 5 > 0 such that |pr,(£) - p M ((;*)! < 2”1 for all '€1 2, |5 - £ ’| < 5 and we shall consider any subdivision O = £ 0 < s l. < . . . < f - ^ = 1 of t2 such that | < 5 , i = 1 , 2 , ..., N - 1 . Then the polygonal lines C±: p = p ,f(|, 1 ), |± - 1 < i < i ± , 1 = 1, 2 , N (subcurves of C) havediameter < 2 ~ 1 and hence are completelycontained in at most one of the systems S' of six segments s^ con­ current at the points (0, 0, 0), (1, 0 , 0), (1, 1 , 0), (0, 1 , 0). By suppressing some of the

281

282

CHAPTER V.

THE FIRST THEOREM

p o in t s ^ we can even suppose t h a t th e end p o in t s pi- 1 = p ' ' ^ i - 1 ' 1 ^ Pj_ = P 1' ( lj_, 1 ) o f each c u rv e

belong to

A* = s1 + s2 +

By (1 7 *5 , i) we know that = C|(S)where C| de­ notes the shortest polygonal line joining and p^ in S; hence Cj C A*.In addition we can suppose that the relative homotopy = CM (S) is performed by a q.l. mapping T M : p = p IT(^)^ £ = (£, T]), il^1 < I < 0 < n< 1 , where p !!(£ ) is constant on each of the segments [6 = ° < T| < 1], [| = 0< ^ where p 11(O = Pj_^ respectively. Thus we have a unique q.l. mapping (Tf1, r): p = p T’(£;), (;er, such that T fI(r) C S, C: (T M , t2 ), C 1: (T !1, t1), and C 1 is a closed polygonal line, [CM] C A*. Now H f transforms T ,! into a q.l. mapping (T!, q3 - q2 ): p = p^(w), weq3 - q°, and C: (T *, q*), CM: (T1, q*). By (17-2, i) we have 0(pQ; CQ ) = 0(pQ; C !) = h and, by 00 we obtain L(Q, T) < 2“1 A[1 (C1) + 1 (C2 )].

+ &n ) [l(C1n) + l(C2n)

NOTE 1. As a particular case of (i) let C: x = x(u), y = y(u), z = z(u), ° < u < 1, be a given curve in E^ and CQ: x = x(u), y = y(u), z = 0, 0 < u < 1, be the projection of C on the xy-plane. Let A = max |z(u)|, ^o:rcian length of C , and (T, Q ): x = x(u), y = y(u), z = vz(u), (u, v)€Q = [0 < u, v < Then V(Q, T) < L(Q, T) < A 1 (CQ ). Indeed, if CQn: x = ^(u), y = yn (u), 0 < u < 1, is any polygonal line inscribed in CQ such that xn (u ) -- ^ x(u), yn (u) -- » y(u) in [0,1] as n -- » if zn (u) is any q.l. function such that zn (u ) -- \ z(u) as n -- > », we shall de­ fine (Tn, Q): p = p(w), weQ,,----- as the vector function p(w) = [xn (w), yn (w), zon(w)], where zon(w) is obtained by applying (1 7 •1 j. i) to the functions zn (w) and 0 . Here the

1 ].

CHAPTER V.

2814-

THE FIRST THEOREM

triangles t 11 having a side on the polygonal line V X = xn (u), y = yn (u), z = z n ( u ) , o < u < 1, are con­ tained in a vertical strip whose height is < A + and whose width is the length of aside of CQn. The same state­ ment holds for the triangle t ! having a side on CQn. Therefore a (Tn> Q) < za(t') + za(tM ) < (A + 5 n ) l(CQn), and hence L(Q, T) < Al(CQ ).

NOTE 2. Another particular case of (i) is the following. Let C: x = x(u), y = y(u), z = z(u), 0 < u be a given curve in E^ and CQ: x = o, y = 0, z = z(u), 0 < u < 1, is the projection of C on the z-axis. Since 1 (CQ ) < 1 (C), we have V(Q, T) < L(Q, T) < A1 (C) where A is the maximum of the func­ tion [x2 (u) + y2 (u) ]2 in [0, 1].

*17 -9 • Bibliographical Notes For general questions on homotopy see the excellent books and papers already mentioned in §8.

§18.

THE

FIRST THEOREM

The theorem in question (see consequence of the statement prove first two lemmas.

18.1.

1 .4 ,

L-| ) shall be given in(18.10,i) as a (18.2, i) of the present§18.In (18.1 ) we

The Set

M(T, A) C E3

Let (T, A): p = p(w), weA, or x = x(w), y = y(w), z = z(w), weA, be any c. mapping from an admissible set A of the w-plane E2, w = (u, into the p-space E^, p = (x, y, z); let (Tp, A), r = 1, 2, 3, be the plane mappings which are the projections of Ton the yz, zx, xy coordinate planes E21, E22, E2^, let r = r(T, A) be the collection of all maximal connected sets of constancy of T in A introduced in (l4 .i) and let r2 = r2(T, A) be the corresponding subcollection of con­ tinua we have considered in (14 .3 ). Analogously, for each mapping (Tp, A) from A into E2p, let rp = r(Tp, A) and r2p = r2(Tp, A), r = 1, 2, 3, be the corresponding collections. Let us denote by r, r2, r , r2p also the sets of points covered by the same collections in A. Finally let M C E^ be the point set M = M(T, A) = T(r21 ) + T (r22^ + Let Dp denote the countable set Dp C E2p defined in (10.5, Note) and (12.5) for the mappings (Tp, A), r = 1, 2, 3 ; let Jp C E2p be the

§ 1 8 . THE FIRST THEOREM

(1 8 .1

)

set of all points P€E2r w h e r e N (p; TP> A) = + ° o , let Fp C E2p be the set of all points P€E2r defined in (16.10, Note). Let Dp C E^ be the set of all points (x, y, z)eE^ whose projections on E2p is the set Dp C E2p, r = 1 , 2 , 3. Thus each Dp isthe countable sum of straight lines parallel to one of the axes, and the measure \T5 \ of Dp in E^ is zero. (i) The set M is measurable in E^ • PROOF. By (1 ^ •3 > Proof of i ) we have r2p = lim (Bn Bn+1 Bn+2 ...) as n where the sets Bn (denoted by A2n in 1 ^-3 ) are open. Since the Images under T or Tp, of compact sets are compact and each open set Bn can be thought of as the countable sum of compact sets, we conclude that all sets T(Bn ), Tp (Bn ), n = 1 , 2 , ..., are Fa-sets [S. Saks, I, p. ^ 1 ]. By the definition of the sets Bn (1 ^4-.3, Proof of i) and by (' Ik. 3, ii) it follows that if a continuum gerp belongs to Bn and Tp (g) C E2p - Dp, then the point Tp (g) belongs to Tp (r2p). Hence Tp (Bn ) - Dp C Tp (r2p) - Dp . Thus we have also T(BR ) - Dp C T(?2p) -I)p and Zn [T(Br ) - Dp] C T(r2p) - Dp . On the other hand r2p = lim (Bn Bn+1 Bn+2 . ..) as n ----- hence T(r2p) C lim Hp T(Bn+p) CznT(Bn ), and also T(r2p) - Dp C zn [T(Bn ) - Dp] . Thus we have T(r2r) - Dr = Zn [T(Bn ) - Dr] = Zn T(Bn ) - Dp, and the set zn T(B ), as a countable sum of F^-sets, is a F^-set (or an F-set, or a G-set) [E. W. Hobson, I, 2nd Vol., p. 261]. Since T(r2p) =. Zn T(BR )+ ^r T (r2p) and the last set has measure zero because it is con­ tained in a countable sum of parallel straight lines, we have proved that T(r ) is measurable in E^. Finally M is measurable as the sum of three measurable sets.

NOTE.

In the lines above we have also proved that the set Mq = M + D, + D2 + D3 = £r [T(r2r) + Dp]

is B-measurable- Here M ) M and MQ - M is contained in a countable sum of straight lines r C E^ parallel to the x, y, z-axes.

2

286

CHAPTER V.

THE FIRST THEOREM

(ii) If T1? T2, T3 are all BV then |M| = 0; the three-dimensional measure of M in

i.e., is zero.

PROOF. Since Tg, are BV we have |Jp | = 0, |Fp | = o, r = 1 , 2, 3 (12.3 and 16.10), andon the other hand, Dp is countable. Thus |D^ + + F^| =0. For every point (x, _ (d ^ + + F^) there is only a finite number k of continua ger (T~, A), these continua are components g of the set T~—1 (x, y), and we have 0 < k < N(x, j ; T^, A) < + 00 (1 4 .4 , iii). On the other hand, since (x, y) is not in F^, the function z(w) is constant on each g above, as well as the functions x(w), y(w). Thus T(g) is a single point of the straight line r parallel to the z-axis through (x, y). This implies that for each (x, y)eE23 - (D3 + J3 + F3) the set T(r23) has a finite intersection with the straight line r parallel to the z-axis through (x, y). Since T(r2p) is measurable and |D3 + + F3 | = 0,we conclude that |T(?23)| = 0. Analogous reasoning holds for T(r2p), r = 1, 2, and thus |M| = 0.

18.2.

A Particular Sequence of q.l. Mappings

We shall denote by (T, A) any c. mapping from an admissible set A the w-plane Eg, w = (u,v), into the p-space E^, p = (x, y, z), (P , F ), n = 1, 2, ...,a sequence of q.l. mappings Fn C Fn+1,

of by

Fn C A> Fn ^ A ° ’ and by A )> (pnp^ Fn )>r = 1> 2> 3 , the plane mappings which are the projections of (T, A) (Pn, F ) on the coordinate yz, zx, xy-planes Eg1, E22, E23(i)

THEOREM. Given any c. mapping (T, A) from an ad­ missible set A C E2 into E3 such that the plane mappings (Tp, A), r = 1, 2, 3, are all BV, then there exists a sequence (Pn, Fn )> n = 1, 2, ..., of a(Pn> ^n )< q.l. mappings such that Pn -- ^T, W(A, T 1) + W(A, T2) + W(A, T3), a(Pnp, FR ) -- ^ W(A, Tp ), r = 1, 2, 3, as n -- ►«>.

The proof is given in the next articles (18.3-9).

§ 1 8 . THE FIRST THEOREM 18.3.

(1 8 .k )

287

Proof of The Theorem (18.2, i )

Let [F ] be any sequence of closed figures invading A0, i.e., Fn C Fn+1, Fn C A, F° fA0. Then for every n, Fn is compact as well as T(Fn ) and therefore there is a closed cube K C E^, say, for the sake of simplicity K = [0, 0, 0, k, k, k], k = k(n), containing T(F ) 1^ in its interior. Let N = N(n) be an integer such that a = k/N < n~ . For the sake of simplicity we shall not display n any more in the notations. We have W(F, Tp ) < W(A, T ) < + «>, r = 1, 2, 3, for each F = Fn « Then the functions N(p;T , F), PeE2p, r = 1, 2, 3, are L-integrable, 0 < N(p; Tp, F) < N(p; Tp, A). Let Jp be the set of all P^E2p where N(p; Tp, F) = + 00; thus |Jp | =0, r = 1, 2, 3. For every real h > 0 let q-^ = qtl(p) denote the square (x - h, x + h, y - h, y + h) C E2p of center the point p = (x, j ) e^ 2r anci s^e-longth 211• Then (oc) N(x, y; Tp, F) = lim (q^)/ N(u, v, Tp, F) du dv as h ---►o + for al­ most all p = (x, y)eE2p. Therefore, if H^ denotes the-set of all peE2p where (oc) does not hold, we have IHpl =0, r = 1, 2, 3. De­ note by Dp the countable set Dp C E2p defined in (12.5) for the mapping (Tp, F); denote by Fp C E2p the set defined in (16.10, Note) for the mapping (T , F) i.e., the set of all points P eE2p such that for at least one component g of FTp1(p) the mapping T is not constant on g. Since W(F, T ) < + », r = 1, 2, 3, by (16.10, Note) we have |Fp | = 0. Finally denote by M Cthe set defined in (18.2) for the mapping (T, F). By (18.2, ii) the three dimensional measure of M in E^ is zero, |M| = 0.

1 8 .4

Let

. Proof of the Theorem (18.2, i) Continued

Aj. • be the closed three-dimensional cell (cube)

Ahij = ~ 1 -x (i - 1 )a < y < ia, (j - 1 )a < z < ja], h, i, j = 1, 2, ..., N. Let B be the system of all edges of the cells Ahij‘ Denote by A^, A^j, A^. the squares which are projections of A ^ j on the three coordinate planes E2^, E22, E21, respectively. De­ note by the linear mapping x ! = x - (h - 1)a, y = y - (i - 1)a, z f = z - (j - 1 )a from onto A 111 . Denote by ^hi [thj' tij] the analogous linear mapping from A ^ tA^j, A^.] on the cell A 11 C E23 [E22, Denote by MQ the subset of the cell A 111 in E^ defined by MQ = ^hij ^hij M3 subset cell A 11 in the plane E2^ defined by ^ h i ^ + H3 + D3 + ^3 ^ and analogously let us define M2, M 1. Then |M0 I = 0, | | = 0, r = 1, 2, 3; i.e., the three dimensional measure of MQ in the cell A 111 C E^ is zero, as well as the two-dimensional measure of each set

288

Mp

CHAPTER V. in the cell

A 11 C E2p,

r =

THE FIRST THEOREM

1, 2, 3

.

Let us denote by N^(x, y), 0 < x, y < a, the function N^(x, y) = Lh l Nfx + (h - 1 )a, y + ( 1 - 1 )a; T^, F] and by N2 (x, z), N 1 (y, z ) the analogous functions defined for 0 < x, z < a, 0 < y, z < respectively. The function N(x, y; T^, F) is zero outside the square = [0 < x, y < Na = k], hence W(T3, F) = (K3 ) I N(x, y; T ^ F) dx dy 2 hi(^hi) / N dx dy = (A11) / N 3 (x, y ) dx dy. Analogously W(Tp, F) = (A11) / Np (a, P) da dp. Let us denote by N(x, y, z), (x, y, z)eAnl = [0 < x, y, z < a] the function N(x, y, z) = N 1 (y, z) + N2 (x , z) + N3 (x, y). Then (A]n ) / N(x, y, z ) dx dy dz = a • (A11 ) / N 1 dy dz + ... = a[W(F, T 1) + W(F, T2 ) + W(F, T3)] = aW, where W is the non-negative number in brackets. Let us denote by M ’ the set of all points (x, y, z)eA1ll where a2 N(x, y, z) < W. Let us prove that |M1| > 0. Indeed, in the contrary case, we would have a N(x, y, z) > W a.e. in A111, and hence, by integration in A^ 11, also (A111) / N > Wa, a contradiction. We have proved that|M1| > 0. Denote by M M the set of all points (x, y, z)€Mf - M which are interior to A111, and such that (y, z ) is not inM 1, (z, x) is not in Mg, (x, y) is not in M3. Thus |MT1| = |Mr - M | = |Mf| > o, and we can choose a point P 0 = ( x 0 , J 0, z 0 ) e M " .

Then

a d d it io n i f we denote b y

xh = xQ + (h -

1 )a,

0 < x Q, y Q, z Q < a , th e p o in t s

y± = yQ + (i -

1 )a,

( x Q,

P^j_j =

y Q, z Q)eM».

In

z j)>

z . = zQ + (j -

1 )a,

and by

phi^ Pij the P°lnts (xh> z j ) eE2 2 ’ & 1 ’ zj ^eE2 1 then we have also PjQij6^ ” anci P^i' pij belong to E2p - (Jp + Hp + Dp + Fp ), r = 1, 2, 3, respectively. Finally we have a2 Z., N(P .r Tl, F) + a2 ^

N(phj, T2, F) + a2

^

N(ph±, Ty F)

&2 V V z0 } + a" N2 (V z0 } + a" N3 (x 0' yQ} = &2 N(V W = W(F, T 1) + W(F, F2 ) + W(F, T3 ). 18.5. Let us denote by h, i, j = 1, 2, straight lines and set R = hence d = (R, B)

=

y0> Z0 } ^

Proof of the Theorem (18.2, i ) Continued

P the system of N3 points P^ij = y±* z j ) ..., N; let us denote by r^, r. ., r ^ . the 3 N2 (x = xh, y = y ± ), (x = xh, z = z.), ( y = y±, z ^ + r, . + rii^* Then we have P C R, BR = 0, and > 0.

For each point p^j -^hj^ us consi^er ^he set A,^. = FT31(p^j) and denote by A.K the collection of those components of which

=

z

.)

§ 1 8 . THE FIRST THEOREM (1 8 . 6 )

289

belong to r2 (T3, F). Set A = + A-hi + >*hj)> A' = zUjj + ^ ^ Since the points p ^ . are not in F ^ T isconstanton each component g of x ±3 and hence ger(T, F). Analogously for the points phl, Pftj • Consequently any two components of the sets \ are either disjoint, or coincide, and hence the components of A are still components g of some X and continua ger(T, F). The same holds for A*. For each g of the image T(g) is a point of r^j and more precisely a point of Mr^j. Since the points P^jjare n°‘t M we have T(g) + P^-y^ h = 1 , 2, ..., N, for each component g of the set \ l .. The same holds 2 ^ for the sets A,^, . Consequently the 3 N sets A. ^ y h, i, j = 1, 2 , ..., N, are disjoint. Note that the number of components g of each set, say x ± y < ^^ij* T3' F) < + 00 (since p^j is not in J^). This implies that the components g of AT are finite in number, say k, and we know, on the otherhand, that the same components g are interior to F and do not separate the w-plane (nor F). Since the set T( a !) is finite, is contained in MR and PT(a !) = 0, the distance d T = {P, T(Af)} is positive and d*1 = min[d, d !]> 0.

18.6.Proof of the Theorem Q8.2, i) Continued By (1 1 .3 , iv) where e = 4 "1d ’1 there exists a subdivision of F into a finite system S of pol. reg. * (simple, or not simple) such that AZtt* = AF* and diam T(*) < 4 ”‘1d M . By (1 4 .4 , ii) we can suppose that the k components continua g of A1 (interior to F and non-separating the w-plane) are in k different simple regions *, say 1 = 1, 2, ..., k, with ArtJ = 0. Let us denote by 1 = k + 1, ..., k all other regions *eS. We shall denote by g-[ the component of A! which is in and by p-^eE^ the point p^ = T(g^), 1 = 1, 2, ..., k. Thus p-^M, p^eR and hence p-^ belongs to some straight line r C R. Since p-^ + ^hij> h, i, j = 1, 2, ..., N, we can conclude that each point p-^ belongs to one and only one straight line r, say r = Phj' rij * Then we can distribute the regions jt^, 1 = 1, 2, ..., k, into families ^ h i * ^ h j ' ^ i j according the point p^ belongs to a straight line p = phi' Phj' Pij * Fop region 1 = ^ 2> • • • > ' denote by Cl' Glr curves ci: cip: (Tr> r = 1, 2, 3, and ob­ serve that C-^ is completely contained in a sphere a of center P^er and radius 4 ”1d !I, hence a contains one segment of the same line r (a diameter of a) and err1 = 0 for all straight lines r ! + r, r 1 C R. In addition, if for instance iceO)^, then Pi61^ ; °(phi* C1 3 ^ ^ °* also we have ( a ) z|0 (phi; C- ^ ) I = N(p^^, T^, F), where 2 is extended over all (1 = 2> • • • > k). A first consequence of (a) is that

t

= minftC-^], P^) = min ^ Cl^ Phi^

290

CHAPTER V.

THE FIRST THEOREM

for all positive, t > 0; therefore for all points p€E23, Ip - Phll < t we have 0 (p; Cl3) = 0 (phi; C13). Hence (p) S |0 (p; C^^)| = N(phi; T^, F) < N(p; T^, P). A second consequence of ( a ) is that for any simple pol. reg. *1 non-overlapping the pol. regions *l€^ h i we have °^Phi^ = results hold for the families 1 = 1, 2, ...y k*

°>

whePe C3: (T3> * !*)- Analogous ^ij reSions *]_>

Let us observe now that the points p ^ are not in H3 and hence |q(h) | " 1 ' (q(h)) / N(p; F)----- *'N(phl, T3, F), as where q(h) denotes the square of side-length 2h and center p^. If J denotes the set of all points peE23 where N(p; T^, P) > NCp^, T^, F), and hence N(p) > N(p^) + 1 then, by (p), we have

h *0

N(phi} ^ N(phi} + Iq-(ti) |_1 |J q(h)| < Iq.(h) |_1 • (q(h)) / N(p), (h < 2_1t ), where the last member approaches N(p^) as h -- ►o. Consequently |q(h)|”1 |J q(h)| --------------------------------------------- ►o as h --► 0; for the set J in E2Q, or p ^ is point of density for the complementary set J 1 of all p where N(p; T3, P) = NCp^, T^, P). A consequence of this result is that for each point pej’with |p - p^jj < tand for each simple pol. region * ’ non-overlapping the regions we have 0(p; C3) = 0.

18.7. Proof of the Theorem (18.2,i) Continued We have now to modify the

mapping

T

on the pol. reg.

jt^, 1 = k

+ 1,

...,kf.

Let us consider first any region it-^,1 = k + 1, ...,kT,which is simple and is such that **A =0. Put as above C-^: (T, itj), Cp: (Tp, Jtj), r = 1, 2, 3, and observe that T(jt) is completely interior to a sphere a of radius 4 "1d fl whose center may be any point peT(jt). If crR = 0, then we have also T(jt)R = 0. If o has points in common with only one straight line r, say r^, then, by (18.6) we have ^3) = °> and, by (17-2, ii), C-^~ o(E3 - rhl) and also, by (i7-2, ii) C1 2,4 0(a Therefore, by (17• 2, iv), there Is a c. mapping (T!, jc1 ) with T 1 = T on Jtj, and T 1(*-]_) C a Consequently also T 1(*1 )R = 0. If the sphere a has points in common with more than one straight line r C R, then a has points in common with at least two straight lines r, say r 1 = r^j_ and r 1! = In any case let us prove that a is contained in a larger sphere o ' of center and- radius d M . Indeed if p !, p 11 are any two points p ’er'cr, p fler!,cr, then |p* - p M | < 2 (4 -1d M ) and since 2”1d* f < a, r 1, r f1 are neither

+,

i.e.,

§1 8 . THE FIRST THEORM (1 8 .7 )

291

parallel nor disjoint hence orthogonal and concurrent in a point

po = P h i j eP- Sln ce I p 0 - p 1 I2 + Ip 0 - p * ' ! 2 = lpl - p ' ' I2 < (2-1d .'1 )2, at least oneiof the two distances |pQ - p !|, |pQ - p M |> say IPQ ~ P TI> is < 2~2 d M ; hence for each point pea we have IP “ PcJ + ” po ^ 2”1(3-M + 2"1d 1 1 = d !|. We have now proved that T(*) C a C or1, where a 1 is a sphere of center P^ij and radius d !f. Hence cr! C A. . .. hij We know already that cl3) - 0, 0(^.5 Cl2) - 0 , 0 (p^j5 C11 ) - 0, By (18.6) we have also 0 (p; )= 0 for every simple pol. reg. it! C it-^ where C^: (T^, it1*) and forevery point peJ!, |p - P^jJ < T* Hence N(p; T^, it-^) = 0 for the same points p and, inaddition ^^hi* T3* *1^ = °‘ ^ere PhieE3 " D3 " F3’ ^hi is a P0:5-nt density for J T, and |D^| = |F^| = 0. Thus p ^ is a point of density of points peE^ - D3 - F^ where N(p; T^, it^) = °* T]le same conclusions hold for the points V ^ y P± j ’ 0 7 -4 , ii) we have now C1 ^ °^E3 " rhi “ rhj ~ P' Hence also * 0 (a1 - cr!R).Thus we can define a c. mapping (T1, it^) with T 1 = T on it* and T 1 ) C a1a !R. Let us consider now any region it-^, 1 = k + 1 , ..., k f, which is not simple and is such that = 0. Then if it-j_ = qQ - (q1 + ... +* * * = Q-o + ••• + Q*> where qQ, ..., q^ are simple pol. regions, then each curve C^: (T, q?) is interior to a sphere cr^ with a^R = 0 [a consequence of (11.3, iv)] . Therefore =*• o(cr^) and also ^ 0 (E^ - R). By (1 7 -3 , iii) there is a c. mapping (T!, it-^) from it into E^ such that T T = T on it* and T(it^) C E^ - R. Let us consider finally any pol. reg. it-^, I = k + 1 , ..., k 1, (simple, or not) with it*A + 0 (if any). Then the points of it*A are on one, or more, of the boundary curves of F*, and diam T(it^) < 4 “1d tf. Then T(*t-|_) is interior to a sphere a of radius 4 ”1d M . If *1 = Qq - (q-i + ••• + *1 = (15 + q.f + ••• + let us consider first any q|, i = 0, 1, ..., *o, such that q?A + 0. Then q£ is the sum of 2m consecutive arcs 11+ 1* + lg + 1^ + ... + lm + 1^, where l y 1 2 , ..., are subarcs of F* and (1 1 + lg + ... + + °> while 1 *, ..., 1 ^ are arcs interior to F, with the exception of the end points which are on F*, and 1 U = 0, j = 1, 2, ..., m. We can j = 2> • • • > m, lm+1 = 1 1. Thus it^ is decomposed into the simple regions it|, ..., it^ and a region it* of the same order as This procedure can be repeated for all i such that q^A + 0. For those i with q?A = 0 we can proceed as in (17.3 , iii). Finally we can define

CHAPTER V.

THE FIRST THEOREM

the c. mapping (T1 in such a way that T 1 = T on each arc for those boundaries q? with - --------------- ---------------------------------------------' i q^A + 0, as well as T f = T on each boundary curve q.* with q± A = 0, and such that (T performs the homotopy of the open curve C: (T, 1 \) into a single point in E0 - R. For those pj boundaries q with q?A = o the corresponding curve is in the interior of a sphere a C E_ R and hence, as in (1 7 *3 > iii), we can define (T, Jtj ) (in a pol. reg. = q{ - (% )° of order 1) in such a way to perform the homotopy of into a point Pi of E3 - R. Finally we can define in the remaining pol. region in such a way to connect the (T 1. various single points pjj by pol. lines in E 3 - R as in (1 7 -3 , iii)• J

Thus (T f. jt^) is defined in any case in such a way that T !(rt-^)R = 0 1 = k + 1, ..., k f, and T r coincides with T along all those open or closed arcs which separate the regions from the remaining part of F, that is from the regions 1 = 1, ..., k, where we shall define T ' as identical with T. Thus T ? is defined on the whole of F and we have T f(F’)R = o, where F* =

18.8.

Proof of the Theorem (18.2, i) Continued

Since T '(F1)R = 0, let dQ = {Tf(FT), R) > 0 and d^ = min Cdff, dQ]. Let us consider any arbitrary subdivision S ’ of F into simple pol. regions ..., *%+]> • • • > where the k regions considered above (18.6) and each of the remaining regions ^kfi' **•■’*k*t verifies the inequality diam T(jt^) < 4 ”1d^, i = k + 1,..., k ,!. Each curve C-., 1 = k + 1, ..., k ,!, is contained —I in a sphere of center a point p-^eT(jt^) and radius khence a^R = o. More precisely {p-^, R} > dQ, radius = 4 1 d^ < d^ and also radius< ^-1a. Let us denote by Thl • the c. mapping from A ^. - phlj. into Ag.y which is obtained by projecting every point P£Ahij. from p ^ . on On each face D of the cells Ahlj denote by pQ the unique point p0 = DR (intersection of D with the only straight line r C R with rD + o). Denote by r the projection of D - pQ from pQ on D*, D* C B. Therefore the various mappings, t j: p' = p'(p), map the system of all curves C-[: (T', jrj), 1 = 1 , 2, ..., k", into a system of curves C^' C B, CJ'1: ( t T, ). In such a way we have a c. mapping T 11, defined on the system K = « * + ** + ... + ,, T M = TThijwhich maps each point weK into a point peB. Let us consider on K all points which are vertices of the polygonal lines

...,

are s

§ 1 8 . THE FIRST THEOREM

(1 8 . 8

)

293

then K is a sum of segments and we can further divide each of these seg­ ments in such a way that, in each of the segments s so obtained, T lf has an oscillation < a, i.e., diam T ff(s) < a. Then T M (s) is an arc contained in the set S ! of at most six segments of B concurring into a vertex of B. By (1 7 •5 , i ) we have (T ff, s ) = (T fM , s ) (B) where T ff1 is q.l. on s, and maps s into a simple polygonal line (T, M , s) con­ tained in S'. Thus each curve C-^1: (T1f, ** ), 1 = 1, 2, ..., k !1, has been modified by homotopy in B into a closed polygonal line C|if; (Tfrf, Jtj) contained in B. If 1 = 1 , 2 , ..., k, let us prove that C-[1T is in the conditions of (1 7 -7 ^ i)- Indeed, C-j^ = C Is con­ tained in a "small" sphere a which leaves outside the points P^ij and whose center is in one of the straight lines reR.Thus If as ^ 0where s is, say, thesegmentbetween the points P ^ j = zj)> ■Phi j+ l = ^xh' ^i' zj+ 1 ^ then o is between the planes z = z ., z = z. and thus C4 is projected by th . . and t-.. . - on a curve _ j+l Jmj ni,j+i C-^ which lies on that part of the boundaries of the cells \ x ± y ^h± j+ i which is between the same planes. Finally, the various mappings t map (T-[ on a curve C-^! contained -in the four sides of the face D = Ahi j+1 and eight segments of length < a orthogonal to D at the four vertices of D. Then also the curve C-J^rf is contained in the same system of segments. Consequently, by (l 7 *7 , I), bound­ ary of a polygonal surface (T*, jt-^), (i.e., T* is q.l. in Jt^), and T* = T rir on **. On the other hand, ^Kp ^i *^31* ^ = ^^hi* C3l^ = By (1 7 «7 , i) we have a(T*, Jtx ) = a^lCKp^, C3l^ = a2|°(Phi' C3l^ [or 0(phj.; C2l), or 0 (p±J.; C^)].

°^phi; G

Let us prove that each curve C-[M , 1 = k + 1, ..., k !1, isin the con­ ditions of (l7 -5 > I)* Indeed, C-^ is contained in a sphere o of radius < (IqAj which leaves outside all straight lines r. If, say, aA-^ij + 0, then the distance of the center of a from P ^ j - do* Then a is contained in a circular cone of center P ^ j and opening cer­ tainly < jt/6, which leaves outside (within A ^ •) all straight lines phi' rhj' pij * Such a °one intersects the boundary of a region which covers partially one, or two, or three adjacent facesD of A ^ • • By applying the various mappings t the curve C-| is eventually mapped Into a curve C|! which is contained in a system. S of segments, S C B, as in (1 7 -5 , I) • Then also the curve C-^1f is contained in the same system S. Thus by (1 7 -5 , i, last line of the proof) theclosed pol. line C^M is the boundary of a pol. surface (T*, Jt^), with T* = T ffr on itj, T*(jt-^) C B, and a(T*, Jt^) = 0. In such a way we have a well determined q.l. mapping (T*, F) from the whole of F into E^, with a(T*, F) = Z a2|0 (ph^; ^32.) I+ 2 a2 l°^Phj^ C2l^ + z a2|°^Pij^ C3l^^ where the sums are extended to all

CHAPTER V.

29^

regions

^hj' a(T*, F)




F)

+

h, i, j = a2 Z

NCp^,

1, T2,

..., N. F)

Hence,

+

+ a2 S N(p±j, T 1, F) < W(F, T,) + W(P, Tg) + W(F, T3). We have finally a(T*, F) < W(A, T 1) + W(A, T2) + W(A, T3 ). Let us observe now that for each both sets T(it^) and T*(*-^) belong to the same cell ^ h ± y 1 = 1, 2, k. Otherwise TU^), T*(*^) belong to at most eight adjacent cells . and T*(*-^) is a single point of the common boundary of these cells, namely, a point of T(jrn ), _j -L with diam T(n^) < k~ a. In any case each point ofT(it^) is at a dis­ tance less than >/~3a from any point ofT*(*-^). We have consequently d(T, T*, F) < N/~3a < n/~3 n“1. If we now denote again F by Fn as in (18.3) and T* by T* we have that the mappings (T*, Fn ), n = 1, 2,... are q.l., that T * ----^T, and a(T*, Fn ) < W(A, T 1) + W(A, Tg) + W(A, T3) Thus the first part of the theorem (18.2, i) is proved.

18.9.

Proof of the Theorem (18.2, i) Continued

Because of (T*, Fn )-- ► (T, A) we have also (T*^ Fn )-- ► (Tp, A), r = 1, 2, 3, and hence (12.4 , i) W(A, Tp ) < lim a(T*p, Fn ), r = 1, 2, 3, as n -- ► 0, thereexists an n such that for each n > n and r = 1, 2, 3, we have a(Tnr' Pn ) > W(A' Tr } " e’ On the other hand, for each

n

(r - 1, 2, 3, n > n).

we have

a(T*, Fn ) = zr a(T^, Fn ) < £r W(A, Tr ). Therefore, for all

n > n,

we have also

W(A, T 1) - e < a(T*,, Fn ) < W(A,

T,

) + £ [W (A> V

< W(A, T 1) +

- a(T£r> Pn )]
00. -

Pn -- »T, a(Pn, Fr ) --------- >L(A, T) Hence (Pnr, Fn ) -- * (Tr, A), r = 1, 2, 3,

as

n -- » °° and, by

W(A, Tp )
00

i) and (5-7), we have , T) =

11 m a(Pnp, T) < ~ n -- »00

and

\ k --- *

w

v ’

n^ ---► 00 as k ---► 00. The set E^ does not depend upon the particular sequence (t]^, ) defining cd and is a subcon­ tinuum of a*. (Cf. B. v. Kerekjarto [I, p. 109]. Also L. Cesari [47]). If cd is a prime-end corresponding to an end t], then w eE^, but E^ may contain other points (accessible and not accessible from a). Different prime-ends cd may have sets E^ not disjoint, even coincident. The family {E^} of all sets E^ is a covering of or*.

NOTE. Though some authors denote the sets by prime-ends we prefer to keep the expressive Caratheodory!s term primeend for denoting a section cd in the collection of the ends. Indeed, while cd determines univocally, the set E^ alone does not determine necessarily the section cd. More details on prime-ends are given in the articles (19 -4 -8 ). For further in­ formation and the bibliography see (19 -9 )- Some examples are given below.

EXAMPLES. Let us consider the sets G 1, ••*, G^ defined above and given in the illustrations. In G 1, if cdq denotes the prime-end corresponding to the only one end r\ with w = (0, 1 ) then E is the segment [u = 0, 1 < v < 3] i.e., E^ con^o o tains the accessible point (0, 1) and all the non-accessible points (0, v), 1 < v < 3. Each other point weG| - E^ is o a point w = w of only oneend ri and w = E for the T] T] CD corresponding prime-end cd. In G2 the circumference C* of the circle C is the set E^ of one prime-end cd

302

CHAPTER VI.

THE CAVALIERI INEQUALITY

and all its points are not accessible .from Gg. Each point w of the line L 2 is the point w^ of two different ends. In G^ (as well as in G^) the circumference C* of the circle C is the set E^ of one prime-end a> and all its points are not accessible from G^ [G^]. In G^, G^ all boundary points are accessible and each prime-end cd corresponds to one end t1i, ECO = wTJ . For other examples see (19.4). 1 9 *4

. Right and Left Wings of a Set

E^

Let us observe that each prime-end co is a section in the collection {tj} of the ends tj ofor, where oo is given by a decreasing sequence (t^, t^t)> n= 1 , 2, ..., of intervals in {t)}• If £21 is any one of the orderings flj, we can use the notation a -< b -< c for expressing the fact that three ends a, b, c are in the ordering a !. Thus we can numerate the ends t^1 in such a waythat t\\ -< -< ... -< tj 3 ^ -< T ig 1 - < -q ] * * Obviously each end t] € ( t i {, t^] f ) belongs to one and only one of the intervals [rj^, t^ +1 ], ^ n = 1, 2, ..., with exception of (a) the ends r\^? t^1, i = 2, 3, ..., themselves which belong to two adjacent intervals; (b) at most one end contained in all intervals (t]1> Til1)* i = 1, 2, ... (if any). Let E^, E^! be the two sets of points wea* which have the following property: weE^CE^1] if and only if there is a se­ quence T)k, k = 1, 2, ..., of ends such that tiA1c+1 ]

(iike[^ +1' ^ ])

where % —

nk—

00

as k-- *-«>. Obviously E^, E^1 are, as E^, bounded, closed subsets of a*. The sets E^, E^? are also non-empty and connected, and hence continua [Cf. H. D. Ursell and L. C. Young, 1, or L. Cesari, 4 7 ]- In addition E^, E^1 C E^ and each point weE^ belongs either to E^, or to E^!, or both. Hence E^ + E^1 = E^ andthen necessarily E^ E^f + 0, since the continuum E^ cannot be the sum of two disjoint continua [G. T. Whyburn, I, p. 16, (10.3)]. (i) Given any two points w 1, ^eE^, there is a sequence 1^, t]2, t)3, t^, ..., ^2k-1, n2k, ..., of ends rjk such that (1) w -- ► w-, w -- w0 as k ---T>2k-1

(2 )

1

r,^]

^ k

with

2

nk +

1

< nk+i;

§19-

ON THE BOUNDARY OP OPEN SETS (CARATHEODORY THEORY) (19-4) k = 1, 2, ... . An analogous statement holds for E^! with nk^A1 > K k+1

k

PROOF. If ^2k^ are any two secluences ends defining w 1 , w2€Ecd' then let *n-j = r \ ^ - Thus t)^€ [tj^,+1 ] and let t}2 = n2k be the first ele­ ment of

(t)i1c) sucil ttLat

n2 > n^ + 1. of

Let

^

(Ti1k) such that

T12k e^ n ' ^n +1 ** with be the first element ^

+1 ] with

n^ > n2 + 1, and so on. The sequence T)k, k = 1, 2, ..., has all the required properties. EXAMPLES. Let Gybe the open set G^ = [o < u < 1, 1 + sin(u“1 ) < v < 3 + sin(iT1)] ; let G q be Gq= [0 < u < 1, 3(2 )"1 (1 + sin(u~1)) < v < 1 + 3 (2 )~1 (1 + sin(u~1))]* let G^ be the (bounded) open set whose boundary G* is the continuum sum of the segments joining the points (-2, o), (-2, 2 ) , (2, 2), (2, o), of the circumfer­ ence r of the ■unitcircle and of the curves

e

= 2~1* ( 2 - p) s i n ( p - 1 )~1 ,

e = 71 1 < p < 2.

In

+ 2 " 1jt(2 - p) s i n ( p - 1 )~1 ,

G^

the point (o, 2.) is the only accessible

point of the segment s = [u = 0, 0 < v < 4 ], and, for the corresponding prime-end c d , we have E^ = s, Ecd= ^u = ° ’ 2 < v < = [u = 0, 0 < v < 2] . Thus E^ E^f is thesingle point (o, 2). In Gq all points of the same segment s are not accessible and s = E^ for some primeend c d , while E ^ = [u = 0 , 1 < v < 4 ] , E ^ ! = [ 0 < v < 3 l « Thus E^ E^ 1 is the segment [u = 0, 1 < v < 3 l- In G^ the point (0, 1 ) is the only accessible point of r (from G^)

CHAPTER VI.

THE CAVALIERI INEQUALITY

and fop the corresponding prime-end

cd

we

have

E^ = r,

Ecd = ^ + y2 = ^ u < ° 1; E(i! = ^ + v2 = 1 , u > 0]. Thus EJ) is a Palp o f points (o, 1 ), (o, -1 ), the former accessible, the latter non-accessible from G^. In the ex­ ample G 1 of (19-1 ) we have E^ = E^ = [u = 0, 1 < v < 3], E^f = w^ = (0, 1 ), where cd is the prime-end corresponding to the only end r) relative to (0, 1 ). In the examples G2, G y G^, we have E^ = E^ = E^1 = C* for the corre­ sponding prime-ends cd.

19 -5 -

Mappings From

a +

{cd}

Given the simply connected bounded open set or, let [t]), (cd) be the collections of all ends and prime-ends of or*. [Since each end t) defines a prime-end oo the inclusion {t|} C {cd} is self'-explanatory.] If 1 Is any cross-cut of or,then 1 divides a into two connected parts , a , and we say that an end -qe {^) belongs to , provided for each end-cut b defining r\, b C a + (w ), there is at least a sub-arc b ! having an end point in w contained in . Analogously any prime-end cdg{cd} defined by a decreasing sequence of In­ tervals [(t^, t^1), n = 1, 2, ...], belongs to ,if, for some n, all ends ^ [ tj^, ] belong to . Given a prime-end cd€{cd}, we say that U(cd) Is a "neighbor­ hood" of cd provided there exists a cross-cut 1 such that, if or1 is the only one of the two parts of a separated by 1 which contains cd, then U is the sum of and of all ends and prime-ends contained in . The reader may observe explicitly that U is a neighborhood of the prime-end cd and not merely of the set E^ corresponding to cd. Indeed, for instance, in G^ where there are three ends t^, t]2, with w = w = w = w , corresponding neighborhoods ^1 ^2 ^3 U(tj1 ), U(t]2 ), U(tj3 ) may be completely different (see illus­ tration). The same in G2 where there are two prime-ends cD y a>2 with E^ = E^ = C*, neighborhoods U(cd1 ), U(cd2 ) may be completely different. u( ,

Let us consider now, as we shall do below, a mapping t from the collection a + (cd) u(r|3) into a closed circle C, which maps each _________ - point wea into a point peC° and each prime-end cdg(cd) into a single point peC*. The concept of

§1 9 . ON THE BOUNDARY OP OPEN SETS (CARATHEODORY THEORY) (1 9*5 ) continuity of such a mapping t at each point wea, or at each prime-end a>e{a3}, can he defined as usual, or by using the neighborhoods U above. We shall now prove the following: (i)

THEOREM. Given any open-bounded simply-connected set oc of the w-plane E2, w = (u, v), and any closed circle C C E2 there is a mapping t from oc + {} into C such that (1 ) t is bicontinuous and one-one between oc and C° and quasi linear between corre­ sponding closed regions in oc and C°; (2 ) t is bicontinuous and one-one between the points weC* and the prime-ends 03e{a)}; (3) If (T, a) is any given c. mapping from a = oc + a* into the p-space E^, p = (x, y, z ) , and T is constant on each set E^, then the mapping (t, C), t = Tt”1 is single-valued and continuous in C [where we mean E^ = t”1(w) in­ stead of oo = t””1(w) for all weC*]. PROOP. The proof is divided into the following parts a, b, c, d. (a) Let w be any point of a, let bd > o be the u * distance of wQ from a , let nQ > 1 be the minimum integer n such that 2“n < d. Por every n > nQ let us consider the subdivision Sn of the (u, v)-plane into squares q of side-lengths 2~n (5*6, Note 2). Por each n > nQ let Dn be the finite collection of all squares Q.£^n completely contained in a. Then Dn is not empty since it contains at least a square q_0€Sn containing w . Let D^ be the subcollection of all squares Q.€Dn which have the following property: there is a finite chain qQ, Q.-]> •••> ^ = 0. of squares of Dr any two con­ secutive of which are adjacent. Let nn be the compact set covered by the squares O^D^. Then n* is a sum of segments s which are the common sides of pairs q, q T of adjacent squares of Sn with qeD^ q C a , q ’a* £ 0, (q, q* closed). We may orient each segment s in such a way that q is at the "left” of s and q 1 at the "right" of s; that is, in such a way that q, s, q* can be mapped by the same linear mapping with positive determinant

305

CHAPTER VI.

THE CAVALIERI INEQUALITY

onto [ ° < u < 1, o < v < 1], [o < u < 1, v = o], [° < u < l, -1 < v say Cn± = z ^ z ^ , joining and C*, let us perform a subdivision [Qf] of C° into a countable collection of simple pol. regions Q 1 analogous to the previous ones, since we divide Q 1 and each partial arc of Qg, Q^, ..., in the same number of equal parts in which we have divided the corresponding «*, and each partial arc of**,5t^, ... . Then the nets E 1*’* + 2 bni, £Q’* + zCnj_ ape topologically equivalent and we can define arbitrarily a q.l. one-one mapping t between corresponding arcs. Thus such a mapping can be extended between correspond­ ing regions and Q S and t satisfies the re­ quirement (1 ) (6.1, i ). Since the ends ^ the points 2 n j_€ Q * are equally ordered and everywhere dense, then each prime-end cd of or [section of ^ q J corre­ sponds to an analogous section in Q * [i.e., a point zfeQ*] and vice versa. Thus t implies a one-one order preserving correspondence between Q* and which we still denote by t . If we consider any interval (Tini> tj .)in * (ti1] then one of the two arcs wm .wnj. of *n together with bni, ^n y defines a crosscut of a such that one of the two parts, say A, in which a is divided, is contained in a - jrn and contains all ends Tl£(Tlni^ ^nj^’ Analogous­ ly let us define the part B of C corre­ sponding to the arc ( z n ± z n p ^ c** Thus A and B are corresponding sets under t. If cd is now any prime-end and z the correspond­ ing point zeC*, if ^nj^ in^0P“ val containing cd [i.e., containing a decreasing sequence of intervals defining c d ], if (zni> z n j) the corresponding arc of C*, let

§19-

ON THE BOUNDARY OF OPEN SETS (CARATHEODORY THEORY) (19-5)

309

A, B be the regions in or and C defined above. Then A is a neighborhood of o> and B a neighborhood of z in C. The fact that A and B, (-nni, Tinj) and ( z n ±z n ^ ape corresponding in t , shows that t is continuous as a mapping between a + (cd) and C. Thus (2) is proved. (d)

Suppose that the c. mapping T be constant on each set E , then the mapping (t, C), —1 o t = T t is single-valued not only on C but also on C since t(w) = T(E£0) for every weC* and where cd is the prime-end image of w under t ~1. Let cdq be any prime-end and zQ be the corresponding point on C*. Given e > 0 there is a p > 0 such that |T(w) - T(wf)l < e for all w, w fea, |w - w f| < p ; on the other hand, there is an interval (r\1, tj11) of {r\) containing cd as above, such that each point w , ^e( 1, t]ti)> is at a distance < p from E . Then o IT(w^ ) - T(Em )| < € for all t)€ (tj », n 11). Let n be an°integer such that d^ < p. We can suppose also n sufficiently large in order that we can restrict (V, ^ f') to be an interval r\! = T)n^,. r\11 = t^. . Then, if A is the corresponding part of a as above, A is divided into countably many regions = [W m , h + l wm+ l,hwm+1,li+ 1]’

m - n'

Each of the points w ^ is connected to a point w^€a* be a segment b of length < 2 ^ < 2dn < p;hence IT(wmi) - T ^ ) ! < e. Each of the points w ’e*'* is on an arc (wm h V h + l )' or on an arc all of diameters < 21 d^ < 21p; hence |T(w!) < 21 e. Each point we*’ is at a distance < 2 (3^ < p from * ’*,i.e.,from some point w T€*!* (otherwise w would be at a distance > 2dm from a*); hence |T(w) - T(wT)l < €. Consequently IT(w) - T(Ea))| < |T(w) - T(w»)| +|T(w') T ( w m i )|

+

^ ^ m i5 "

T(wJ l

< 1 + l T ( w J1 - T ( E m o e + 2 l € + € + e = 2 4 e, for allweA. Conse­ quently, 1 1 (z ) - t(zQ )| < 2 k e for all zeB and thus the continuity of t in Cis proved. Thereby (i) is proved.

CHAPTER VI.

THE CAVALIERI INEQUALITY

NOTE. The regions Jtn of the previous proof have also the following properties: *n C a, *n C (*n+1)°, | a and ||Cn, r|| -- o as n ---- ► ~ where Cn :(T, it*), n = 1, 2, ..., and r: (t, C*), or r:(T, t^}), where (E^) is the order­ ed collection of the sets E^, 19-6.

A First Extension to Sets a Simple Jordan Region

oc

Open in

Let J be a closed simple Jordan region of the w-plane E, w = (u, v), and let a be any connected set open in J, a C J. Thena , a * C J, ococ* C J*, and the set F = $ ( a ) = a* - aa* = a - oc(boundary of a in J ) is a closed, bounded set, and hence the components of F are continua. Let 7 be a component of F. We will suppose in this section that 7 is the only component of F. Thus the definitions of endcut of oc relative to a point w e y , of cross-cut of oc relative to twopoints w.j, w2e y , of pointw ey accessible from a, of end t] of 7 in a, of prime-end m of y in a, remain un­ changed as in (19*1-3); the statements that there are points we7 accessible from a and that they are everywhere dense remain unmodified, as well as the property of separation of four ends ^ of 7 in a; only it may happen here that there is a cross-cut, c = b 1 + cQ + bj in oc, relative to two points w.j, w 2€7 dividing a into two parts one of which contains no end-cut b relative to points W€7 . In this case we can suppose b ^ b2 to be subarcs of J* defining two ends t^, t\2 of y in oc and the ends r\ and prime-ends cd of 7 in a can be linearly ordered (as the points of a closed interval, namely the interval [t^, tj2]. Let us denote by &2 the corresponding two linear orderings of the collections {^3 7,u . {cd)7,oc of the ends j] and prime-ends cd of 7 in a. In the first and third illustration {cd )7, Ct can by cyclically ordered (orderings n.f, 1 fld*); In the second illustration (cd) y ,ex can be linearly ordered (orderings fl2 ). The case where 7 is a single point (fourth illustration) is exceptional and trivial, because the collections {tj}_ 7 yOC. {cd)/ j CC contain each only one single element. The extension of all considerations of (1 9 *1 -4 ) to each component 7 of the set S (a) does not offer difficulties provided & (oc) has only finitely many components and a is connected and open in a closed simple Jordan region J, oc C J .

ON THE BOUNDARY OP OPEN SETS (CARATHEODORY THEORY) (1 9*T )

§19-

19-7 -

As i n

( 1 9 *6 )

w -p la n e

P = ft (a) 7

of

F

The G e neral Case

l e t J be a c lo s e d s im p le

E2

Jo rd a n r e g io n o f th e

and a any con n e cte d s e t open i n is

a c lo s e d , bounded su b se t o f

fo rm a c o l l e c t i o n

{7 }

J;

J

component

7

7,

and th e components

more e x a c tly th e y h o ld ,

o f F such t h a t

{7 , F -

p a r t i c u l a r f o r each component fin ite

th u s

may be u n c o u n ta b le .

c o n s id e r a tio n s o f ( 1 9 * 6) h o ld even i n cases where th a n one component

311

7

of

F

has more

e.g.,

f o r each

7) > 0 ( i f if

F

A l l th e

a n y );

in

F has o n ly

a

c o l l e c t i o n o f com ponents.

L e t us now suppose t h a t no r e s t r i c t i o n i s made on th e components 7. a

N e c e s s a r ily th e r e a re p o in t s

weF

w h ic h a re a c c e s s ib le fro m

and th o s e p o in t s a re everyw here dense i n

be some components or.

An

7

example may be g iv e n as f o llo w s . L e t

where

C

-1 , n * + n— 1 < e < n i t (p, e ) polar coordinates whose pole F = ft (or) = C* + h 1 + h2 + ..., (7} point W€7 = C* is accessible from second exercise.] n

of

P, where b y

1

+ 2it-n

(i)

(ii)

5

= max { ( w ) ,

7)

= max { ( w ) ,

7 1}

G ive n any n = 1, 2, (1 ) (2 )

( iii)

— 1] ,

n = l , 2 ,

o f s e p a ra tio n in J *.

[See

p . 122. ]

for a ll

wel

fo r a ll

[or wel] .

7 ^ ( 7 t h e r e e x is t s a sequence ..., such t h a t &n — ► 0 where l n+1 s e p a ra te s

6n = max { ( w ) , l n and 7 i n

7}

l n, w e ln ;

J; (3) for every 7 x e{ . 7 ) a > 7f + 7 (if* any) there is an n such that ln separates 7 and 7f in J for all n > n T here i s a sequence n = 1, 2, . . . , of fin ite (1)

system s

[U n

[ 1 ] n c [ 1 ] n+1i

...,

o f th e

i s meant a s im p le c lo s e d

Given any two distinct 7>7l€(7)a (if con~ tains more than one element), there exists an 1 C ocseparating 7 and 71in J, and, given e > 0, we can also suppose 5 < e, where 6

+ h 2 + . . . ),

is the center of C. Then = {C*, h ^ h2, ...). No a. [Cf. Newman, I, p. 162,

c u rv e , o r a s im p le a rc whose end p o in ts a re in s ta n c e B. v . K e r e k ja r t o , I ,

(h 1

h i s th e a rc

L e t us r e c a l l h e re th e f o llo w in g p r o p e r tie s 7

b u t th e re may

a = C° -

i s th e c lo s e d u n i t c i r c l e and

hn = [ p = 1 -

components

F,

whose p o in ts a re a l l in a c c e s s ib le fro m

d is jo in t

1

such t h a t

fo r

31 2

CHAPTER VI. (2 )

g iv e n any

THE CAVALIERI INEQUALITY 7 € (7)a

l n , n = 1, 2, in g ( i i ) . Let

7

be any component o f

a new s e t

A = A (a,

o n ly component o f a l l p o in ts The s e t

7)

th e r e i s a sequence

...,

F,

open i n

F,

le t

s a tis fy ­

i.e .,

"We s h a ll d e fin e

J

If

as f o llo w s .

A = a.

7* + y> l e t us d e n o te by P f = P ' ( 7 f >7) w e j w h ic h a re s e p a ra te d by 7 ’ fro m 7

p!

may be empty and, i f

and n o t n e c e s s a r ily co n n e cte d .

n o t, i s

a

A

i s open i n

d e f in e ,

as i n

{(d ) 7, iia

o f th e ends

[n o t i n

J

(19•6 ), o r i n

a] .

F = a - a, t]

and i t s b o und ary

y 1 =)= 7.

7.

of

cd

7)

a ",

any a rc

7,

o f th e

even f o r or.

We s h a ll

f u r t h e r s ta te m e n t: any c lo s e d s im p le Jo rd a n r e g io n , or C

wQ

any p o in t o f

A = A( a,

7),

J

J, F th e boundary 7 6 ( 7 ) ^ any component 7

a c c e s s ib le fro m

b C A + (wQ), h 7 = (wQ),

d e f in in g an end

t]

of 7

in

7, A* Then (1 ) ba i s a non-em pty s e t, open i n (2 )

th e 7

b u t w h ic h a re i n th e la r g e r

and t h i s i s done f o r a l l

J is

F,

7 in

w h ic h we may d e n o te f o r th e sake

cd

a conn ecte d s e t open i n F = a - a o f a in J, of

dense i n wQ; i f l n , n = 1, 2,

. . . , is

b

A, and

any sequence

r e la t iv e to 7 as i n ( i i ) , th e n th e re e x is t s an nQ such t h a t 1R b f 0 f o r a ll (3 )

J

we have d e fin e d a com plete c o ll e c t io n o f

th e

th e s e t

in

T h e re fo re we can

th o s e whose p o in t s a re a l l u n a c c e s s ib le fro m If

$ (A )

( 1 9 * 1 - 4 ), th e c o lle c t io n s

^ and p rim e -e n d s

and p rim e -e n d s

A = A( a,

a ls o need (v )

7'elr)^,

I n such a way, f o r each component

o f b r e v i t y " w it h re s p e c t to set

J,

+ £(7' + p f)

has o n ly one component, th e component

set

set o f J.

The s e t A = A ( a , 7) C J is open i n J, is connecte d and AA* C J * , $ (A) = ~K - A = 7.

Now th e s e t

ends

th e in

c e r t a in ly open i n

where th e sum i s extende d o v e r a l l elem ents The f o llo w in g s ta te m e n t i s e s s e n t ia lly known:

A

i s th e

Let

A = A(a, 7) =

(iv)

7

O th e rw is e , f o r each

n > n Q;

fo r

n > n ,

any VG

sequence [wn J o f

have

wn -- as

p o in ts

n ------- ^°°*

set

§19-

ON THE BOUNDARY OP OPEN SETS (CARATHEODORY THEORY) (1 9 *8 ) PROOF.

Let us consider any end-cut b = w ’w , w 0 €7 ; b C A + (w ), b7 = (wQ ). Let 1 R be the sequence defined in (ii) of polygonal lines 1 and let us consider any point w ^ a separated by 1 ^ from 7 in J. By connecting w 1 to w^ by another simple arc, we can suppose that b is an end-cut wo w o > wo€7' b C A + (w0 ), b 7 = (w0 ). Then, by (ii), we have blR + 0 for each n, and we shall denote by wn the last on b of the points webln [i.e., the arc wn wQ does not contain points of ln besides wn ]• Then, by (ii), the pointsw 1, w2, ... are distinct and ordered on b. Thus wn -- ► w as n -► oo, where w is a point between w^ and wQ, or w = wQ . Suppose, if possible, that the first case holds and denote by 5 > 0 the dis­ tance of the arc c: (w^ w) from 7. Then for all n with dn < 5 we have wR e (w^ w), 5 = {c, 7} < (wn,7 } < max [w, 7} for all w e l n ; i.e., 5< dn, a contradiction. Thus the second case above must hold, w = wQ, i.e., wn -► wQ, and this proves (1) and (2). Let us observe that each point wn €bln+1 is in the arc (wn-1 w Q ) of b and since wn_1 ---► wQ, we have also w ^ ► wQ, and (3) is proved. Thereby (v) is proved.

1 9 •8 ■ Extension of (1 9 •5 , i ) to The G-eneral Case Suppose, as before, that J is any closed simple pol. region that a is any connected subset of J open in J and S (a) is thecompact set % (a) = a - a. Foreach component 7 of $ ( a ) and for the corresponding set A = A(a, 7) defined above we can formulate a statement analo­ gous to (1 9 *5 , i). The following three cases must be discussed separately: (1) J*A = 0; (2) J* C A;

J C E2,

(3)

J* A* +

0,

J* - A +

0

.

In the first case we haveA C J°

and since

8

(A) = A - A

=

A* - AA* = 7, where AA* is a subarc of J*, and 7is a continuum, we have A* = 7 and 7 separates A from Eg - J . Also A is a simply connected bounded open set. Thus the statement (19-5, i) is immediately applicable and hence there

313

CHAPTER VI. exists a mapping

THE CAVALIERI INEQUALITY

from A + { t^), n = 1, 2, ...] is a decreasing sequence of open intervals defining c d ; (2) 1 divides or into two regions r , r^ and ln separates rn from the fix end r\ = °° in a; 1 < n“ (3 ) diam ln n-1, alm ln = 0, m * n, Qfln+1 C i*n, m, n = 1,, 22,, ... . The regions rR above are particular neighborhoods of cd in ^ion a (1 9 -5 )Let us observe that if a point w is the limit of a sequence [w ] of points n = 1, 2, ..., wn -- *-w as n -- ► 00 then by (3), w is also the limit of any other sequence [w^] of points wn€ln> n = 1, 2, ... . We have also w = lim inf 1 = lim sup 1R ( 1 0 . 2 ). We say briefly that w is the limit of a fundamental sequence [1 ] of cross-cuts relative to c d . According to Caratheodory, by principal part E^°^ of E^, is meant the set of all points w which are a limit of some fundamental sequence [ln ] of cross-cuts relative to c d . (i) For e v e r y prime-end cd there is at least one fundamental sequence Ll^] of cross-cuts rela/ \ tive to c d . The principal part E^ ' of E^ is a / (non-empty) continuum E^°^ and \ CD ECDv ' C ECD! ECD11 C ECD . The first and second part of this statement are due to C. Caratheodory [1] the inclusion E^0 ^ C E^ E^1 is obvious since w -- ^w, w ! -- ►w. See also for a proof of (i) L. Cesari ^n ^n [^7] • If the prime-end cd corresponds to an end t) then w T] = E^0 ^ C E ’ E " C E (see for instance the examples G 1 1, CD CD a) CD v ^ (jrj, G^). In the examples G2, G y G^ and for the prime-ends cd discussed above we have E^°^ =CDE /’ = E T! = ECD ; in GQ / \ CD \CD O = [ u = 0, 1 < v < 3], E'° = E 1 E 1T C E . CD ' ~ ( n ) CD CD CD The example G^ shows that E^ may be only a part of the intersection E^ E^T.

CHAPTER VI.

316

THE CAVALIERI INEQUALITY

*1 9 .1 0 . Bibliographical Notes For the concepts discussed in this §19 see the papers and books quoted at the beginning of the section: C. Caratheodory [1], B. v. Kerekjarto [I], H. D. Ursell and L- C. Young [1], and, in connection with the questions discussed in §2 0 , also L. Cesari [4 4 , 4 7 , 4 8 ]. For the concepts of right and left wings of a set (1 9 -4 ) see H. D. Ursell and L. C. Young [1 ] (the exposition of (1 9 -4 ) is closer to L- Cesari [18J). For the proof of (1 9 -5 , i) see B. v. Kerekjarto, I, pp. 110-115. The extension (1 9 •7 ) of prime-end theory to every component 7 of the boundary S (a) = a - a of any bounded set a C J (open in a simple closed Jordan region J) through the con­ sideration of the auxiliary set A = A(a, 7) is given above in view of the application to contours in §2 0 . See for further details L. Cesari [4 4 , 4 7 , 4 8 ]. On the general problem of the boundary of plane open sets see also G. Choquet [1 ], F. Frankl [1 ], E. Hopf [1 ]. On the extension of prime-end theory to open sets in E^ see B. Kaufmann [1 , 2, 3] and S. Mazurkiewicz [1, 2, 3]. On the general problem of compactification of topological spaces and groups see, e.g., H. Freudental [1, 2, 3], H. Hopf [1], and also the recent book S. Eilenberg and N. Steenrod [I].

§2 0.

CONTOURS OF A CONTINUOUS SURFACE AND THE CAVALIERI INEQUALITY 20.1.

Contours of A Continuous Surface

Let (T, Q):p = T(w), wej, be any continuous mapping(surface) from the simple closed Jordan region of the w-plane E2,w = (u, v), into the p-space E y p = (x, y, z). Let f(p)> peE^, be any realsingle-valued continuous function in E^, For any real t, - 00 < t < + 00, let C = C(t), D+ = D+(t), D~ = D“(t), be the sets of all points wej, where f [T(w)] = t, or > t, or < t, respectively. Since f [T(w)], wej, is continuous in J, then C is closed, and D+, D” are open in J (or empty). In any case we have C 3 S(D+ ) = T)+ - D+, C ) 5 (D“) = D ” - D~. If M(t) is the subset of E^ where f(p) = t, then we have also T(C) CM.

NOTE 1.In elementary cases C is simply the contour (a single line, or a finite system of lines) corresponding to the value t (level). In the general conditions above it may happen that the two sets 3f(D+ ), $ (D ) do not coincide and that C has interior points, besides the customary complications of the boundaries of open sets. _Thus T(C) is a general closed set of E3. As we approach C from D , or from D*, we can say that we approach the "lower border", or the "upper border" of the contour C, but we will not attempt to introduce a terminology which is only suggested as a help to the reader.

NOTE 2. It is not restrictive to suppose that f is defined only on the compact set [S], even in the hypothesis, we shall consider later (20.5, i) that f satisfies a Lipschitz condition

§2 0 .

CONTOURS AND THE CAVALIERI INEQUALITY (2 0 . 2 )

317

|f(p) - f(p!)l ^ K|p - p f|. Indeed it is possible to extend the domain of definition of f to the whole space E3 in such a way that continuity holds in Eo, or the Lipschitz condition above holds in Eo [E. J. McShane, 8].

20.2.

The Generalized Length

In the conditions of (20.1 ), let {a) = {a)t be the collection of all components a of D ~ ; hence each a is a bounded connected subset of J, open in J. For each a e [ a ) let be the collection of all components 7 of the boundary ft (a) = a - oc of a in J; hence each 7 is a subcontinuum of J. For each c c e { a ) and 7 e {7) let A = A (a, 7) be the set defined in (1 9 -7) ^ and let (n) A, {cd} A be the collection 7) a 7f of all ends, and prime-ends of 7 in A. We shall suppose that an order­ ing Q has been chosen for the collections (tj) {c d } ^ [q = n-j, n2, or ft = nj, n£] (1 9 -4 ; 19*6). Let [r)J = [t]1> ti2, •••, T)n ] be any finite subfamily of ends ^ ( ti) a ordered asthey are in {-n3 7 , -tX A. Let us consider the corresponding set [w ] = [ w-, wn, ..., w 1, w. = w €7, i = 1, 2, ..., n, of points of T) 1 ^ n ^i 7, each w^ being accessible from A [not necessarily from a ] , and let S be the sum S = z |T(w^) - T (wi+1)|, where z is extended over all the values i = 1, 2, ..., n-1, if ft is one of the orderings n.j, n2, and 1 = 1, 2, ..., n, wn+] = w ^ if n is one of the (cyclic) orderings n*, Finally let X = Sup S, where Sup is taken for all finite ordered subfamilies [ r \ ] C {i]}^ We shall denote X also by the more complete notation x ( y , ctr). We have 0 < X < + 00. The number

(1)

K t ) = K tj t, j, f) =

y,

X a)

ae{Qf}t 7€fr)a shall be denoted by the generalized length of the image of ft (D~) under T briefly, of the image of thecontourC(t). Analogous definition for ft (D+ ). We could denote l(t)also as the (generalized) length of the lower (upper) border of the image T(C) of the contour C of level t. We shall prove below (iv) that the generalized length is essentially a sum of Jordan lengths of continuous curves. Let us observe explicitly that in (1) the sum withrespect to 7 may be uncountable. It can also be observed that if a component } is a single point, then the collection ft]}^ ^ contains a single element and, according to the defi­ nition above, we have X = 0. Thus all components 7 which are single points have no influence in the value of l(t). The following statements hold: (i) x ( y , oc) = 0 if and only if 7 is a continuum of

31 8

CHAPTER VI.

THE CAVALIERI INEQUALITY

c o n s ta n c y f o r T i n J . In p a r t ic u la r f o r a l l 7 w h ich a re s i n g l e p o in t s . PROOF. I f T f i n i t e system x = Sup S = o.

X = o

i s c o n s ta n t on 7, th e n f o r each [ tj] we h ave S = 0 and hence V ic e v e r s a i f x = o, th e n we must

h ave S = 0 f o r a l l I t\]; hence |T(w ) - T ( w ’ )| = 0, T(w T) ) = T ( wT[f ), f o r a l l ends Ti w' Vl elU ) . Thus T i s c o n s ta n t on th e s e t o f th e p o in t s we7 a c c e s s ib le from A = A (a , 7) and, s in c e T i s c o n tin u o u s , T i s c o n s ta n t on 7. l ( t ) < + 00 im p lie s t h a t a l l numbers x(y, a ) a re f i n i t e and t h a t a t most f o r a c o u n ta b le su b ­ f a m ily o f s e t s 7 we have A.(7, a) > 0.

(ii)

PROOF. In d e e d , f o r each in t e g e r n, n = 1, 2, . . . , o n ly f i n i t e l y many term s x ( y , a') o f th e sum (1 ) may be > n " 1. Thus a t most c o u n ta b ly many term s o f th e sum (1 ) may be > 0 . A l l o th e r term s must be z e r o . T h e re b y ( i i ) i s p ro v e d . ( iii)

X (7, a) < + 00 im p lie s t h a t , f o r each p rim e end coefo) 7,ai,i T i s c o n s ta n t on th e s e t ECD .

PROOF. L e t E ^ , E ^ ! be th e s e t s d e fin e d i n ( 19*10 ; th u s E^ = E^ + E ^ ’ , E^ E ^ ! 4 0. I t i s enough to p ro v e t h a t T i s c o n s ta n t i n E f as w e ll a s i n E ^ ! . Suppose, i f p o s s ib le , t h a t T i s n o t c o n s ta n t on E ^ . Then t h e r e a re two p o in t s wQ l , ^ o 2 e^ su ch t h a t |T(wQ l ) - T (w q 2 )| = 31 > 0. Then we can d e te rm in e a sequence [t^] o f ends iln € ^ l 7 a ' n = 1, 2, . . . , su ch t h a t , i f wR = w^ , n = 1, 2 ,

. . .,

W2n-1---" woi'

-< tj2 -< r\^ < •• •> and

we h ave

W2n---^wo2

as

n -- i}•

T h e re fo re each f i n i t e system [ t^ , i s a f i n i t e f a m ily [ rj ] as a b o v e .

t)2 ,

. . . , t)2^ +1 ] On th e o th e r hand

we can suppose t h a t a l l p o in t s w2n_i> w2n c lo s e to wQ l , wq2 i n o rd e r t h a t

|T(w2n-i} " T(wol)l
|T(wQ l ) - T ( wq 2 )| - 1 - 1 = 1 > 0 .

§2 0 .

CONTOURS AND THE CAVALIERI INEQUALITY (2 0 . 2 )

319

Finally the sum S relative to [t^, t]2, • 'n2N+1 ^ satisfy the relation S = z |T(wn ) - T(wn+1)| > 2 N1 , where z is extended over all n = 1, 2, ..., 2N, and N is any arbitrary integer. Hence X = Sup S = + 00, a contradiction, and thus it is proved that T is constant on E^. The same holds for E^! and finally for E^. Thereby (iii) is proved. (iv)

If X = x ( y , a ) < + 00 then T Is constant on each set ECD, cdcIcd) 7,aA and C: (T, E C )D , cd€{cd}^ a is a continuous curve C of Jordan length 1(C) = x .

NOTE 1. The proof which follows holds also in the case X = + provided T is constant on each set E^. In the statement (iv) the collection Ccd37 ,Aa, . is thought of as ordered in any one of the orderings ft [ft*, ft.^, or ft1, ftg]. PROOF OF iv. Let A = A(7, a ) , 1 = A + 7. By (iii), T is constant on each set E^ C7. For the sake of simplicity let us consider onlyone of the three cases defined in (19-8), namely, the first case JA* = 0. Then there is a continuous mapping t, as in (1 9 *5 , i), from A + (cd) *into a closed circle Q, which is one-one between A and Qo and between (cd) fl 7> and Q*. If (t, Q) is the mapping defined by -1 t = T t -1 , where tmaps each point weQ* into the set E^ corresponding to cd, then (19-5, i) t is single-valued and continuous on Q, in particular on Q*. Hence C is the continuous curve C: (T, Q*) image of Q* under t. By the definition of X = x (7, a ) we have x = Sup S 1 , S ’ = z |t(w^) - t(w^+1)|, where Sup is taken for all finite systems [w^, i = 1 , 2 ,..., n], of points w^eQ* which correspond under t to points of 7 accessible from A (ends of 7 in A), while the Jordan length 1 = 1 (C) of C is given by 1 = Sup S, where S denotes a sum like S ! and Sup is taken for all finite systems [w^] of points of Q*. Therefore we have X < 1 .On the other hand, the points of Q*corresponding under t to

320

CHAPTER VI.

THE CAVALIERI INEQUALITY

accessible points of 7 are everywhere dense in Q*; hence each sum S can be approached as we wish by sums S* and, therefore, 1 < X. Thus the equality 1 = X is proved and thereby also (iv) is proved. (v)

If l(t) < +00, then the numbers x ( y , a ) countable collection the countable sum of continuous curves r

This statement is a consequence

for each- aefo'}^ y e { y are all zero but for a of continua 7, l(t) is Jordan lengths of (ordinary) in E^. of (1) and of (i), (ii) and (iv).

NOTE 2. In the previous lines we have supposed implicitly that neither D” = D”(t), nor C = C(t), is empty. If either D~, or C is empty let l(t) = 0. Also, wehave considered only D”(t) and the correspondingcollection (a)^ ofthe components a of D~(t). Identical considerations hold for D+(t). We could denote by l“(t), l+(t) the two functions l(t) de­ fined by using D~(t), or D (t) for all t. By examples it is easy to show that it may happen that l~(t) =)= l+(t) for some t. In the follow­ ing lines the function l(t) is always the function l~(t).

20.3. Properties of The Length

l(t)

In the present number we denote by J, JR simple closedJordan regions of thew-plane Eg, w = (u, v), and by (T, J), (Tn, Jn ) c.mappings from J, Jn into the p-space E^, p = (x, y, z). (i) Givena c. mapping (T, J) and any sequence f(p), fn (p)> P e^ , n = 1, 2, ..., of c. functions in E3 such that fn (p) > f(p), fn (p) = $ f ( p ) uniformly In E^ as n -°o, If l(t) = 1 (t; T, J, f), ln (t) = 1 (t; T, J, fn ) are the functions defined in (20.1), then l(t) < lim ln (t) as n -^ for all - 00 < t < + 00. PROOF. If l(t) = 0 the statement is trivial. Suppose l(t) > 0. Then D”(t), and C(t) are non-empty sets In J. Given e > 0, by the definition of l(t) there is a finite system [ a] of disjoint components

§2 0 .

CONTOURS AND THE CAVALIERI INEQUALITY

(2 0 .3

)

of D“(t), ae{o'}“, and, for each a , there ex­ ists a finite system of continua such that X

zL

Qf€[or]

ye[y]a

x(7, or) > l(t) - e ,

[or > e“1], according as l(t) < + «>, or l(t) = + 00. Let M denote the total number of components 7 of the systems ^7^a > oce[oc]; hence 1< M < + For each are[a] there must be a finite system [ 13 of polygonal lines as in (19-T) (simple closed pol. lines, or simple pol. lines whose end points are on J*), such that 1 C a for each letl]^ and such that, for each y e [ y ] a , there is a line le[l] separating 7 from all other 7Te[7 ]Qr, y x + 7, in J ( [ 7 ] ^ ls finite). For each ye [ 7 ] there must also be a finite system [t\] of ends t\ of 7 in the set A = A (or, 7) [i.e., [t]] C {ti) a] suchthat, if S is the corresponding -1 sum (20.2), we have X > S > X - eM if X < + «>, S > €~1 if X = + °°, where X = x(7, a ) . For each t j € [ t) ] we consider also an arc b such that b^ = Cw^), b C A + (w ). If ^-s collection of the arcs so determined we can suppose b b f A = 0 for any pair of arcs b, b f€[b]^, b + b*. Let N be the maximum number of elements of the M collections [ ti] and [b] . 7

I

*7

7

Let 5 > 0 be a number such that If [T(w)]- f[T(w')]| < e M-1 N-1 for all w, w'ej, |w - w 1| < 5 . We can suppose that all arcs b have diameter < 5 . Let us choose in each or apoint w • Thenfor each 7c [7 ] , we can suppose that the polygonal lines letl]^ separating 7 from all 7? ^ y> 7!e[7]^, separates also wQ from 7. We can also suppose that all points wel are at a distance < 6 from 7 and that 1 crosses all the arcs be[b] . Finally, we can suppose thatfor each be[b]^ we havebl =(w), where w is the end point of b not in 7. Thus 1 separates all arcs be[b]^ from wQ and fromall 7* + 7> 7*^7]^ in J.

321

322

CHAPTER VI. Let t C

THE CAVALIERI INEQUALITY

be asimple pol. line joining wQ to 1 , and whose points are separated by 1 from 7 in J. Let k be the continuum k = w + Z t + zl, hence k C a C D” and, In consequence, if t a = max f [T (w)] for all wek, we have t < t. Let 2a be the minimum of the differ­ ences t - ta for all a e [ a ] . Then we have t < t - 2 c r < t for all t

a,

ae [or] . Let t2 and t ln, tp be the min and max of f[T(w)], and f [T(w)] for all wej. Then we have t 1 < t 1 , t2 < t2 , t1 < t2 and also, since D”(t) and C(t) are both non-empty, also t^ < t < t2 . In addition, t ln-- ►t.j, t2 -- ► t2 as n -hence there is an n^ . < t < t0 o such that t-1 < t in 2 < t~ 2n for all n > nQ . As a consequence the sets D”(t) and Cn (t) analogous to D”(t) and C for the function fn, are non-empty for all n > nQ .Let n 1 be the smallest integer n 1 > nQ such that |fn (p ) - f(p)| < min [ a , e M “ 1 N"1] for all peE3 [or at least for all peT(J)]. Hence fn [T(w)] < t - o < t for all wek, n > n 1,ore [a]. On the other hand, we have fn [T(w)] > f[T(w)] = t for all w €7, 7 e [7 ] , are[or], and all n. All this implies that the continuum k Is contained in a unique component & n of the set D“(t), while each y is neither in a n nor in a*. This implies further that the set Fn = or* - Q?na* separates an from all 7e[ 7 in Q;hence for each y a certain component 7n of Fn must separate k^ from y .Consequently b7n + 0 for each be[b]^ relative to y and, if wn is the last point of b7n on b leaving 7, then wR is an accessible point of yn from A(a , 7n ) and the subarc b 1 of b from wR to 1 defines an end t] of 7 in A (a , 7 ). Each collection ° ? these ends t] is ordered as [tj], and, if Sn denotes the corresponding sum and we observe that each arc b has diameter < 5 , we have Sn > S - N[2 - eM-1N-1]; I.e., —1 Sn > S - 2eM . The components y n just now obtain­ ed are certainly distinct (since they are separated in J by the same lines letl]^). Therefore we have

§2 0 . CONTOURS AND THE CAVALIERI INEQUALITY (20.3 )

X(7 n> «n ) > S 1n (t) ^

2 eM-1 >

X-

eM-1 -

Z Z V ane[an ] ’'ne[?'n]

2eM-1,

>

(2 ) >

Y,

X

oreta]

y e [ y ]

X( y>

«) - M • 3eM-1 >

> l(t) - c - 3e = l(t) -

he

for all n > where t7n l denote the systems of regions orn C a , and of components determined above. If l(t) = + °°, or at least one of the numbers A.(7, os) = + oo, then (2) has to be replaced by ln (t) > • All this proves that l(t) < lim ln (t) as n ---> « and (i) is proved.

yr

NOTE. The statement (i) holds even if the parameter n is a continuous parameter nQ < n < + °° with n -- > + 00, or n < n < n- with n -- ► n1 - 0. o i

(ii)

Given a c. mapping (T, J) and any c. function f(p)> P 6^ , then, if l(t) = l(t; T, J, f), we have l(t) < lim 1(t) as t -- > t - 0 for all - 00 < t < + 00.

PROOF. For every t < t let a = t - t , f(J(p) f(p) + cr, peE3’ Tben we have f0(p) z ^ f ( p ) uni­ formly in E^ as a -- * 0 + and fa(p) > f(p) for all Pe^ and a > 0. We have also, with obvious notations 1 (t) = la(t) = l(t; T, J, f0)* Therefore, the present statement is a consequence of (i) and the note above, if only we observe that instead of the sequence [fn (p), n = 1, 2, ...] we have the system tfa(p), cr > 0] ordered for decreasing cr as a -- > 0 +. (iii) Given the c. mappings (T, J), (T , Jn ), n = 1, 2, ..., with TR -- > T [i.e.,

32^

CHAPTER VI.

THE CAVALIERI INEQUALITY

J° f J° as n > oo] , and given any c.function f(p), psE3, if l(t) = l(t; T, J, f), ln (t) = l(t; Tn, Jn, f), then for all - oo < t < + oo ve have (3)

Kt)
t-o n -- > oo

PROOF. Given € > o let us proceed first as in the proof of (i) till the determination of the number a = a (e ) > 0 and let a * be any arbitrary number 0
o where 5 n =max |f[Tn (w)] - f [T(w)J |for all weJn Let n 1 be the smallest integer such that 5 n < min [a1, N“1] for all n > n 1. Then, as in the proof of (i), we have f[Tn (w)] < t - 2a + a» for all wek^, a e l a ] and f[T (w)] > t - a* for all W€7, y e [ y ] , ae[cc] • This implies firstthat, for all t suchthat t - 2 a + a , < T < t - a f and all n > n1 we have E)”(t ) + o, ^n (T) + °> where D”(t ), C (x) arethe sets of all wej where f[TR (w)] < t , or = t , respectively. Proceeding now as in the proof of (i) we have 1(t) > l(t) for all n > n1 and t - 2 a + a f < T < t - a f, provided l(t) < + ». If l(t) = + », then the second member has to be replaced by e“1 as in the proof of (i). This assures that, if cp(r) = lim ln (T) as n --- >00, we have also cp(t ) > l(t) [or > e”—1 ] for all t - 2a + a!< t < t - a 1, whereaT arbitrary number

o < a ! < a.

cp(t )> l(t) - b e for all t l(t) < lim cp(t ) as t --- -

for

is any

Therefore we have

- a < t < t, and finally o. Thereby (iii) is

proved.

NOTE. In (iii) the function l(t) is a function l”(t). For a function 1 (t) = l+(t); i.e., determined byusing the components oc ofD+(t) instead of D~(t), (iii) is replaced by l+(t) < lim lim where the interior limis taken as n -- =>oo and the exterior lim is taken as x -- > t + o.

§2 0.

(iv)

CONTOURS AND THE CAVALIERI INEQUALITY (20.4)

LEMMA. Every real single-valued function F(t), a < t < b, - o o < F(t) < + 00 satisfying the relation (a)F(t) < lim F(t ) as t -> t - 0 for all a < t m. Now cp(t, m) takes only the values 0 and 1. Let us prove that cp(t, m) satisfies (a). By (a) for the function f we have that, for every t, a < t < b, and e > 0 there exists 5 > 0 such that f(£) > f(t) - e for t - 6 < | m and we choose e > 0 small enough so that f(t) - €> m, then we have also f(|) > m for t - 5 < |< t; hence cp(t; m) = 1 , cp(|;m)=l for all t - 5 < £ < t; if f(t) < m, then cp(t; m) = 0, cp(|, m) = 0, or 1. In any case cp(t, m)satisfies (or). Let us prove that cp(t, m)is measurable. If, for a certain t, we have cp(t, m) = 1, then, by ( a ) , it follows that for all g of a left neighborhood of t we have cp(j*, m) = 1. Hence the set E lm of the points a < t < b where cp(t; m) = 1 is a sum of maximal disjoint intervals open at the left, closed at the right, and E lm is thus measurable. Consequently, the set Eom = (a, b) - E lm where cp(t; m) = 0 is measurable. This proves that the function cp(t, m) is measurable. Hence, the point set where f(t) > m is measurable and therefore f(t), as well as F(t), are measurable. Thereby (iv) is proved.

(v)

Given the c. mapping (T, J) and any c. function f(p), P€^ , the function l(t) = 1 (t; T, J, f) is measurable in y ( -

CO

+

00 ) .

This statement is a consequence of (ii) and (iv).

20 .4 . Two Elementary Lemmas (i)

If if

A is a triangle of the w-plane E2, w = (u, v), cp(w) = au + bv + c is a linear non-constant

325

CHAPTER VI.

THE CAVALIERI INEQUALITY

f u n c t io n , if t ^ , t g a re th e cp(w) i n A, if x(t), t1 < t ment o f A on w h ich cp(w) = t d en o te s a ls o th e le n g t h o f A. ( t J,

(a

p

P

+ b )

2

a re a A =

r*tr2

J

m in and max o f < t 2 , i s th e s e g ­ and x ( t ) ), th e n \ ( t ) dt

t 1

PROOF. The linear function cp(w) has its min and max In two vertices w ^ w2 of A; hence, if w^ is the third vertex and t^ =cp(w^), we have t1 < t^ < t2 . Each segment \ ( t ) is on the straight line r(t): au + bv + c - t =o of E2; hence all segments x(t), t1 < t < t2, are parallel, and x(t^) divides A into two triangles A'1, A2 where one may be reduced to a segment. In addition the function \ ( t ) is continuous in tt^, tg], linear on each of the two intervals [t^ t^], [t^, tg], and x(t1) = x(t2 ) = o. Therefore "2

2~1 X ( t 3 ) ( t 3 - t , ) = 2_1

[ \ ( t 1) + X ( t 3 )]

( t 3 - t 1) =

to A(t) dt, and also area A 1 = 2“1 *.(1^) 1{r(t1), r(t^)) = A 2~1 - t1 ) a"1 where a = (a2 + b2 )2 and (r(t1), r(t^)) denotes the distance between the parallel linesr(t-), r(t ). By comparison of the two previous * 3 uo relations we have a area A 1 = f , A(t) dt. Analogously r 1 a area Ap = J A.(t ) dt and by adding these two re3

lations we have (4 ). ii)

Given € > 0, a compact set K C E^, and any Lipschitzian real function f(p), peE^, with If(p) - f(pr) l < M | p - p l| for every p, p'eEj^ and some constant M, then there exists a q.l. function cp(p), P€% > with |cp(p) - f(p)| < e, grad cp < M + € everywhere in K.

PROOF. Forthe sake of simplicity let us suppose N = 2. Let f^(p), peE2, p = (x, y), be the mean value integral of order n relative to the func­ tion f (32.1). Then f ^ ( p ) is continuous in E2,

§20.

CONTOURS AND THE CAVALIERI INEQUALITY (20 .5 )

has partial derivatives f\Xin ^, f,j ln ^ continuous in E2, and is Lipschitzian in E2 with the same constant M (32.2, ii). Hence both partial derivatives, as well as the derivative in each direction of f^n ^, are in absolute value < M. In addition, we have also grad f < M. On the other hand, if Q, is any closed square containing K, we have f^n ^ ^f uniformly in Q. Thus we can fix n in such a way that |f^n ^ - f| < e/2 in Q. Let us divide Q into a finite number of equal squares q by means of straight lines parallel to the sides of Q, and let us suppose that the squares q are so small that the function f a s well as the partial de­ rivatives f^n ^ have an oscillation < e/2 in each q. Let us divide each q into equal triangles t by means of one of the diagonals and let cp(p), peQ, be the q.l. function which is linear in each triangle t and takes the same values of f^n ^ at the vertices of each t. Then the partial derivatives q>x, ep^ of cp in each t are constant and equal to the derivatives f x i n \ fTin ^ of f^n ^ y at convenient points of the legs of the (right) tri­ angle t (5 -1 3 , Note 2). If A, B are the values of the same derivatives at any point pet we have |cpx - A| < e/2, |cp - B| < e/2 everywhere in t. Consequently Igrad cp(p^) - grad f^n ^(p)| = |(cp2 + cp2 ) 2 - (A2 + B2 )2 | < I® - A| + I® - B| < e. Finally, ^ grad cp < grad f (r\) + e < M + e, y and also |

t2. If we consider the sets D“(t), D+(t), C(t) in Q, then the boundary of D"(t) in J is the sum of seg­ ments X(t ) contained in triangles t and whose images are the segments l(t), plus sides of triangles t of constancy for T, plus segments \ ( t ) as above whose images are single"points. Each of the segments x(t) of the first category separates D”(t) from D+(t) in t. For all other segments f[T(w)] is constant on each of them. Hence there is at most a finite collection [t] of values of t such that x.(t) may not separate

§2 0 .

CONTOURS AND TEE CAVALIERI INEQUALITY

(2 0.5)

D~(t) and D+(t) in t . For all t not in [t], the number l(t) = l(t; T, Q, f) is equal to the sum El(t) of the numbers l(t) for all t€[t]• By (2 0 .4 , i) we have

l(t) dt =

Thus (5) is proved for both f(p) quasi linear. (b)

(T, Q)

and

Let us suppose (T, Q) is still quasi linear in a simple pol. reg. Q C E2 and f(p), peEy be any Lipschitzian function in E^ with constant K > 0 . By using (20.4, ii) there exists a sequence f^(p), P€E3' n = 1 , 2 , ..., of continuous functions, q.l. in E~, such_that |f*(p) - f(p)| < n“1, j . j ii grad ^(p) < K + n , everywhere in E^. If fn (p) = ^(p) + n-1' PeE3> we have f(p) < fn (p) < f(p) + 2 n _ 1 , grad f = grad f^ < K + n-1, for each p€E3- By (a) we have (K + n” 1 ) a(T, Q) > / ln (t) dt where the range of integration is (- », + «) and ln (t) = 1 (t; T, Q, fn ). By (20.3, i) and Fatou!s lemma [S. Saks, I, p. 29] we have, as n -- > 00, K a(T, Q) > / l(t) dt. Thus (5 ) is proved for all Lipschitzian functions f(p) and (T, Q) quasi linear.

(c)

Let us suppose now that (T, J) is any c. mapping and that f(p), PeE3 anY Lipschitzian function of constant K > 0. If L(J, T ) = + 00, or K = 0, or both, (5 ) is trivial. Therefore let us suppose K > 0, L(J, T ) < + 00. By (5*6, iv) there is a sequence (T , Qn ) of q.l. mappings from simple pol. regions Qn such that

329

330

CHAPTER VI.

THE CAVALIERI INEQUALITY

On C On--C>0, . By (b) we have now KTn ) > / ln (t) dt, where the range of integration is (- °°, + »). As n — and by Fatou!s lemma, we have K L(J, T ) > / 9 (t ) dt. For every h > 0 the substitution t = t r -h, gives K L(J, T) > / cp(t - h) dt and, since, by (20.3, iii), l(t) < lim cp(t - h) as h -- * 0 +, by Fatou!s lemma we have finally, K L(J, T) > / l(t) dt. Thereby (i) is completely proved-

*20.6.

Bibliographical Notes

For formulas analogous to (20.5; 5) involving k - and (k + 1)-dimension­ al Hausdorff measures (2.13), see S. Eilenberg [1], S. Eilenberg and 0 . E. Harold [1], L. C. Young [1, 10], A. S. Besicovitch [10], E. R. Reifenberg [1]. In the last three papers such formulas are extended to parametric continuous surfaces [k = 1] and in the last two papers formulas in­ volving Lebesgue area are also given. On this subject see H. Federer [8]. The present discussion concerning (20.5, i) and the concept of generalized length defined in terns of ends as the Jordan length are published here for the first time (Abstract Bull. Am. Math. Soc., 5 7 , 1 9 5 1 , p- 168). For further discussion see L. Cesari [ k k , k l , 4 8 ]. For application of (20.5, i) see, besides §§20, 2 k , 36, also R. E. Fullerton [1, 2] on the existence of special regular representations (9*6) of non-degenerate sur­ faces (§3 5 )- For smoothing processes of contours see R. E. Fullerton (loc. cit.) and L. Cesari (loc. cit.)Formula (20.5; 5) is an extension of elementary formulas often used. for instance L. C. Young [1], L. Cesari [ k 2 ] , H. Lewy [3].

See

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

**************************************************************************

§21.

THE EQUALITY

V = U 21.1.

The Indices

d, m, a

Throughout the present chapter we shall denote from an admissible set A of the w-plane E2, p-space E^, p = (x, y, z), and by (Tp, A), mappings which are the projections of (T, A) xy-planes E21, E22, E23*

by (T, A) any c. mapping w = (u, v), into the r = 1, 2, 3, the plane on the coordinate y z , z x ,

Let F be any figure (5 - 1 ) , F C A, and S = S' + S M be any finite sub­ division of F into simple pol. regions qeS1 and non-simple pol. reg. ReS'1, R = (pQ, p ^ ..., p^), (5 -1 ), and let dp, ap, r = 1, 2, 3 , be the indices of S with respect to (Tp, A) (12.6). Hence dp = max diam [Tp (q), Tp (R)] for all qeSf, ReS11; q*)|, where z^ ranges over all regions qeST with q C F°; ap = zM z* diam Tp (p*), where Z* ranges over all boundary curves p*, ..., p* of R and Z !! over all regions ReS1!. Let us denote by indices d, m, cr with respect to T of the subdivision S f + S 1r of F the non­ negativenumbers d= max diam [T(q), T(R)] for all qeS1, R€SM , m =maxmp, r = 1, 2, 3, and cr= z ff z* diam T(p*). Obviously we have dp < d, < m, cxp < a, r « 1, 2, 3. (i) If the mappings (Tp, A), r = 1, 2, 3, are all BV, then for every figure F C A and positive number e there are decompositions S = S 1 + S ,! of F with d, m, a < e . PROOF. Since F C A the mappings (Tp, F), r = 1, 2, 3, are BV. Let = Tp (F) and Fp C be the subset of defined in (16.10, Note), r = 1, 2, 3, Since all three mappings (T , F), r = 1, 2, 3, are BV, we have |Fp | = 0, r = 1, 2, 3, (16.10, i). Let Hp C Kp be any closed totally dis­ connected set such that - H^l < €, Hp Fp = o (12.6, Note), where we can suppose vacuous If |Kp| < s. Finally, let « = T^1(H1) + T"1(H2 ) + T”1(H3)• Since HpFp = 0 each component of the closed set Tp1(p), psH^ is a continuum gcr(T, F), and since 331

332

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

each Hp is totally disconnected also, the components of the closed sets T~1(Hp) are continua ger(T, F). The same holds for $ . Therefore we can apply (1 1 -3 , ii) to each of the N (disjoint) components Q, (pol. regions) of the figure F, where we replace € by N~1e. Then there is a subdivision S = S f + S f! of F into simple pol. regions qeS1 and not simple pol. reg. ReSM , R = (pQ, p ^ ..., p^), such that diam T(q) < €, diam T(R) < e, z 11 Z* diam T(p*) < e, where z!t, z* are extended over all R€SM and over all boundary curves p*, p*, ..., p* of R, and such that Tt* # = it* F* $, R* £ = R* F* $ . We have d = max diam [T(q), T(R)] < e, a = Z 1 1 Z* diamT(p*) < and finally nip = |Tp (z^ q*)| < |Kp - Hp| < e, r = 1, 2, 3, where Z^ ranges over all qeS!, q C F° and finally m = max mp < €. Thus d, m, a < €, and (i) is proved.

€

NOTE. Because of it* £ = it* F* § , R* £ = R* F* $, it followsthat, if M is the set M = z! q* + z!! R*, where z 1, Z M range over all qeS1, R€St!, we have Tr (M F°) C Kr - £ , r = 1, 2, 3, and hence |Tr (M F°)| < €, r = 1, 2, 3.

For later use we add here also the following observation. By using the no­ tationsof (5 .4 ) and (5*11), let t : q = t ( p ) , be any orthogonallinear transformation from the p-space E^, p = (x, y, z), onto the q-space q = t), £)> (change of orthogonal cartesian coordinates) and let (T1, A) be the mapping defined by T f = tT. Let (Tp, A), (T^, A), r = 1, 2, 3, be the plane mappings which are the projections of T on the yz, zx, xy-coordinate planes In E^ and of T ! on the t](;, coordinate planes in E^. For each figure F C A and subdivision S = S1+ S ,! of F as above we have certain indices d, m, a of S with respectto T and d 1, m S cr1 of S ! with respect to T 1. We have already proved (18.10, vi) that the mappings (T ,A), r = 1, 2, 3, are all BV if and only if the mappings (T^, A), r = 1 , 2, 3, are all BV. (ii) If the mappings (Tp, A), r = 1, 2, 3, are all BV, then for every figure F C A and positive number € there are decompositions S = S r + S M of F with d, m, a < €, and d !, m !, a1 < €. PROOF. First let us observe that t is defined as in (5*7) by real numbers a, b, c and an orthogonal matrix H with determinant |H| = + 1, hence + 0.

§21

. THE EQUALITY V = U (2 1 .2 )

Therefore the functions x(w), y(w), z(w) (components of T(w)) are constant on a set I C A if and only if the functions£(w), t] ( w ), £(w) (components of T f(w)) are constant on I; hence the two collections r(T, F) and r(T!; F) [1 0.k ]coincide. Now let us proceed as for (i) where we consider certain sets H^ C Tp (F), r = 1, 2, 3, and analogous sets HJ, C Tj,(F), r = 1, 2, 3 > defined quite independent­ ly, and whose properties are listed above, i.e., Hp, are closed, totally disconnected, |Tp(F) - Hpl < e, |T£(F) - HJJ < €, and HpFp = 0, H^F^ = 0, where Fp, F^ are the sets defined in (16.10, i) relatively to the mappings Tp, T^, r = 1, 2, 3 - Now let « = T”1 )+ zrT^“1(H^). The subdivision on S, defined by making use of this set £ ,satisfy both requirements d, m, o < e, and d 1, m !, o ' < €, as in the proof of (a). Thereby (ii) is proved.

21.2.

The Function V

As A Limit

The statement (12.8, i) for plane mappings can now be stated and proved for all c. mappings from A into E^. (i)THEOREM. If (T, A) is anyc. mapping from an admissible set A C E2 into E^, if the plane mappings (Tp, A), r = 1, 2, 3, are all BV, then for every sequence [Fn J of figures Fn C F ^ Fn C A, F° f A0,and for any finite subdivision Sn = SA + S " of Fn into simple pol. regions qeS^ and non-simple pol. regions ReS^1, whose indices c^, n^, an => o as n -- > », we have lim Y, v(q, T) = V(A, T), n -- > oo qeS^ lim X n -- >00 qcS^ v(q, Tp ) = V(A, Tp ), r « 1, 2, 3 • PROOF. The second equalities are already proved in (12-8, i). Let us prove the first one. By defi­ nition of V(A, T) there is a finite system SQ

333

33^

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

of non-overlapping closed, simple pol. reg. * C A such that z v(jt, T) > V(A, T) - €, where the sum EQ is extended over all *eSQ and € is any posi­ tive number. By (9 *1 , Note) wecan suppose it C for all it ^SQ . Therefore there is an index n1 such that Tt C F° for all n > n 1and lteSQ.

A°

Since the mappings (Tp,A), are BV, the functions N(p; Tp, A), peE2p, are L-integrable, r = 1 , 2 , 3; hence, given any x\ > 0, there exists a X > 0 such that (h) / N(p; Tp, A) < r\ for every measurable set h C E2p, Ih | < Let B^, B^, BQnp be the sets defined in (12.6) for the mapping (T , A) and the subdivision Sn of Fn « Then and ^Bonr^ < ^an Provid-eci an < 1 * Hence there is an index n 2 such that, if h ^ = B ^ + BQnp we have |hnp| < X for all n > n2, r = 1 , 2, 3. By (12.7, i), for each r = l, 2, 3, for each *eS , and for each point peE2p, there exists an index n = n(r, *, p) such that, for all n > n(r, Tt , p), either peh^, or 0 (p; CQp) = E 0 (p; C^,), where Cor: (Tr' °nr: (Tr'qeSn' and whdre z is extended over all qeS^, q C it. Let ^^(p) = s |0 (p; C^)! if p does not belong to h , = N(p; Tp, A) if peh^. Then |0 (p; CQp)| < n(r, x , p); that is |C(p; CQp)| < lim cp^tp) as n -- >00. By Fatou!s lemma [S. Saks, I, p. 29] we have also v(tf, Tp ) = (E2p) / 1 0 (p; CQp)| < 13m (E2p) / ^ ( p ) Let us now observe that (E 2 r ) I ^ ( P )

< (E2 r ) / 2|0(p; 0 ^ ) 1

+ (hjjj,) / N(p)

and that the last integral is < € for all n > n2Thereiore we have also v(*> Tr ) < lim (E2r) / Z 1 0 (p; Cm )| = lim Z v(q, Tp ). Consequently v(«, T) = [zr v2(«, Tr )]J < [ Zv []JLm z v(q, T ^ ] 2}* = < (11m 2r t£ v(q, Tr )]2}2 < lim 2 l z r v(q, Tp )]2, where In the last step we have applied (2.10, c). We conclude that v(*, T) < lim E v(q, T) as n — *00. Finally V(A, T) - e < e q v(*, T) < EQ lim E v(q, T) < lim E* v(q, T), where E* ranges over all qeS^. Since E ! v(q, T) < V(A, T), we deduce that V(A, T) = lim E* v(q, T). Thereby statement (i) is proved.

§21 . THE EQUALITY V = U (21 .3 )

335

NOTE 1 . Under the conditions of (i), if A ! denotes any admissible set A f C A and we consider all those (closed) regions completely contained in A !, then we have lim Z v(q, T) = V(A!, T) n -- ^ a, qeS^qCA* The proof is just the same as above.

NOTE 2. As in (9-16) and (12.15) we may consider finite subdivisions Sn of Fn into pol. regions R (simple, or not) and we may consider for each R€Sn the functions cp(p; R, T), v*p = v*(R, Tp ) = (E 2 r^ R> Tp r = 1 , 2, 3, v*(R, T) = + v*2 + v*^)i, and the indices dn, rr^ (analogous to the first twoindices d, m of 21.1 ) defined by dn = max diam T(R) for all R£Sn, and m^ = max |Mnp| for all r = 1, 2, 3, where l y = Tp (zoR*) C E2p, and ZQ denotes any sum ranging over all HeSn with R C F° (see 9*16 and 12.15)* Then, under the conditions of (21.2, i), if Sn denotes any finite subdivision of F into pol. regions R whose indices d^, -- > 0 as n -- > T)j n -- ^ 00 ReSn

lim n -- „ oo ReSn

y

V*(R, T ) = V*(A, T ),

r = 1, 2, 3. The proof is the same as for (21-2, i). fication of the next limit theorem (21.3, i) holds.

21.3.

The Equality

An analogous modi­

V= U

This equality will be proved in (ii) together with the following statement (i) which assures that also the function U is a limit. (i)

THEOREM. If (T, A) is any c. mapping from an ad­ missible set A C E2 into E^, if the planemappings (T , A), r = 1, 2, 3, are all BV, then for every sequence [Fn] of figuresFn C Fn+1, Fn C A, F^ A°; and for any finite subdivision 3n = of Fn into simple pol. reg. and non-simple pol. reg. ReS^*, whose indices

T

336

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

d , n^, a -- > 0 as

n -- * oo, we have

11m 2' u (q, T) = U(A, T), n -- > oo

lim £i |u(q, Tp )| = U(A, Tp ), n -- > oo r = 1 , 2 , 3 , where qeSA, q C F ° . (ii)

z l ranges over all

THEOREM. If (T, A) is any c. mapping from an admissible set A C E 2 into E^ and the plane mappings (Tp, A), r = 1 , 2 , 3 , are all BV, then U(A, T) = V(A, T). PROOF OF (i) AMD (ii). The second relation in (I) has been proved in (12.9, i). By (12.9) we know that 0 < U(A, T) < V(A, T ), and that 0 < |u(q,Tr )| < v(q, Tp ), r = 1 , 2 , 3 , 0 < u(q, T) < v(q, T) for every By making use of the inequality (2.10, b) and of the defi­ nitions of u(q, T) and v(q, T) (12.1; 12.9) we have

M

i “ u(q, Tr )|j | + [sr u2(q, Tp )j >

v(q, Tr )

>

[ z t ' r2(q>


0 +,

p -- * 0 +• Hence there is < 2_n for all p < pnp. p^] . Thus I H ^ I < 2 ~n =1, 2, 3. Let HpnQ be where z ranges over all

IH^I A

z

- iT”* 1,

and, If Kp is the intersection set Er3Q ..., we have IK^J = 0.

Kp = Hpl Q H ^ q

For each n let sn = Sn + Sn* a finite sub­ division of FR intosimple pol. reg. an(i non-simple pol. reg. whose indices d , n^, an satisfy the following relations: dn < min [2~n, pn], < 2~n, an < 2“n . Thus z(mn + 0. Let us consider, for instance, the mapping (T^, A). By (1 2 .11, ii) we have Z ! |0(p; C^ )I ---*N(p; T^, A) as n --- > 00 for all peE23 where is the set of measure zero in E23 considered in (12.11, ii), and C3: (Ty q*), qeS^. Let p be any point P£E23 - K3 - J3. Then there ex­ ists an index n = n(p) such that P€E23 ” J3 ~ H3no for all n > n(p); hence PeE23 “ J3" H3np 8,11(1 also {p, H3n) > pR . Let z* be anySiam ranging over all qeS^ such that o(p; C3) +0, C3: (T^,q*)If q is any of these regions qeS^ with o(p; C3) + 0, then there is at least one point w0eq° with T3(wQ ) = p (8.6, ii); hence T^(q) is contained in the dn-neighborhood of p with d^ < pn As a consequence T3(q)H3n = 0 and also qfc^ = 0. Since q C F qh = 0 and hence n . h„ n = F nh, we✓have .\ ^ q C A. for some j.If Z^J' denotes any sum ranging over all qeS^ with qC A., we have s' + ...

10(p;

C3)| =

Z* =

+

2 ^

< Sj N(p; T3, A j )

for all n > n = n(p). Since the limit of the first member as n -» » is N(p; T3, A), weconclude

+

§21

. THE EQUALITY V = U (21.4)

339

that N(p; T^, A) < Z . N(p; T^, A .)• This relation holds for all points P€E23 i-e., almost everywhere in E2^ (1 2 .1 4 ). Since the contrary re­ lation holds for all PeE23 (1 2 .1 4 ), we conclude that N(p; T^, A) = z. N(p; T^> A.) a.e. in E2^; hence by integration, we have W(A, T^) = Zj W(Aj, T3^* The same relation holds also for T 1 and Tg. Thus we have proved that W(A, Tp ) = z . W(Aj, Tp ), r = 1, 2, 3; hence by (12.8, ii) we have also (1 )

V(A, Tp ) =

Z.

V(Ay

Tp ),

r = 1, 2, 3 .

By (9 *1 , i ) we have now (2)

V(A, Tp ) < V(A, T) < V(Aj, Tp ) < V(Aj, T)
0 there is a number 0 > 0 such that (I) / N(p; Tp, A) < e for all measurablesets I C E2p, |I| < a, r = 1 , 2, 3. Let nQ be the smallest integer

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AEEAS

such that 2~n < cr for all n > n . Let us consider, for each n, the decomposition no of the collection S^ we obtain SA nj + S_ by putting each region Q.eS^ in accordSnj' no' ing q C A . or qh + o, and let ZnQ any be sum ranging over all

< (Hrnp ) / for each Sno
nQ such that ( z - Znj + ZnQ) v(q, T) > V(A, T) - e for all n > n 1. Then for all n > n1 we have Zj T) > Zj Znj v(q, T) > V(A, T) - he, where e > o is any given number. Thus z. V(Aj, T) > V(A, T). By (9 -5 , i) the contrary relation holds and thus it is proved that V(A, T) = z. V(A., T). Thereby (i) J J is proved.

§2 1 . THE EQUALITY V = U 21.5 (i)

.6 )

(21

A triangular Inequality for

3 In

V

If (T, A) is any c. mapping from the admissible set A C E„ into E~ then ^ 1

r

Iv 2(a ,

t ,)

+

v 2(a , t 2)

n2

+ V2(A, T3)

< V(A, T)
, r = 1 , 2 , 3. The second, inequality has been proved, already in (9-1, i). Let Fn be any sequence of figures Fn C A, PR C Fn+1, F° t A°, let ^n = ^n + ^nT an^ subdivision of Fn into simple pol. reg. qgSn and non-simple pol. reg. ReS^1 whose indices d , -- > 0 as n -- ► 00. if z1 denotes any sum ranging over all Q.eS^ we have, by (21,2, i), z'v(q, Tr ) -- >V(A, Tp ), r = 1, 2, 3, S ?v(q, T) -- >V(A, T) as n -- >00. On the other hand by definition of v(q, T) (12.1 ) and by making use of the inequality (2.10, c ) we have A.

2

jz:rJs'v(q, Tr )J

j

JL

r i2 < Z'j2rv2(q, Tr )] = £'v(q, T) .

As n -- =*00 we have immediately V(A, T). Thereby (i) is proved.

2 (A, Tp )] ^
i) are proved in L. Cesari [12] by means of an analogous procedure.

3^2

§22 .

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

SOME LIMIT THEOREMS FOR THE FUNCTIONS 2 (i)

2.1

U

AND

V

Some Theorems on Plane Mappings

Let (T, A) be any c.plane mapping from anadmissible set A of the w-plane E 2 into the p-plane E^, let (Tn, Fn ), n = 1 , 2 , ..., be a sequence of q.l. mappings from figures Fn C A into E^ such that Tn -- »T (5 -3 ), let it C A0 be any simple closed pol. reg. and C: (T, **), CR : (Tn,it*), and suppose that lim v(*, T ) = v(jt, T) + X., X > 0 , as n --- >oo. Then, given anye > 0 , there exists a finite system [q] of non-overlapping simple closed pol. regions q C itsuch that T(q) C [C]€ and lim Zv(q, Tn ) > X, where [C]€ is the e-neighborhood of [C], and where z is extended over all qe[q].

PROOF. Since * is compact, T is uniformly con­ tinuous on it; hence there is a number 6 = 5 (e) such that |T(w) - T(w*)| < 3 ~1e for all points w, w fe*, |w w 1| < 5 . Let[q] be any finite subdivision of n into simple pol. regions q all of diameter < 5 . Let [q ] 1 be the subsystem of all qe[q] such that T(q)(E^ - [C]€ ) + 0 and put [q] 1 1 = [q] - [q] 1 - Then T(q) C E ’ - tC] 2 € /3 for all qe [q] f and T(q) C [C]£ for all q€[q]M . Let E, L 1, 1 denote any sum extended over all qe[q], [q].!, [q]M , respectively. We want to prove that (a) lim E,!v(q, TR ) > X as n — > oo. Suppose this is not true, therefore, (p) z,!v(q, T ) < X - o, a > o, for infinitely many n. Let n be the smallestinteger such that * C Fn, d(T, Tn, Fn )< 3 ~1€ for all n > n (5 *3 )and let C: (T, **), Cn : (T , **), n > n. Then,by (8.3, i), we have (7) 0 (p; C) = o(p; Cn ) foralln > n and peE^ - [C]€^. For each n > n andqe[q] let c: (Tn, q*) and let I C E^ be the set sum of all sets tc]. Since the mappings Tn are q-l«, each set [c] has measure zero- Hence |I| = 0 and, by (8 .6 , i), (5) 0 (p; Cn ) = Z0 (p; c) for each n > n and P€E^ - I; that is, a.e. in E^. We have

§22.

SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (22.1 )

v(«, Tn ) = (Ep /

10 (p; Cn )| = |(E| - [C]6/3) + [C]e/3 ]/ I0 (p; Cn )| = J, + J2 .

Because of (7) we have J 1 = (E» -

^°h/3)

I0 (p; C)| < (E») /

I

1 0 (p;

C)| = v(it, T

Now for each qe[q.]! the set T(q) is completely contained in a circle r of center in a point peEJ, - [C]£ and radius e/3; hence r C E^ - ^^2e/3’ Consequently 0 (p; c) = 0 for all points p outside r and hence for all P€[ d 2e/3' (q.eCqJ1, n > n). Because of (5 ) and of the last remark we have J2 = ([C]€/3) / =

2

1 0 (p;

1! n. Therefore v(it, T ) = J1 + J2 < v(*, T) + x a > 0, for the same n, and finally lim v(*, T ) < v(it, T) + X - a, where lim is taken for all integers n -- » a contradiction since the same lim is v(it, T) + X. Thus (a) is proved and the system [q] of the statement (i) is the system [q]11 - Thereby (i) is proved. (ii)

Let (T, A) be any BV plane mapping from an admissible set A of the w-plane E2 into the p-plane E^, let [Fn ] be any sequence of figures Fn C A0, Fn C Fn+1, F° f A°, let Sn = + S^1 be any finite subdivision of Fn into simple pol. reg. and non-simple pol. reg. RcS^1, whose indices dn, mn, crn -- > 0 as n -- * «)} and denote by z! any sum extend­ ed over all q. C F° (12.6). Let (P^, ft ), t = 1, 2, ..., be any sequence of c. plane mappings such that P.j_ -- > T as t -- » + 00;hence ft C A, f^ C ft+1, T A°,

cr

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEdCZE, PEANO AREAS

d(T, P^, ft ) -- > 0 as t -- > + oo, and suppose that W(ft, Pt ) -- *W(A, T) as t -- >+ oo. Then, for every integer n there is another integer t(n) such that, whenever t > t(n) we have Fn C ft and (1 )

lim S’|u(q, P,. )l= U(A, T), n -- » oo u

(2)

lim s' |u(q, Pt ) - u(q, T)| = n -- > oo

0

uniformly with respect to t > t(n). PROOF. By(2 1 .3 , i) we have (a) lim Z*|u(q, T)| = U(A, T) as n — > oo. Since W(A, T) < + oo, W(ft, P^) -- > W(A, T) as t -- >00, there is a tQ such that W(ft,) < + 00 ^or aH t Since Fn C A0, Fn is compact, and f° f A° as t -- >00, thenfor every n there is an integer t1(n) such that Fn C f^ for all t > t^n). Since (Pt, ft ) -- > (T, A) and W(ft, Pt ) -- > W(A, T) as t -- >00, by (12.11, iii and Note), given € > 0, there exists 8 > 0 such that (h)/N(p; P^, ft ) < 6, (h)/N(p; T, A) < e for all measurable sets h C E^, |h | < 5 and all t > t . Since mn -- > 0there is an nQ such that < 5 for all n > nQ . Now let us consider, for every n > n , the compact sets Bn, B^ defined in (12.6), i.e., Bn = zrq* C E2, Bn = T ^Bn^ = T (Z1q*) ,C E^ . Thus = |B^| < 5 for all n > nQ . For any p > 0 the p-neighborhood, say (B^) , of B^ has the following properties: (Bn)p ^ Bn> ^Bn^p " Bn -- ” 0 as p -- * 0 +; hence |(B* n )p - Bn*| -- > o as p -- * 0 +. Therefore there exists a number Pn > 0 such that |(B^) - B^| < 5 and hence

I(Bn^pI = I I

+ I

" B^|

nQ . Let Hn denote the set Hn = (B') C E ’ . Because of d(T, P.u, f,u_) --> 0 n p^ d as t -- > oo, there exists an index t(n) such that t (n) > tQ, t (n) > t1(n), d(t, Pt, ft ) < pR for all t > t(n). Consequently, if C: (T, q*), C._: (P^, q*), where qeS^, q C F°, n > nQ, we have J|Ct, C|| < pn for all t > t(n) and, by (8.3, i), also 0 (p; C) = 0 (p, C^

§2 2 . SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (2 2 .2 ) for all 0 (p; C) q C F°, we have £'|u(q, Pt ) -

peEJ> t > t(n). Consequently = 0 (p; Ct )for all peE^ - Hn, q t > t(n). Thus for every n > nQ, t> t(n),

u(q, T)| = Z* |(Ep /

=

Z '\(\)

< (Hn ) / z! T as

t -- »«=• d(T, P^,

f

hence ft C A, f*t C ft , f° f A°, t ) -- * 0 as t -- > » and denote by

(Ptr-' ftr}' ** = 1 , 2 , 3, th e p la n e m appings w h ich a re th e p r o je c t io n s o f ( P^, f ^ ) on th e c o o r d in a t e p la n e s . Suppose f u r t h e r t h a t a ( f t > ) < M, t = 1y 2 y . . . , f o r a g iv e n c o n s ta n t M < + 00. Then th e re a re two se q uen ces of in t e g e r s su ch t h a t F „ C f . , t v -----> nv ----k

and

*°° Z!u(q,

lim k —

7t

tk

K

) > U(A, T) , k

k

E^

3 1+6CHAPTER

VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS k

11m = + oo

r = 1 ,2 , 3 , where q C F° • k NOTE i. r = 1 ,2,

S' |u(q, Pt _)| > U(A, T_), zkr

~

r

£f is extended, over all

Because of v(q, Pt ) > u(q, Pt ) > 0 , (12.9) and U =V (21.3, ii) we

3,

qeSi, ^

,

v(q, Ptp) > |u(q, Ptr)L have also

lim £! v(q, P^ ) > V(A, T) , k -- > 00 lim £» v(q, Pt k = > 00 Lkr r =

1 ,2,

)

> V(A, T ) , " r

3 .

NOTE 2 . In the statement (i) we can add the further requirement that, for each k, t^ is as large as we want; i.e., given any sequence ofintegers t(n), n = 1 , 2, ..., we can require that t^ > t(n^) for all k = 1 , 2, PROOF OF (i). It is enough to prove that, given any positive number e, there exist some integers n and t, as large as we want, such that F^ n C f,u and (3 )

Z 1 u(q, Pt ) > U(A, T) - e , u(q, Ptp) > U(A, Tp ) - e , r =

1,

2,

3

.

First of all letus observe that, since P^ -> T as t — > 00 anda(Pt, ft ) < M, we have L(A, T) < lim a(Pt, ft ) < M as t -- >00 (5.8). Hence L(A, T ) < + 00 and, as a consequence, V(A, T) < + 00, V(A, Tp ) = W(A, Tr ) < + «, r = 1, 2, 3, (9 -^, ii; 9 -1 > i)- The mappings (Tr, A), r = 1, 2, 3, therefore, are BV and the functions N(p; Tp, A) are L-integrable in E2p, r * 1, 2, 3. As in (12.6; 21.1 ) let B^, B ^ be the compact sets Bnr = z»q*, B ^ = Tp(Bnp) = Tp (z!q*) where E1 ranges over all qeS^, q C F°. By (21.3, i) there exists a finite system [Q] of non-overlapping simple closed pol. regions Q C A° such that

§2 2 . SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (2 2 .2 )

I

u(Q, T) > U(A, T) - lT’e ,

Qe[Q] (b)

Z Iu(Q, T )| > U(A, T ) - ^_1€, Qe[Q] P P r =

1, 2 , 3

•

For each Qe[Qj let us denote by Sn = ^ (Q) the system of all regions Q^S^ such that q. C Q° (if any). Let K be the number of regions QetQ,] and nQ any integer such that Q, C F° for all n > nQ and Qe[Q]. Let a = (20K)“1e and let N be any integer such that No > M + 1. Since the functions N(p; Tp, A), r = 1, 2, 3, are L-integrable, there is a number tj > 0 such that (h)/N(p; Tp, A) < cr for every r = 1, 2, 3, and all measurable sets h C E2p, I h | < t)• As usual let (B^) denote the p-neighborhood of the compact set C E2p and observe that ( B ^ M B^,, (B^) o, ^^Bnr^ “ Brirl -- * o as p -- > 0 + . Consequently, for every pair of integers n > 1, k > 1, there is a number p = p(n, k) > o such that I^Ar^p ' BArl < 2" N 5 hence r = 1, 2, 3 .

KB/^I

< |B*r | + 2“\ ,

Since dn, m ^ crn > o as n ----- > by (21.3, i, and Note) we can define by induction, for each integer k = 1, 2, ..., an index n^. and a positive number p^. such that dn

< €’ “n 1

< 2~1ir>'ni - n0' Pi =

1

X Q.€S (Q)

1

T)

U1

(5 a) > U(Q, T) - a,

M < 1 > Tr )l > U(Q, Tr ) - a, r =

Y

1,

qeS (Q) “1 < min [Pk-1, e],

< 2“ r|,nk > k

(5b)

k = 2, 3, •••

pr

= P(nk, k)

2, 3 ,

3^7

CHAPTER VII.

IDENTIFICATION OF LEBESGTJE, GEOCZE, PEANO AREAS

By definition of

we have thus

< t^. Since all the regions cl€S>ol (q. C Q, Qe[QJ ),where, for eachk, we consider only the terms with t> t^. We shall provethat for each Qe[Q] and r = 1, 2, 3, we have X

[

lim

qeSok; Lt €[ t] , t —>«>

|u(q, P. )|

1

J

> U(Q, Tr ) - (^K)"1€ for all but finitely many k. Suppose indeed that this is not true. Then there would be some Qe[Q] and r = 1, 2, 3, say r = 1, such that X T lim |u(q, Pt1 )I ] qeSQk L te[t] ,t — » » J < U(Q, T 1) - (^K)"1€ for infinitely many k. Let us consider the first N values of k for which (8) holds and, for the sake of brevity, suppose that these values are k = 1, 2, 3, ..., N. Let tQ = maxtt.j, t2, ..., t^] . Then for each t > tQ, t€[ t], all numbers u(q, P^.r ) exist, q€Sok> k = 1, 2, ..., N, r = 1, 2, 3 -

350

CHAPTER VII. For each

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS Qe[Q]

let

S11 = S11(Q) = SQl,

S2 1 = s21 ^ = °' and for ever^ k = 2 , let us define as follows the collections 3 ,k

■ 3 ,k < «"

S2k '

S 2k ■ 3 0k -

S ,k-

3,

•••, N,

Let

3 ,k

be the subcollection of all q € S w h i c h are con­ tained in some region qeS-, ^-1 and that T 1(q)[E^1 - H ^ j ^ ] * 0. Since diam T.,i(q) < diam T(q) < r -, we have - d^ nk c pl k-i T 1(q) C E ^ -H-^J 1^1for all

qeS1k* On

the

other hand, for every qeS2k = ^ok " ^ik we have either q C ^k-l^i k-1 and then T-] (q.) C or q C q.k_1 c’qk_2> *k-leS2,k-l'’ % - 2 eS1,k-2 and then T,(q) C or q C q ^ C qk_2 C qk_3> qk-l eS2,k-l ’ then T(q) C

q■k-2eS2,k-2, ^-k-S53!,k-3' or

anci

* C ^k-1 C *k-2 C’-" c *1' % eS2Bu = k-1, k-2, ..., 2, q ^ S ^ , and then T(q) C Hj *1• If HQk denotes the set HOk = H iI’ + H21' + ••• + Hk-l,1' then we have T ](q) C HQk for every Q€S2k and, by (8.6, ii), also o(p; C) = 0 for all P e^21 - HQk where

C: (T^ q*),

cl€S2k’

the otheP band,

|H k I < ^ { { ' 1 + ••• + |H“ j J < 2 (2 _1 + 2-2 + \

2-*+] h

Z

and finally

|u(q, T )| =

qeS2k

Z

l(E') /

=

Z

qes„v

l(Hk ) / 0K

< (Hok)

By (6b) we havenow, since

Z Iu(q, T

(9)

q€S1k Since

I

S1k

0(pj C)|
tr -

>Qe[Q] Z qeSok Z |u(q, P.tr > z )

U(Q, Tr ) - (lHC)_1e

Qe[Q]

(11)

Z

> |u(Q, T )| - It- 1e > U(A, T ) - lT1e - U-1e > ~ Qe[Q] r r

> U(A, Tp ) - e,

r =

1, 2 , 3

•

Analogously, by (2.10, c), we have, for the same

t,

3

Z'u(q,Pt ) > Z Z [Z u2(q, P.) Qe[Q] q e S Q k L i>=1

(12 )

Z {Z Z

,e[Q] I r=1 Lq£Sok

|u(q, Pt )|

tr

1 } S Z Qe[Q] I(Q), j J

where each sum in brackets is > U(Q, T ) - (4 K) 1€, and the same sum is also non-negative. Therefore, if bp = min [U(Q, Tp ), (itK)"1e], i ap = U(Q, Tp ), we have I(Q) > [2r(ar - br )2] • By (2.10, b) we have now

§22.

SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (22.2) V

(E r 4 >

I(Q) >

]% IV

ar ~ br + br

1“ Mrav r - br )2| J

>-

(2

i 2 )2 rar

]♦

ar -

(V r J

srU (Qi T r )

Srbr > U (Q> T) -

'e

3(^0

Therefore, by (4 ) and (12), ^'u(q,

Pt)>Q Z[Q]I(Q) > Q€[Q] Z

u(Q,

T) -

3

( W

g

> U(A, T) -

-

3 (4 ~1 e)

= U(A, T) - e

By (11 ) we conclude that relations (3) hold for each n = nk, k > k and, correspondingly, foi all t large enough. Thereby (i) is proved. (ii)

Under the same conditions as in (i) the sequences tn^] considered in (i) can be required to have the following further property: for each r = 1, 2 , 3, and k = 1, 2, ..., there is a subsystem S(k, r) of simple pol. regions qeS* , q C F k ■Ak such that

Z

lim |u(q, T_)| = U(A, T ) , k -- * 00 qeS(k, r )

Z

lim |u(q, Pt ) - u(q, T )| = o k -- >°°qeS(k,r) kr r PROOF. Let us suppose r = 1. It is enough to prove that, given € > 0, there exist some integers n and t, as large as we want, such that relations (3) of the proof of (i) hold, and, furthermore, I

2 1u(q,

Ptl )| - U(A, T 1)| < e

s|u(q, Pt1) - u(q, T 1)| < € ,

>

353

.

35^

CHAPTER VII.

IDENTIFICATION OP LEBESGUE, GEOCZE, PEANO AREAS

where z ranges over a convenient subcollection of simple pol. regions q.GS^, q C F°.

S

First of all the same argument of the proof of (i) may be repeated. Let kQ be the integer determined in the proof of (i) such that relation (7 ) holds for all Qe[Qj, r = 1 , 2 , 3 , k > kQ, and let Tc denote any integer IF > k . For each of the values k = k + 1 , k + 2 , ..., k + N and for each Qe[Qj let us define the collections Slk =S 1k(Q), S2k = S2 k (Q) as in the proof of (i). Thus = 3 , ^ + 1 (Q) = SQ^ +1, S2 ,k+ 1 = °' while S 1k’ S2 k = Sok " S 1k’ U + 2 < k < k + N, are defined as in the proof of (i ). Finally, let S(k) = Z S (Q), Qe[Q] 1k

k+

1
oo

Indeed, suppose the contrary is true.

qeS(k)

.

Hence

lim u(q, P. ) - u(q, T 1) > cr te [t ], t - + » 1

for all k + have also

1

< k < k + N.

X [ lim qeS(k) [ te[t],t — »

Then by (8.7, ii) we

v(q, Ptl) - v(q, T- )

> cr -*

for all I c + i < k < l I + N , and we can apply (22.1, i) to each region qeS(k) while we take for e the value e = p . Then we obtain N k _ systems ! k + i < k < k + N, of non­ overlapping simple pol. regions n for which (10) holds, as in the proof of (i). Finally, we can repeat the further argument of the proof of (i) till the contradiction is shown. Thereby we

§2 2 . SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (2 2 .3 ) have proved that (1b ) holds for at least one value of k, k + i < k < I c + N. Thus there is a t large enough, te[t], such that we have also

Z, |u(q,

qeS(k)

Pt ) - u(q, T )| < a < e , 1:1 1

and, therefore, the second relation (13) is proved. On the other hand, for each Q,e[Q] and the same number k, relation (9) holds. By addition of such K relations and by ( b ) we have

Z |u(q, T , ) | > Qe[Q] X U(Q, T. ) - l+Kc > > Z |u(Q, T 1. ) | - 5_1€ >~ U(A, T1)- l T 1e ‘ QetQ] qeS(k)

5_1e •

Thus we have

Z

U(A, T-) > |u(q, T, ) | > U(A, T. ) - e , 1 ' qeS(k) 1 1 i.e., the first relation (13) is also proved. The same argument as for (i) assures that rela­ tions (3) are also satisfied. Thereby (ii) is also proved.

22.3. (i)

Invariance of The Areas V and U Orthogonal Linear Transformations

For

For every c. mapping (T, A) from an admissible set A C E2 into E^, the function V(A, T) is invariant for linear orthogonal transformations in E^ (change of orthogonal cartesian coordinates). The same holds for the function U(A, T) provided V(A, T) < + 00.

PROOF. Let t be an orthogonal linear transforma­ tion of the p-space E^ into the p* -space E^ , P* = (l, T|, £)• (T!, A) be the mapping T ! = tT from A into E^ and let (Tp, A), (T^, A), r = 1, 2, 3, be the plane mappings which are the projections of (T, A), (Tf, A) on the coordinate yz, zx, xy, 51, |t]-planes

355

356

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

E21, E22, E23, E^, E|2, E^3 - By (18.10, Note 2) we know that V(A, T) < + °° if and only if V(A, T 1) < + 00- Therefore, we have to consider only the case V(A, T) < + °o, V(A, T 1) < + hence V(A, Tr ), V(A, T p < + », r = 1 , 2, 3 Let

[Fn]

be any sequence of figures

Fn C A0,

Pn C Fn+i> Fn t A° and Sn = S^ + S " beany finite subdivision of Fn into simple pol. regions and non-simple pol. regions such that; dn9 mn9 an9 with respect to

dn' mn* an are the inciices T and T 1, we have

sn

V V V dA’ aA — > 0 as n “ (21.1, ii)• Then, by (21.3, i ), we have

(1 5 )

lim S'u(q, T) = U(A, T), n -- > 00 where z? q C P°.

is extended over all regions

Let us consider now the sequence of q.l. mappings defined by (18.2, i), say (Pt, ft ), t =1, 2, ... . Then for the figures ft we have f^ C A, ft C ft+1, f° t A0, and d(T, Pfc, ft ) -- » 0 as t -- > oo, i.e., Pt -- > T. In addition, if (?tr^ ^ t^ r = 1; 2, 3, are the plane mappings which are the projections of (Pt, ft ) on the coordinate planes, we have a(Ptp, ft ) -- * W(A, Tp ) as t — ^ oo, r = 1, 2, 3, and a(P-^ ^t ^ M where M = ZpW(A, Tp ). Let (P£, ft ) be the q.l. mapping P£ = tP^, t =i, 2, ..., and let (P£r, r*= 1, 2, 3,be the corresponding plane mappings which are the projections of (P£, f^ ) on the coordinate planes . Since L is invariant for transformations t (5 -1 1 ) we have L(A, T) = L(A, T f)> and, by (5 »7 ), we have also a(ft, Pt^ ^ = 1 9 29 '* *9 hence &(?-£> < M for every t. Since a(ft, Ptp) -- > W(A, Tp ) as t -- ^ oo, r = 1, 2, 3, we can apply (22.1, i) and we deduce that for each n = 1, 2, ..., there is an integer t(n) such that

§2 2 . SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (2 2 .3 ) lim

(1 6 )

z! |u(q, Ptr) - u(q, Tp )| = 0 ,

r = 1, 2, 3, uniformly with respect to t > t(n), where z* ranges over all q C F°. Since a(P£, ft ) < M, t = 1, 2, ..., we can apply (22.2, i, and. Note 2) to the mapping T T and. we deduce that there exist two sequences of integers such that n^ -- *°°, t^ -- *«>, 'nn, c f +- > and

\

lim

(1 7 )

where

(18)

S'u(q, P£ ) > U(A, T 1) , k

sf ranges over all regions

qeS^ , q C F] k (Pt, ft ) are q.l., by

Finally, since the mappings (8.11, i) we have 1 u(q. Zvu 2 {q_, P^r ) ] 2 =

[ Sru2(q, Pt

By (17) and. (18) we have first, as U(A, T') < lim

S'u(q,

P£

k -- > °°,

) = lim

s'u(q,

bk

where

= lim

z*

= lim

z«

= u(q, Pt )

Pfc

bk

2r(u(q, Tr ) +

5 p )‘

)

=

]*•

Sp = Sp (q; t k ) = u ( q , Pt p ) - u ( q , Tp ).

By inequality (2.10, b) we have now U(A, T') < lim < lim

2* [ (spu2(q, Tp ))2 + z*u(q, T) +

( 2p62 ) 2 ]
t(n^), k = 1, 2, ..., by (16) we have EP l5r l -- * 0 as k -- ► while the ex­ pression under lim is < U(A, T) (9 *9 )* Thus we conclude that U(A, T !) < U(A, T). By exchanging the use of T and T 1 in the previous proof we have also U(A, T) < U(A, T !) and hence U(A, T) = U(A, T')- By (21.3, ii) we

357

358

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

have also V(A, T) = V(A, T !)completely proved.

22A.

The Indices

Thereby (i) is

d, m, n

Let (T, A) be any c. mapping from an admissible set A C E2 into E^, let (Tp, A), r = 1, 2, 3, be the plane mappings which arethe pro­ jections of T on the coordinate planes E2p, r = 1, 2, 3, and let us suppose that the mappings (T , A), r = 1, 2 , 3, are all BV. Let S be any finite system of non-overlapping simple pol. regions it C A and let d = max diam T(it) forall iteS.Let z denote any sum extended over all x e S , let Bp be the point set Bp = ZTp(**) = Tp(Zir*) C E2p, r = 1, 2, 3, let mp = IB I, i.e., is the (two-dimensional) measure of the compact set Bp in the plane ^2p, and let m = max [m1, m2, ]. Obviously, the indices d, m now defined coincide with the indicesd, m defined in (21.1 ) when S is the system of all regions qeST, q C p0, for a subdivisionS ’ + S 11 of a figureF, as in (21.1). Now let \i = max [U(A, T) - zu(*# , T), U(A, T ) - z|u(it, Tp )l, r = 1, 2, 3]. The non-negative numbers d, m, p. are said to be the indices of the finite system S of non-overlapping simple pol. regions n C A. (i) For any c. mapping (T, A) from an admissible set A C E2 into E^ such that the plane mappings (Tp, A), r = 1, 2, 3 ) are BV, given any positive number € there are finite systems S of non-overlapping simple pol. regions n C A, whose indices d, m, \1 are < e. This statement is a consequence of (21.1, i) and (21.3, i). (ii) For any c. mapping (T, A) from an admissible set A C E2 into E^ such that the plane mappings (T , A), r = 1,2, 3, are all BV, and for every finite system S of non-over­ lapping simple pol. regions n C A of indices d, m, n, there exists a B-measurable set H^ on each plane F2p, r = 1, 2, 3> such that (a) Z|o(p; Cp )| = N(p; Tp, A) at each point peE2p - Hp and the same relation holds for N+ and N”; ("fr) I I 5:r = 1, 2, 3 * This statement is a consequence of (12.13, i).

§2 2 . SOME LIMIT THEOREMS FOR THE FUNCTIONS U AND V (2 2 .5 ) (iii)

359

Under the same conditions of (ii) we have 0 < V(A, T) - Zv(jt, T) < n, V(A, Tp ) - Z v ( jc, Tp ) < n, r = 1, 2, 3.

PROOF. Since U = V (21.3, ii) and 0 < u < v (12.9), we have 0 < V - Zv < U - Zu < ji, and an analogous argument holds for V(A, Tp ), r = 1, 2, 3.

NOTE. As in (21 .1, ii), if t is any orthogonal linear mapping from the p-space E^, P = (x, y, z), into the q-space E^, q = (|, tj, 5) (change of orthogonal cartesian coordinates), let (Tf, A) be the mapping T 1 = tT from A into E£.Let (Tp, A), (T£, A), r = 1, 2, 3, be the plane mappings which are the projections of (T, A) on the yz, zx, xy-planes E21, E22, E^, and of (T!, A) on the t)(;, £{-, £T)-planes E^, E*2, E^. Then a system S of non-overlapping simple pol. regions n C A has indices d, m, n with respect to (T, A) and certain analogous indices d !, m T, p 1 with respect to (T!, A). The following statement holds: If the mappings (T , A), r = 1, 2, 3, are BV, then, given € > 0, there are finite systems S of non-overlapping simple pol. regions it C A such that d, m, \x, d f, m 1, p * < € . This statement is a consequence of (21.1, ii) and (21.3, i)«

22.5-

Further Properties of The Indices

d, m,

n

Let (T, A) be any c. mapping from any admissible set A C E2 into E^ such that the planemappings (T , A), r = 1, 2, 3, are BV; let S = [ir], S ! = [itf]be any two finite systems of simple pol. regions *, jt1 C A. Suppose that the regions jteS are non-overlapping, and also that the regions jt!eS! are non-overlapping, and let d, m, |i, d !, m f, be the indices of the two systems S, S T. Let Bp = Tp(Zjt*), Br = r = 1> 2 > 3' where z, z1 are sums ranging over all jt€S and JtfeSf. Let C, Cp, C T, C ^ ' r = 1, 2, 3, be the curves images of jt*, Jtf* under the mappings T, Tp, T 1, T^, r = 1, 2, 3. Hence Bp = z[Cp], Bp = zf[C^], r = 1 , 2 , 3 - Let Hp, H^ be the sets defined in (22.h , ii) relative to the systems S and S !; let p = d ! = max diam T(jt! for all jt!eSr and Bpp be the set of all points peE2p at a distance < p from B . For any JteS let S*(jt) denote the subcollection of all itfeSf such that Jtf C Jt°; let S*"" be the subcollection of all Jt! such that jtfjt* + 0 for some jteS and let S^ be the (remaining) subcollection of all it'eS1 such that jtf(Zjt) = 0. Thus the subcollections S*(jt) for each jteS, S*, S| are disjoint and zS(it) + S* + S| = S*. Let

360

CHAPTER VII.

IDENTIFICATION OP LEBESGUE, GEOCZE, PEANO AREAS

Jp be the set = (Brp + B^ + + H p . Let and any sum extended over all jt!€S(*) and all j^eS* + S^. (i) Under the conditions above we have (1) 0(p; Cr ) = 2'(n) 0(p; Cp ) for every *€S and for almost all

2"

denote

peE2r “ J r> p = 1' 2> 3; o(p; C p = 0 for every jt!eSj + S^

(2)

and for almost all PeE2r " Jr* (3 ) N(p; Tp, A) = Z |0 (p; Cp )| = Z'|0 (pj C p | for almost all J , r = i, 2 , 3; (.4 ) n(p; Tp, A) = Z 0 ( p ; Cp ) = Z » 0 ( p ; C p for almost all peE*2r- vJr :’ W ,;u(*S Tp ) + Sp (it) (5 ) if u (jc, Tp ) = Z!!V for each ireS and 6^ = ztfv(jt!, Tp ), we have z|5 p(*)| < 2(Jp )/N(p; Tp, A), 0 < 5 ^ < (Jr )/N(p; Tp, A), r = 1, 2, 3. This statement is a consequence of (12.13, ii and iii). (ii) Under the same conditions of (i), if ar = ar ^ = Tr^: r = i , 2 , 3 , for every ne S and D(n) = Z'(n)tu(«', T) - zraru(*', Tr )], then we have 0 < ZitD(Jt) < n + 2Zp(Jp )/N(p; Tp, A), Z"v(*«, T) < Zp (Jp )/N(p; Tp, A) .

NOTE. In (ii) and (iii) below we suppose all *eS, i t reSr.

u(jt, T) =)= 0, u i * 1 , T) + 0

PROOF. The last inequality is an immediate con­ sequence of the statement (5) of (i) above. By the Schwarz inequality (2.10, a) we have now for each it1eS (it) 1 1 2 ? r Zru2(*', Tp ) 2 = u(*', T) Z ct~ |£raru(*', Tp )| < r r and hence hand

D(*) > 0 for each 1(it)

0
t(n; r, it).

Finally, by (22.2, i) applied to each mapping (T!, J t ) , and the sequences S^, n = 1, 2, ..., and (P£, Jt), t = 1, 2, ..., there are two sequences integers n and t such that (M

U(«, T p
- 00 ^

2 |u(q, P£r )| ,

r = 1, 2, 3. As we observed in (22.2, Note 2) for each k we may suppose t^ as large as we want, namely t^ > t(n-, , r, it) for all r = 1, 2, 3, k = 1, 2, ... . For convenience we may suppose

365

366

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

that the sequence t^ is the same for the N regions jtefjt]. Indeed, first let t(n) = max t(n; r, *) for all r = 1, 2, 3, and jte[jt]. Then let us apply (22.2, Note 2) successively to the N mappings (T1, J t ) by taking each time t^ > t(n^). By N successive extractions we obtain a unique sequence [n^J and correspondingly a unique sequence [t^J with t^ > t(n^) satisfying the above conditions for all it€[it]. In such a way we can now determine a unique n = n^ and a unique t = t^ such that, for each jte[it], we have (5 ) d*n < p' mnn < N"1^ t ^ t(n, r, «), (6) 2qu(q, T) > U(n, T) - N" 1^, (7 ) £q lu(q, Tr )l > U(«, Tr ) - N'1t, r = 1, 2, (8)

3,

2q lu(q, Ptr) - u(q, Tr )l < N-1a, r = 1, 2, 3,

(9 ) U(«, T^) < Sq lu(q, P^r )l + N-1 cj, r ■= 1, 2,

3

-

The relations (6) and (7 ) above simply state that nnn „

TP >

V

Tj) < 1 = 1, 2, where T 1 on th e r)£, ££,

T f = t^T, | t]-p la n e s .

Z^(n,

(,t’ T3> >

T) > V(A, T) - ct4,

Y(-A >

T) "

°k

,

,

and

T^, r = 1,

2 , 3,

a re th e p r o je c t io n s

of

Let [it] be any f i n i t e system o f sim p le p o l. r e g io n s as ab ove, and l e t M denote th e number o f r e g io n s n g [ jt] . F o r each jre[jt] w it h V(jt, T^ ) = 0 l e t T* = T Q = T i n n and suppose t h a t th e c o ll e c t i o n s m entioned i n ( i i ) r e l a t i v e to n a re a l l em pty. F o r each ire[ jt] w it h V(jt, T ^ ) > 0 l e t t = = m in 2 ~ 1 [M“ 1 a, M V(jt, T ^ ) a ] . F o r e v e ry jt e [ jr ] and b y ( 2 3 . 3 ) we have V(jt, T ! ) V(jt, T ) < + 00 and, ( 1 2 . 8 , i i ), ( 1 2 . 3; 9 . 1 ), a ls o ( E ^ ) / N ( p , T ’ ,it) = ¥( j t , T ’ ) = V(jt, T » ) < V(it, T f ) < + 00. T h e re fo re th e f u n c t io n N(p; T ^, jt), p e E J^ , is

by

SOME ANALYTICAL PROPERTIES OF CONTINUOUS MAPPINGS (2 3 .3 )

§2 3.

L-integrable, and there exists a 5 q > 0, 8q =5 q (jt ), (h)/N(p; T^, jt) < t for every measurable set h C suppose 0 < SQ < 2~1.

373

such that |h| < SQ . We may

Let P ! C E ^ be the set of all points peE^ where N(p; T^, it) > Thus P 1 is bounded (since P ! C T^(it)) and open (12.2) and thus 0 < |P 1| < + 00.

0.

Let i|r(p)> P€-E^.3, denote the characteristic function of the set P 1. Let D 1 C E ^ be the countable set of points P€^ 3 defined 0 °*5 , Note 2) relative to the plane mapping (T^, jt); let D2 be the set of measure zero defined in (16.10, Note) such that for each point P€^23 ” ^2 the components of the closed set T^~*1(p) C jt are continua of constancy for T 1 and therefore for T; let be the set of measure zero of all points PeE23 w h e r e N(p; T^, jt) = +00. Given any compact set I C E ^ and a point P€E23 we cieno’ te parameter of regularity r of I with respect to p the ratio r = |I| : |R| where R is the minimum square R C E ^ of center p and sides parallel to the coordinate axes covering I. Thus if, for every t > 0, we denote by Q and Q 1 the closed squares of center p,sides parallel to the £- and T]-axes, and side-length 2 t, 2(1 - a)t, respectively, then the two compact sets Q and Q - Q !° have parameters of regularity 1 and 2 2a - a with respect to p; hence independent upon t. As a consequence, by [S. Saks, I, p. 1 1 7 , 6.1; 6.3], almost all points p^E^ have the following properties: IQI ( 13)

-1 •

IQP'I

— *■

IQ - Q T

*(p)>

1•

(Q

I QI -

Q')

as

” 1•

(Q) / N

T y

/ N( p;

t

— >

T^,

*)

* ) — *• N(p;

T^,

3

N (p^ T >

n)

it),

0 + .

Let be the set of measure zero of all points peE^ where at least one of these relations does not hold and let P = P 1 - (D1 + D2 + ). Then |P| = |Pf|> |PQ| = |PfQ| for every Q, and hence IQI”1 • IQPI—— > 1 as t --- >0 + for every point peP. Thus for each jt€[jt] andcorresponding t > 0 we have determined a number 0 < d Q < 2_1 and a set P C E ^ with the properties described above. By (13) and for each point peP there is a number d1 = d^p) > 0 such that for each t with 0 < 2t < d.j(p) we have IQI-1 • IQPI > 1 (

11+)

- 80,

||Q|_1 ' (Q) / N(p) - N(p)|<
^y (9 *^ i), (T7• 8,Note

1)

and

(21 . 4 ),

V(sj', T*) =

4 t2,

V(sj_ - s!', T) = 0, V(si - s|, T*)
0, there will be an integer v = v(p, w) such that weE2 - An for all n > v(p, w). Since the collection finite, let v(p) be the maximum of the integers v(p, w) for the k points w chosen, one for each ge{g)pAs a consequence we have g(E2 - AQ) + ° for all n > v(p). If n > v(p) and Tt is any simple pol. reg. jtC Aq then, necessarily, g(E - it) + 0 for all ge{g}p. Then by (14.J+, iii) wehave 0 (p; C) = 0 where C: (T^ Tt*). This proves that we have also N(p; T]; = ° for all n > v(p) and hence lim N(p; AQ, T1 ) = 0 for all peE21 - (D1 + B1 ), i.e., almost everywhere in E21. By (a) and (12.8, ii) we deduce V(An, T1)= W(An, T1 ) = (E21 ) I N(p; T1, An )-- * 0 as n -- > ». (ii)

If (T, A) is any c. mapping from A C E2 into Eg with V(A, T) < + 00; if (Tp, A), r = 1 , 2 , 3, denote the plane mappings associated to (T, A) (5 -4 ), if K is any set K C A,such that all three sets Tp(K) C E2p, r = 1, 2, 3, have measure zero, then for every sequence of ad­ missible sets C A with lim A^ = K as n -- ^ oo, we have lim V(AQ, T) = 0 as n -- > 00.

PROOF. The same proof asfor (i) where for each we consider the set Bp + Dp + Tp (K). (iii)

r

If (T, A) is any c. mapping from the admissible set A of the w-plane E2 into the p-space Eg, w = (u, v), p = (x, y, z), and V(A, T) < + °°;

§2 5 . THE LEBESGUE AREA AS A MEASURE FUNCTION (25-1) if C A, i = 1, 2, open in A and each A^ (1 b .1 ) then we have

is a sequence of sets is a sum of sets ger(T, A)

VCS-A^ T) < S1V(A1, T). PROOF. (a)

Let us consider first only two sets A 1, A2 as above. Since V(A, T ) < + oo and A 1 + A2 C A, we have V(A1 + A2, T ) < +00 (9.1). By (9 •1 ) the^e is a finite system «2, ..., *^ of simple closed non-overlapping simple pol. reg. C A 1 + A2, such that N £ 1

=

T) > V(A1 + A2, T) - e. 1

Let us consider the sets F^ = A(a £ - AiAi^ = A(A^ - A^), closed in A, and let F = F-|F2 ‘ Let us prove that F*-^ = 0 , 1 = 1 , 2 , NIndeed, if F k^ + 0 for some it-^, take a point weF*-^, i.e., weF.^^, hence weFi^i, wcF2 icr ThenweA, weA^ - A^, w not in A^, i = 1, 2, w not in A 1 + A2, what contradicts C A 1 + A2 . ¥e have proved that Fjt^ = 0, 1 = 1, 2, ..., N. From F^ = = °> it followsF-,^2. * ^2 * 1 = °' i.e., the sets are disjoint (for and 1 ). Let us prove further that F2*l not contain points belonging to the same set ger(T, A) (1 4 .1 ). Let us suppose that there are two points w ^ F ^ ^ , w 2 €^2 Trl' W 1 9 W 2 6^' geT; gF^ + 0 , i = 1 , 2. This implies gA^ = 0 because, If gA^ + 0, then, since the sets A^ are sums of sets g, we would have g C A . and hence gF^ = 0 , a contradiction. Thus gA^ = 0 , i = 1, 2, and as a consequence, g(A1 + A2 ) = 0, git-^ = 0, which contradicts w^gtt-^, w^€g, I = 1,2. We have proved that Fi*i, F2*1 do not contain points belonging to the same set ger(T, A).

398

CHAPTER VII.

IDENTIFICATION OF LEBESGUE, GEOCZE, PEANO AREAS

For any 1 let us consider the mapping (T, n^) ; i.e., the c. mapping from the compact set induced by (T, A) on Let r(T, ) be the corresponding family of maximal continua g* of constancy for T in Then each continua g*€r(T, jt-^) is contained in a set ger(T, A) [which may be larger than g f and contain even infinitely many disjoint sets g !er(T, ir^)]. Consequently, for each 1 , the two sets F2jt1 cannot contain points belonging to the same continuum g !€r(T, *-^). Also let us point out that both sets A2Jt-^ are both open in and sums of continua g !er(T, tc^). Let C the minimum set i = 1, 2, whichis sum of continua g fer(T, ) (10.2, iv). Thesets M^, i = 1, 2, are closed (10.2, iv). Letus prove that M^A^ = o, i = 1, 2. Suppose for instance M 1A 1 + o and take a point weM1Al. Then weg!, g'er(T, jt^), and since both M 1, A1jc-j^ are sums of continua g !, we have g 1 C M 1, g 1 C A 1, and by the definition of also g^F^jr^) + 0; hence g !F1 + 0. This is impossible since, from g 1 C A ]; we deduce g fF-| = 0. Let us prove that M 1M2 = o. Suppose M 1M2 + o and take w e M ^ . Then wegf, g*er(T, Jt^) and, since both M 1, M2 are sums of continua g r, also g 1 C M ], g f C M2, and g tF1 + o, g !F2 + o, that is F2*1 have points belonging to the same g ?er(T, jt^), what has been excluded before. By M 1M2 = o where both M 1, M2 are sums of continua g !€r(T, ic-^) we deduce (10.7, vi) that tji = {M1, M2)l > o where, for the sake of simplicity we have used a subscript 1 instead of the more complete subscript (T, *^). Now let € be any positive number. By (21.1, i), (21.2, i) there exists a finite collection of non-overlapping simple closed pol. reg. *lk ^ *1* k = diam T(*lk) < also

■'*' ml' such that k = 1, 2, ..., m^,

and

§2 5 . THE LEBESGUE AREA AS A MEASURE FUNCTION (25-1 )

399

ml X > ( * i k, T) < V(*l’ T) " N_1s' k=i

£

(3 )

We have

C

C A 1 + A2 , k = 1, 2,

...,

•

Let us prove that for each k and 1 we have either C A1, or C Ag, or both. First if = 0 , then Jtlk C A2; analogously if *11^2 = °> then C A 1 • Let us suppose now that * ^ ^ 1 ^ *1 1 ^ 2 ^ *lk ” ^1 ^ Then *11^1 ^ 0> "lk^i ^ 0 and diam T ^ik^ < 2""1^i^ (M1, )i = t]1 > 0 , where is surely connected. By (10.7, viii) we have *3.1^2 = °* From here it follows *11^2 = ° anc^ since *1^2 ^ °' also itlk C Ag. Analogously, if we suppose ^lkA 1 + 0, *lkA2 + °; *lk - A2 + 0, we prove it^k C A^. We have proved that, for any k, we have either *llc C A1, or C A2, or both, k = 1, 2, ..., 1, and this statement holds for any 1 = 1, 2, ..., N . Let us divide all the pol. reg. k = 1, ..., m^, 1 = 1 , ..., N, into two classes putting a reg. in the first class if C A1, and otherwise in the second class. By the above statement we have ir-^. C A2 for all of the second class. Denote by z1, z11 any sum relative to the regions of the two classes, respectively. By (2), (3) and (9 - 0 we have N N V(A, + A_, T ) < £ v ( i t , , '

d

1 =1

T) + s
' 1=1

(c) Let A = A1 + A2 + ... . By repeating the first part of the argument in (a) we can define a finite system of simple closed non-overlapping pol. reg. ^ C A, 1 = 1, 2, ..., N, such that N X 1=1

T) > V(A, T) - € •

The set J = Jt1 + *2 + ... + is compact, J C A, and each point wej belongs to at least one of the sets A^, i = 1, 2, ... . Hence by Borelfs theorem, there is a finite m such that J C A 1 + . . . + A m . By (9 - 1 ) and (b) we have now m V(A, T) < 1=1 X v U ,x T) + e < V(A.i + ... + Am , T) + e < m

oo

< X V(A., T) + e < X V T) + e. i=i i=1 Since e > o is any given number, (iii) is proved.

NOTE. The statement (iii) does not hold necessarily if the sets Aj_ are not sums of sets ger(T, A) as we can prove by using the same mapping (T, Q) of the example (1 3 •1 > A). Indeed, using the same notations of (1 3 -1 , A), if J1 = [0 < u
0;

i.e., V(J1# T) + V(J2, T) < V(Q, T).

b/l,

§2 5 .

THE LEBESGUE AREA AS A MEASURE FUNCTION (2 5 .3 )

25.2.

401

The Concept of Measure-Function

A family 21 of sets is called a RING if 21 is closed with respect to the operation of finite sum and finite intersection; W is called a FIELD if $1 is a ring and 21 is closed with respect to the operation of subtrac­ tion; 21 is called a a-ring, or a a-field, if 21 is a ring, or a field, and 31 is closed with respect to the operation of countable sum. A set function cp(A) defined in a family % of sets A is called MONOTONE INCREASING in 21 if q>(A) < cp(B) for all A, Be 2f, A C B; ADDITIVE in A if cp(A + B) = |G| - € [or > e“1 if |G| = + 00] where € > 0 is any given number. It is also enough to prove that, given e > 0, t > 0, there is a measurable set H C G such that |H| > IG| - e [or > e ” 1 ] and such that, for every (|, rj)eH, there exists aset J as above with IF(p ) - F(p0 ) - A| < t |p - pQ| for all p e j . By [S. Saks, I, p. 298] we know that F x . FTyr as well as F are measurable in G. Hence there is a closed bounded set H C G, with |H| > |G| - e [or > €~1] such that Fv, F__ exist ateachpoint of H and F, F„, F__ are ^ j A j continuous in H. Therefore the same functions are uni­ formly continuous in H and thus, given t > 0, there is a 8 > 0 such that |F__(p) - F__(p!)| < t for all p, p*eH, Ip - p fl < 5 and analogously for F and F. «y For every integer n, let A^, Bn be the sets of all points (|, T])eHsuch that, respectively, |F(x,

T)) - P(|, T)) - Fx (l; n)(x - l)l < n_1 lx for all

| - n-1 < x < | +

si n-1,

§2 6 . REGULAR APPROXIMATE DIFFERENTIALS (2 6 .2 ) lF(t, y) -

f (i , ti)

14-09

- Fy (|j n)(y - ^i)l < n_1 ly - nl for all

ri-n- 1 < y < r i + n -1.

Since F, Fa . FTyr are continuous in H, both A n, n are closed subsets of H. In addition C An+ n we have | H - A nl, |H - Bn l < e and, if Cn = A^B^ also |H - Cn l < 2€. Let n be the smallest integer such that n > n, n”1 < 5 , and let a = n~1, A = AQ, B = Bn, C = Cn . Let C 1 C C denote the set of all points U, r))eC which are points of density 1 for C as well as points of linear density 1 with respect to x and to y. By [S. Saks, I, p. 298], C 1 is measurable and|C 1| = |C |• For each (g, T^eC1 denote by i the set of all numbers h > 0 such that (£ + h, r])eC, (|, t} + h)eC, 1h| < 5 . The set i is of density 1 in h = 0 and symmetric around the same number. Denote by J = J(|, r]) the set of all points (x, y )eG such that x = I + h, |y - rjI< h, or |x - g | < h, y = ^ ± h. J has all properties above and is of density 1 in (g, t))• On the other hand, for every p = (x, y)eJ, say p = (x, y) = (| + h, y), we have peG, (g, T))eJ, (g, -q)eG1, and IF(p) - F(p0 ) -

A| = IF(x + h, y) - F(|, r))

+

hFx (g,

+ |F(g, y) - F(|, t]) - (y - ri)Fy (i, r])I + |Fx (g, y) “ px (^ T])|h< Th +

t

|y - r|| + Th
, or = zF(I!), where z ranges over all I !eS, where we use the usual conventions for the sum of real numbers (even + and - 00) and where we suppose explicitly that never may happen to add + 00 and - 00. All subadditive, overadditive, and additive functions are said to be NORMAL functions. Obviously the additive functions are both subadditive and overadditive; if F is subadditive [overadditive] then - F is overadditive [subadditive].

NOTE 1. More generally the set A above could be any ad­ missible set; (G) the collection of all sets G C A open in A. The collection (I) could be the collection (it) of all simple closed pol. regions it C A, or of all simple and

§2 7 . INTERVAL FUNCTIONS (2 7 -2 ) non-simple pol. regions x C A. the collection of all intervals such that l/X < l/lT,

The collection (q) could he Ie{I) of side-lengths 1 , 1 ! 1 /1 1

< X

where 1 < X < + oo is any given number. In this case we shall denote {q.) as the collection of all intervals Ie{I) of co­ efficient of regularity X [Cf. S. Saks, I, p. 106]. The collection {1 } could also be the family (I)R of all inter­ vals I C G such that a,a + 1 , and b,b + 1 * are elements of given sets {{■}, £-q) of real numbers everywhere dense. In this case {qj could be the collection of all Ie{I)R of a given coefficient of regularity X > 1 . Por other possible cases see the original paper of S. Banach [1 ]. Also, cf. the recent book: H. Hahn and A. Rosenthal, Set Functions, I, Chapter V, where the problem of differentiation of general set functions is deeply investigated.

NOTE 2 . For numerous previous applications of normal functions of interval to questions connected with length and area, see the books of L. Tonelli [I] and of S. Kempisty [];]. Consistent use of normal functions in questions of surface area has been made by T. Rado [19, II] and by L. Cesari [12, 28].

27.2.

BV and AC Functions

Given any function F(I), Ie{I), as in 2 7 -1 , let S denote any finite system of non-overlapping sets Ie{I), I C A. By total variation of F(I) in A we shall denote the number V(A) = V(A, F) = Sup X |F(I) |. S IeS Since the same definition holds if we replace A by any of the sets G€{G), or Ie{I), we have defined a set function V(G), Ge(G), or V(I), led). Obviously we have IP(I)I < V(I) and V(I) > 0 for all Ie {I). Moreover, V is monotone increasing in (I) and in (G) - We shall denote the set function V(I) as the total variation of F(I). The function F(I) is said to be of BOUNDED VARIATION in A, or BV in A, if V(A) < + 00. The function F(I) is said to be ABSOLUTELY CONTINUOUS in A, or AC in A, if

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

(a) given e > o there is a 5 > o such that for finite systems S of sets Ie{I) we have z|F(I)| < €, provided z|I| < 5 ; (b) F(I) is additive in A.

all

NOTE. The case where (I) is the collection of all simple pol. regions k C A and F is the function v associated to any c. mapping (T, A) has been considered in (9 * 3 )• For other par­ ticular cases see (9 *4 ), (9*6), (12.1). In the definition of absolute continuity S. Banach [1] requires only condition (a), but the condition (b) is added explicitly in the main statements [we have seen in (13 •1 ) that the conditions (a) and (b) are independent]. In the account which follows of the Banachtheorems of derivation (and only forthe sake of brevity), we have associated both conditions (a), (b)to the concept of absolute continuity of an interval function, in harmony with the definition of AC c. plane mappings givenin ( 1 3 •1 )• For amore detailed account of Banach1s theory, in connection also with the concept of Burkill!s integral, see the book of S. Kempisty [I].

27*3

(i)

(ii) (iii)

• •Some Preliminary Statements

If F(I) is any arbitrary interval function and V(I) the corresponding total variation, then for every sequence G, Gj_, i = 1, 2, ..., of sets G, Gie(G) (27.1) with G±Gj = 0, i + j, z ±(J1 C G, we have ziV(G1 ) < V(G). As in (i), if G = z±G, then Z±V ( G ± ) = V(G). If F(I) is any arbitrary function, then the corresponding total variation V(I) is over­ additive .

The same proofs hold as for (9 -1 * i* ii, iii)* Note here that, if a function F(I) is non-negative and overadditive, then F(I) = V(I) (iv)

Each BV normal function F(I) is the difference of two non-negative overadditive functions.

PROOF. If F(l) is overadditive, then F(I) = [V(I) + F(I)] V(I) where both F + V and V are non-negative over­ additive. If F(I) is subadditive, then - F(I) is over­ additive and F(I) = V(I) - tV(I) - F(I)], where both V and V - F are non-negative and overadditive. (v) (vi)

A function F(I) satisfies (a) if and only if V(I) satisfies (a). Any AC function F(I) is BV, provided IA | < + 00.

§27-

INTERVAL FUNCTIONS (27-4)

The same proofs hold as for

2 7 . k.

(1 3 •1 ,

i* ii)•

Derivative of An Interval Function

Given the interval function F(I), led), (2 7 - 1 )> let pQ he any point pQ€A°, let q denote any closed square qe{q) with pQeq and let 5 = s(q) he the diameter of q. Then by upper and lower derivatives F !(p0 ), F 1(p ) of F(I) at the point pQ we denote the numbers F T(p ) = lim — 0 5= 0 |q|

-

F !(p ) = "lim 0 6 -- - 0 |q|

}

- «> < F' < F ’ < + ^ f ! = F ! then their common value is de­ noted as the derivative F !(p0 ) of F(I) at pQ . LEMMA 1 . If a set J of points p = (x, y)eE2 has the property that for each point pej there is a closed square K with peK, K°J = 0 , then J is measurable and has measure zero. PROOF. If peJ, then peK* since K°J = 0 . Thus the outer density of J at the point p is < 3 A ov < 1/2, according p is a vertex of K* or an interior point of a side of K. As a consequence J has no point of outer density 1 . This implies |j| = 0 [S. Saks, I, p. 129]. LEMMA 2 . If the set J of points p = (x, y)eE2 has the property that for each point pej there is a closed square K(p) with peK(p), K(p) C J, then J is measurable. PROOF. Since J° (subset of all interior points of J) is open (or empty) and hence measurable, it is enoughto prove that = J - J° is measurable. Let peJ1 be any point of J 1 . Then peK(p) for at least one square K(p) and the following statements hold: (1 ) p is a boundary point of K(p); (2) all points p 1 interior to K(p) are not in J1; that is peK*, K°J = 0 . By Lemma 1 the set J-jhas measure zero; hence «T- as well as J = J° + J„ are measurable.

CHAPTER VIII. (i)

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

For every function F(I) the upper and lower derivatives F !(p), F f(p) are L-measurable functions in A0 .

PROOF. For every a > 0 let J denote the set of all points peA0 where F T(p) > a. Then for each peA0 there is a sequence K^p), n = 1, 2, ... , of squares K^p) such that peK^p), K^p) C A°, FtK^p)]: IK^Cp)l > a - n"1, |4(p)| < n~1,n = 1, 2, ... . For each n let Jn denote the setJn = ZK^p) where z ranges over all points pej. Let H = J1J2 ••• • Obviously J C Jn for every n; hence J C H. Let now p be any point of H. Then P€Jn for all n andhence PGKn (Pn ) ? o v at least one point pneJn ’ ¥e have ^ ^ n ^ n ^ ; - a “ n”^ and since PeKn ^Pn ) we have F f(p) > a. Thus each point peH is also a point of J, that is H CJ. Thereby we have proved that H = J. Now each set Jn satisfies the condition of Lemma 2, and hence Jn is measurable. As a consequence, also, H = J is measurable and F*(p) is measurable. Analogously for F !(p). (ii) Each BV normal function F *(p) a.e. in A0.

F(I)

has derivative

PROOF. By (2 7 *3 , iv) we can suppose F overadditive and non-negative. Therefore o < F T < F* < + °o everywhere in A0. By (i) the set J CA of all points peA0 where F l < F f ismeasurable.Suppose, if possible |J| > 0. If (i^, vn ), n = 1 , 2, ..., is any sequence containing all couples of rational numbers 0< < vn < +oo, and is the subset of J of all pej where ° < F ! < Un < vn < < + °°> then J = As a consequence there is at least one n such that IHJ > 0. Let H = |H| = IHJ = m > 0, u = un, v = vn; hence 0 < F ! < u < v < F ! < + oo for all peH, and |Hi = m > 0 . By [S. Saks, I, p. 218] there is in H at least one point P0eH of outer density 1. If q is any square with P0€(l> q C A°, we have lim |qH| : |q| = 1, lim F(q) : |q| < u as 5 (q) -0. Therefore given e > 0 thers is a square q = K with the following properties: (a) PQ£K, K C A0, iKHl : IK| > 1 - e, F(K) : |K| < u. For every point

§27.

INTERVAL FUNCTIONS (27-4)

peK0H let us denote by q any square such that peq, q C A0; thus lim F(q) : | q| > v as 5 (q) --- ►O. Therefore there Is a sequence qn (p)> n = 1 , 2, ..., of squares with p€qn (p), qn (p) C K°, &[qn (p)3 -- ► 0 as n -- F(qn ): |q | > v. By Vital!fs covering theorem [S. Saks, I, p. 109] there is a finite system of non-overlapping squares Kj, K2, ..., such that O) K± C K C A°, Pd^) : |K±| > v, i = 1, N, 2|K^l > |K°H|- €jK| • The system [1^, 1 = 1 , N] can now be completed by some system of non-overlapping intervals [I., j = 1, 2, ..., M] in such a way that [K^] + [I.] is a subdivision of K into a finite sys­ tem of (non-overlapping) intervals. Since F is non­ negative and overadditive, and because of (a) and (p) we have now, successively, u|K| > F(K) > EiF(Ki ) + SF(I .) > > v| KH|

Z±F ( K ± )

> ^±VI K±|

- ev|K| > (1 - 2e )v| K| .

As a consequence u > (1 - 2e)v, where e is any positive number, and hence u > v, a contradiction. Thus it is proved that |J| = 0 and thereby (ii) is proved. (iii)

For each BV normal function the set of points peA0 where F r = + co, or F r= — 00, has measure zero.

PROOF. By (2 7 -3 , iv) we can suppose F non-negative and overadditive. We have only to prove that the set J of the points peA0 where F 1 = + » has measure zero. By (!) we know that J is measurable. Suppose |J| = m > o, if possible. For each point pej and any number a > o there is a sequence qn (p), n = 1, 2, ..., of squares such that peqn (p), 5 [qn (p)] -- 0, F[qn (p)] : |qn (p)| > a. By Vitali*s theorem, given e > 0, there is a finite system of non-overlapping squares K^, i = 1, 2, ..., N, such that (a) K± C A, F(K± ): IKJ > a, z|K± | > |J| - e [or > e-1 if |JT| = + oo] . Consequently we have + oo > V(A) > ZiF(Ki ) > 2:±a|K± | > a[|J| - e], (or > ae“1), a contradiction, since a, e are arbitrary

416

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

numbers. Thus we have proved that (iii) is proved.

2 7 .5

(i)

|J|

= 0

and

. Banach!s Theorems

Theorem. For each BV normal function F(I), the derivative F !(p) exists a.e. in A° and is L-integrable in A0 . If F(I) is overadditive and non-negative then V(A°) > (A0 )/ F f(p), and for every Ie{I), also V(I) = F(I) > (I) / F T(p)•

PROOF. By (2 7 -3 , iv) we can suppose F(I) overadditive and non-negative in both parts of the theorem. Let Q, denoteany closed square of the p-plane E2 and m any positive number. Let g(p), peA°, be the function de­ fined as follows: g(p) = m if F*(p) exists and F T(p) > m; g(p) = F*(p) if F !(p) exists and 0 < F*(p) < m; g(p) = 0otherwise. Thus g(p), peA, is bounded and measurable in A and g(p) is also L-integrable in the bounded set QA°. Let U(I) be the additive interval function U(I) = (I) / g. For almost all points peQ°A° we have: lim U(q): |q| = g(p); lim F(q): |q| > g(p) as 5 (p) -- >-0 and q denotes any square with peq, q C A0 . The first statement is proved in [S. Saks, I, p. 118]; the second one is a consequence of (27.4, ii) and of the definition of g. Let J denote the set of the points peQ°A which have both properties; thus |J| = Given e > 0 , for each point peJ there is a sequence of squares qn (p)> n = 1 , 2 , ..., such that U[qn (p)]: |QX1 (P)I < g(p) + F[qn (p)]: |qn (p)| > g(p) - €, 5 [qR (p)] -- ^ 0 as n -- ►«>. By Vitalirs covering theorem (applied to the set J C Q°A°) there is a finite family [p^] of points p^, i = 1 , ..., N, and a corresponding system of non-overlapping squares such that (on) PieKi' Ki C W 0 , U(K± ): |K±|< g(p± )+ e; F(K^_): |Kj_| > g(pj_) - e, i = 1 , ■■•) N, and (p) IQA° - £KjJ < m“1e. By (a) and (p) we have now, successively,

|QA°I*

(J) / g(p) =

+ (QA° -

< s1[g(p1 ) + €] | | < ^F d ^)

/ g(p)


o. As m -- + 00 we have (A0 ) f P*(p) < V(A°). Theorem (i) is proved. (ii) Theorem. If F(I), Ie{I) is a non-negative, overadditive, BV interval function, then a necessary and sufficient condition in order that V(A°) = (A°) / F'(p) is that F(I) is AC in A0. In these conditions we have also V(I) = F(I) = (I) / F'(p) for all led). PROOF. By Theorem (i) the function P l(p) exists a.e. in A and is non-negative and L-integrable in A0; hence the interval function U(I) = (I) / F f(p) is non­ negative, additive and AC in A0 . By Theorem (i) we have also V(I) = F(I) > U(I) for every Ie{J). NECESSITY.Suppose V(A°) = (A° ) f F 1 and let K be any closed interval, K C A°. Let Dv^, n = 1, 2, ...] be any sequence of figures (5 * 1 ) such that K C ^ C A0, f A°, and each ^ is a sum of intervals I. Then for each n we can divide into a finite system Sn of intervals I, one of which is K. By Theorem (I) we have V(K) = U(K) + a, a > 0 , V(I) > U(I) for every ^€Sn - Hence, if z denotes any sum ranging over all- (K), we have V(A°) > VCM^) >V(K) + ZV(I) > U(K) + a + ZU(I) = a + (M^) / F»(p). As n -- ►» the last integral approaches (A0 ) / P f(p) = V(A°); hence V(A°) > a + V(A°), a > o, and finally a = 0. Therefore V(K) = F(K) = U(K) and this relation holds for all intervals K C A0 . Thus F(K) as well as V(K) are additive and AC in A0 . SUFFICIENCY. Suppose F(I) be AC in A0. Given € > o let 5 > o be the number defined by condition (a) of (27*2). Let K be any closed interval K C A0 . Almost all points peK have the following properties: peK°; F ’(p) exists and o < F !(p) < +

k F !(p) - e, F[qn (p)]: lqn (p)l < F !(p) + c, B[qn (p)]-- ► () as n -- ► oo. By Vitali!s covering theorem applied to the set J, there exists a finite system of points p^J, i = 1, 2, ..., N, and a corresponding system of non-overlapping squares such that K± C K, U(K._): |K± | > F ,(pj_) - €, F(K^): |K± | < F t(p± ) + €, i = 1, ..., N, |K - ZKjlI = |J - ZK± | < 6. Let us denote by [I ., j =1, ..., M] any finite system of intervals such that [K^] + [Ij] is a finite subdivision of K into (non-overlapping) intervals. Thus and> (a )> ZF(Ij) < e. On the other hand, by condition (b) of (27.2) we have U(K) = z±U(Ki ) + F(K) = Z^FdC^) + ZjF(Ij). Now we have U(K) > Z-UO^) > Zj_|Ki l[Fl(pj_) - e] > > Zi |Ki i[F!(p± ) + €] - 2 e |K| > EiF(Ki ) - 2e|K| = = [F(K) - ZF (I* )] J

2 €|K|

> F(K) - € - 2e |K| .

Since € is any positive number we have U(K) > F(K), while, by Theorem (i), F(K) > U(K). Hence U(K) = F(K) and this relation holds for every interval K C A0 . Let e be again any positive number. By definition of V(A) there exists a finite system S of non­ overlapping intervals I C A0 such that 0 < V(A°) - ZF(I) < g , where Z ranges over all IeS. Hence V(A°) < ZF(I) + e < z(I) f F 1 + e < (A°)/Ft + €, and finally V(A°) < (A°) / F r since g is any positive number. On the other hand by Theorem (i) we have V(A) > (A0 ) / F T; hence V(A°) = (A0 ) / F*. Thereby Theorem (ii) is proved.

*27.6.

Bibliographical Notes

Besides the original paper of S. Banach [1], see the already quoted books S. Kempisty [I] and H. Hahn and A. Rosenthal [I]

§2 8 . GENERALIZED JACOBIANS (2 8 .1 ) for references; the latter especially for the deeply discussed question of the derivative of general additive set functions. For further studies on some types of non-additive interval functions including the normal functions see L. Tonelli [30]/ and S. Faedo [1]. On Burkill integral and derivation of in­ terval functions let us quote here I. C. Burkill [1 , 2, 3], L. C. Burkill and U. S. Haslam-Jones [1, 2], S. Saks [2, 6].

§28.

GENERALIZED JACOBIANS 28.1. The Generalized Jacobian of A Continuous ------------ riane ------------

Let (T, A) be any continuous plane mapping from an admissible set A of the w-plane E2, w = (u, v), into the p-plane E^, p = (x, y). Then, by (9 *1 ), the functions v ( jt, T), v + ( tc, T), v”(jt, T) are defined in the class Cit) of all simple pol. regions jt C A, the functions V(A!, T), V+(A!, T), V~(A!, T) are defined in the class of all admissible sets A ! C A; in particular in the class { j t }, and V, V , V” are the total variations of v, v+, v" in the sense of (27*2). [The total variations of V, V+, V“ are the functions V, V+, V~ themselves (2.7.3, iii)]. Since V(A, T) = V(A°, T) and the same holds for V+, V~, we shall restrict ourselves to the open set A° of all interior points of A. The family {q) of all closed squares q C A with sides parallel to the u- and v-axes is con­ tained in {n}» By using the notations of (2 7 - 1 and 2), we shall say that v, v+, v~, V, V+, V” are non-negative functions, that V, V+, V" are monotone-increasing and overadditive in {*), that v, v+, v~, V, V+, V" are BY functions in {*} if and only if (T, A) is BV (9* 1 ), and that v, V are AC if and only if (T, A) is AC (13-1 )- Therefore, by (2 7 -5 , i), if (T, A) is BY, the derivatives Y !(w), Vj(w), V*(w) exist and are finite a.e. in A0. It is convenient to put V T = 0, V| = 0, V^ = 0 at all points weA - A° and where Y f, Yj, V [ are not defined, or + 0 0 . Thus the functions V !, Y|, are defined everywhere in A and finite­ valued. By (2 7 .5 , i) and by V+ (q, T) + V~(q, T) = V(q, T) (12.8, ii), we have (i) For any BV plane mapping (T, A) from an admissible set A C E2 into E^, the non-negative functions V f(w), Vj(w), V|_(w), weA, are L-integrable in A, V* = Vj + V^ a.e. in A and V(A, T) > (A) ; V'(w), V+(A, T) > (A) / VJ(w), V“(A, T) > (A) / V^(w). For every BV plane c. mapping (T, A) as above we shall denote by GENERALIZED JACOBIAN 3 (w) = 3 (w; T) the function 3 (w) = VJ_(w) - V_^(w) for all weA. We have |3 j = |V_“!T - V “1 1| = V ! a.e. in A. Hence, “| < — |V| + + V—

42 0

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

(ii) For every BV plane c. mapping (T, A) fpom an admissible set A C E2 into E 1, the general­ ized Jacobian ^ (w), weA, is L-integpable in A, we have 3 = V| ICs| < V_J_ + V_J_ = V T a.e. in A, and (A) / |3 (w)| < (A) / V|

+ (A) /


o r V| = 0, V|_ = V 1; hence 3 = + V 1 a.e. in A.

PROOF. Since V f = V| = = 3 = 0 in A - A0 we may pestpict oupselves to A°. F o p evepy Integep n = 1, 2,..., let us denote by 1 ^, I ? , I1n, I2n the subsets of A° whepe V| > 0, > 0, V| > n-1, V [ > n~1 pespectively. Let € be any positive numbep. By (12.9, ii) thepe are two finite systems M m = [q.lnL M2n = [q2n] of closed pol. regions q, all non-overlapping, such that S 1iu(q, T )I > V+ (A°, T) - n-1e, Z'»|u(q, T)| >V"(A°, T) - n-1e,

where zf, zrf denote sums ranging over all qe[q]ln, qe[q]2n- If M 1n = z’q, M2n = z!tq denote also the sets of all points of the regions we have M°nM2n = 0 and sets k° - M1n, A0 - M2n

q,

lM*n l = °> lM2n^ = °* us nov observe that the are open, hence admissible. By (12.14 ), we have

V+(A°, T) > £+V+ (q, T) + V+(A° - M1n, T) >Z'|u(q, T)| + V+(A° - Mm , T), and hence V+(A° - Mm , T) < V +(A°, T) - E'|u(q, T)| < V+ - (V+ - n-1 e) = n-1e. Analogously we have we have

V-(A° - M2n, T) < n -1e.

On the other hand, by (28.1, i)

§2 8 . GENERALIZED JACOBIANS (2 8 .2 )

421

0 < (A0 - M m ) / V; < V+ (A° - M m , T) < n-1 e, 0 < (A° - M2n) / V_^ < V"(A° - M2n, T) < n-1 e, and finally, 0 < |(A0 -

M1n) l m |n

1 < (A° - M m )

/ V| < n 1 € ,

0 < |(A0 - M2n)I2n|n_1 < (A° - M2r) / th a t

i s , I(A

we have

AC T

T

M 1n)I1nl < 6,

-

- M2 n )I2 n l < e.

|(A

(A ° - Mm ) + (A ° - M2 n ) + M*nM*n

in 2n

= T

T

In 2n

A°

I 2n[]'l n ^

W a n K “

V'_

< n_1e;

S in ce

=

=

0

and a ls o

- Mm> * (A° - M2n> * M*nH2n] ‘

1n f] + ■t ln [I 2n^A “

’ ] + ^ ln I 2n ^ MlnM2n^’

l ^ n ^ n l S U,„l * K n < n l < 2 t ’ where € is any positive number. n and finally |I-jI2 I = 0.

This implies

= 0 ^or ever^

Therefore V|V^ = 0 a.e. in A° and since V r = VJ_ + V [ a.e. in A°, we must have either V| = V ! , V^ = 0,or V^ = V 1,Vj = 0 a.e. in A0 . Finally 3 = V* a.e. in A° implies 3 = +V 1 a.e. in A0 . Thereby (i) is proved. (ii)

For every BV plane set A C E2 into (A)

f

mapping (T, A) from an admissible the equality |3 (w)| = V(A, T)

holds if and only if (T, A) is AC. Under these conditions the functions V+, V~ are also AC and (A) / V 1 = V(A, T), (A) / Vj = V+ (A, T),

(A)

f V'_ =

V"(A, T).

PROOF. Since V 1 = VJ. = V'_ = 3 = 0 in A - A0 and V(A°, T) = V(A, T) we can restrict ourselves to A°. Since (T, A) is BV the functions V, V+, V" are BV and 3 = V* - V'_, V 1 = Vj +V'_ a.e. in A. By (2 7 -5 , i) and (28.1 ) we have (A°) /

13 1

= (A0 ) / |V» - V | < (A°) / V ' + (A°) / V*

(1)+

If


0 2 sin 0) + (ag cos 0 + b2

2

> h2,

|T(w)

-

T(w* ) |

possible - u! = is the minisin 0) 2 as

> h|w

-

w! | •

Por any square q with q* C E denote by 1 the side-length of q, by A the parallelogram which is the image of q under T; hence, |a| = |J|1 2, |q| = I2. Let 0 < o < 1 be any number and denote by q 1 the square of center wQ with sides parallel to the u- and v-axes, and side-length 1(1 - a). Then q ! C q, (q!*,q*) = 2~1crl. Denote by A 1 the parallelo­ gram which is the image of q f under T; hence, |A1| = (1 - a )2 |J| l2, {A1*, A*) > 2~1hcrl. On the other hand, |T(w) - T(w)| < pe(p) < 2le (2 1 ) for all weq. If 1 is small enough, we have e(2l) < 4 “1ha, hence, |T(w) - T(w)| < 2~1hlcr for all weq. If we denote by C the curve C: (T, q*), the relation above Implies ||C, A*|l < 2“1hlcr. Thus the curve C is outside A 1• Analogously we could prove that C is inside the parallelogram A*1 which is the image of the square q !* of center w and side-length

k2k

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM (1 + cr) 1 . On the other hand, since, for every peA*, we have {p, A*} > {A1*, A*) >

>

2 -1h(Tl

> ||C, A* II,

and by (8.3, i), we have 0(p; C) = 0 (p; A*) = + 1 for every peAT. Therefore, (3 )

V(q, T) > (E|) / | 0 (p; C)| > |A' | =

(l

- ff)2 |J||q| •

This relation holds for all q with q*C E and 1 small enough. As 1 --- ►() we have V !(w) > (1 - a)|J|, and since cr is any number 0 < o < 1, we have also V f(w) > |J|. (b)

In the reasoning above if + 1

for every

J > 0 we

have

0

(p; C) =

(p; A*) =

peA1 and hence, instead of (3) we have

V+(q, T) > (E|) /

0 +(p;

C) > |A'| = (1 - a)2J|q|,

and hence Vj_ > J. Analogously, if J < 0, we have 0 (p; A*) = - 1 for every peA*, and, hence, V"(q, T) > (Ejp I and, hence,

0

0 '(p;

C) > |A'| =

(1

0

(p; C) =

- |J| .

(c) Let us suppose, If possible, that V 1 > |J| in a set I C A° of positive measure |I| > 0. As in ( 2 7 . k , ii) we can determine two real numbers a, b, 0 < a < b < + °o, such that the set I C I of all points weA°where V r(w) >b > a > |J(w)| haspositive measure |I | = m > 0. We may suppose that at all points weIQ we have either V| = V f, V [ = 0, or V_|_= 0, V'_ = V 1 (28.2, i). We may also suppose that each point weIQ is a point of density for IQ. Since we satisfy these conditions by disregarding subsets of measure zero in IQ, we have still |IQ| = m > 0. Let 0 < e < l be a number small enough so that a < a 1 < b r < b, where a f = a(l + e ) + e , b* = (b - €)(l -e). Let us ob­ serve also that N(p; T, A) is L-integrable in E^ and hence there exists a t > 0 such that (h) / N(p; T, A) < me for every measurable set h C E^, IhI < 2mi.For each point wel0 we shall consider a sequence [qn ] of closed squares of center w, diameter 5(Q.n ) -- ^ an-d such that q* C E.

§2 8 . GENERALIZED JACOBIANS (2 8 .3 )

425

Thus all points weIQ have now the following properties: (1 ) |J| < a; V' > b; (2) |qn r 1V(qn )-- ► V* > b; (3) (4 )

lq.n l_1 • U 0q.n l — Ii the curve Cn : (T, q*) iscontained, in an annular region between two parallelograms C Q^1 C of center p = T(w); (5 ) |qn l_1 IQ^I -- H J I < a, 1q.n I“1 IQ^1I -- ► |J| < a, lqn r 1lQl!ll - Q^l -- *-o,as n ----- » (6) 0 (£; C) = 0 in E^ - Q£»; 0(|; C) = + 1 and constant in Q^. By Yitali!s covering theorem [S. Saks, I, p. 109] there exists a finite system [w] of Ndistinct points weIQ and a corresponding finite system [q] of N non-overlapping squares q of centers we[w] withq* C E, such that, if C: (T, q*), then C is contained in Q ff - Q 1 where Q ! C Q 1T are parallelograms of center p = T(w), we[w] and (a) V(q) > b|q|, IQ !I> IQ11I < a|q|, lQM - Q 1I < t Iq I for all qe[q]; ((3 ) either Y+ (q) > b |q|, V“ (q) < e |q|,or V+ (q) < e|q|, V~(q) > b|q|; (7) I q P 1 |IQq| > 1 - 2~1e; ( d ) |IQ - Zq| < me where Z ranges over all qe[q]; (e) 0 (|, C) = 0 in E^ -Q M ; = + 1 and constant in Q 1. By (7), (s) we have first

U q l > l 2 l 0ql = | I 0 - ( I Q -

= l^ 1 >

|zq| =

Z |q|

m - m€ = (1


« + Q') + (Q" - Q.')] / n(p; = (E| - Q " + Q') / = (Q') /

Consequently since

|n| < N,

|V+ (q) - V"(q)|
) / N(pj T, q),

and (p), forevery

T, q) =

qe[q]

we have

426

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

(b - e)|q| < a|q| + (Qf1 - Q,f) / N(p; T, q). By adding all these relations relative to the squares and observing that zN(p; T, q) < N(p; T, A) (12.1 4 ),

qetq], we have

(b - e)z|q| < az|q| + (z(Q*» - Q «)) / N(p; T, A). Here (1 - e )m < z|q| < (1 + e )m, and Iz(Q!1 - Q !)I < Z|Q!1 - Q,1| < xz|q| < (1 + e )mt < 2mT, and finally, because of the choice of t , also (b - € )(1 - e )m < a(l + € )m + me;

(d)

that

is, b ! 0, V_J_ + V^ = 0; hence VJ. = V^ = 0, 3 = V* - V|_ = 0 = J a.e. in AQ. This proves that 3 = J in A . Statement (i) Is completely proved.

*28.4.

Derivative of The Function cp(M) of (1 4 .8 )

Let (T,A) be a BVplane mapping from the admissible set A CE2 into E^ and let M be any B-measurable subset of A. Then the functions v(p; T, M), v+, v“ are L-integrable in E^ (1 4 .7 ) and non-negative; hence the set functions cp(M) = (E^) I v(p; T, M), cp+(M) = (E^) / v+, cp~(M) = (E|) I v~ are non-negative and finite. By (1 4 .7 ) we have = + and is additive in the collec­ tion of all B-measurable sets M C A . If for every simple pol. reg. jt C A, we consider cp(it°) as a function of * [i.e., F (it) = q>(jt°)], then cp(*°) is overadditive in the collection {it) of all simple pol. reg. it C A (1 4 . 8 ). In addition cp(jt°) is BV in A. Analogously for cp+> Consequently (2 7 -5 , Theorem i) the derivatives cp1, cp|, cpj_ of the set functions cp, cp+, cp exist a.e. in A0 and we can define them as equal to zero in all remaining points of A0 as well as in A - A°. Obviously,

cp cp+ cp~

cp

a.e.

§2 8 . GENERALIZED JACOBIANS (28.4) 0 ! < (Ej«) J v‘.

PROOF. For any square M = q C A, for M = A°, as well as for any open setM= G C A° the statementis a consequence of (2 7 *5 , i)- Since 0 < v < N, o < cp1< V f where N, V r are L-integrable functions in E| and E2and analogous relations hold for v+, v“, by (1 4 .7 , i and ii) the statement is immediately extended to all B-measurable sets M C A . (ii)

If (T, A) is any BV plane mapping from an admissible set A C E2 into E^, then we have cp! = V 1, cpj = VJ_, c p = VJ_ a.e. in A°.

PROOF. For every simple pol. reg. x C A we have 0 < cp(jt) = (E,p f v < (E£) / N = W(x, T) = V(*, T). Consequently, 0 < cp! < V 1 a.e. in A and analo­ gously 0 < cp| < VJ_,0 < cp^ < V^ a.e. in A°. Suppose that we have cp1 < V 1 in a set I C A° of positive measure. Then (see proof of 27 . k , ii) there is also a pair of real numbers a,b, 0 < a < b < + °o, such that the set J of all weA°where cpr < a < b < V r has positive measure |J| = m > o. For every integer n let {QJn be the family of all closed squares q “ [a± < u < a±+1, bj < v < b.+1J], a± = i2~n + 3~r\ bj = j2~n + 3~n, i, j = 0, + 1, + 2, ..., which are contained in A° (Cf. 1 4 .5 , proof of ii). For each point wej and for all n large enough there is one and only one square qe{q)n with weq0 and, on the other hand, for all n large enough we have cp”(q) < a|q|, V(q) >b|q|. If t > 0 is any integer, by Vitali!s covering theorem, there exists a finite system o f non-overlapping squares q, each be­ longing to some CQ.^n with n > t, such that

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

cp(q) < a|q.|, V(q) >a|q|, |J - ZJq| < t~ 1 . Therefore, = ZV(q) - Zcp(q) > (b - a)z|q| > (b - a)|zjq| > > (b - a)[|J| - t"1], where Z ranges over all qe[q]. . Since |J| = m > 0 , for all t > 2 m“~1, we have A^_ > 2 —1 (b - a)m. Now we have Afc= ZW(q) - Zcp(q) = (E|) f [ZN(p; T,q) Zv(p; T, q)] dp, Afc = (E^) / [*t (p) -tt (p)] dp, where^ = ZN, = Zv. Here we have o < ^ < < N(p; T, A), where N(p; T, A), peE^, is L-integrable In E^. Let us denote by H the set of all points peE^ where N(p) = + °°, and by D = D(T, A) the countable subset of E f defined in (12.5)• Then |H + D| = 0 . By (1 4 .4 , iii), for each peE^ - (H + D) the collection Cg] of all continua ger (T, A), _j p Sk

those

ge[g]p

which are proper

continua, if V l > v2, •••, are the numbers consider­ ed in(1 b .4 , iii) we have |v1| + ... + | I = N(p). On the other hand (lb. 7) |v | + ... + |v, | = v(p; T, A),

1

and finally

v1 = j(w1),...,

= j(wk ) where j(w)

is the local index defined in (1b .6 ). Let 6 = 5 (p) the minimum diameter of g^ + 2m”1, *T22~^ < 5 for all t > tQ . Since all squares qetq]^ have diameters *f22~n < *J~22~^ < 5 no one of the continua gk

*’*' ^k€ ^ p

+1’

can

completely contained in a

square qe[q]^.. Let us consider now the difference ^ and observe that, for all t > tQ, both ^ and ^ are sums of only those numbers |vs |, s = 1, ..., k^, relative to elements w = getg]^ with weq° for some qetq]^. Consequently, ^ ^ = 0. This proves that ^ - tj- -- ► 0 as t -- ► + 03 for each peE^ - (H + D); that is a.e. in E^. As a consequence A, -- 0 as t ---- ► + °°, while we have proved that A^. > 2 1 (b - a)m > 0, a contradiction. This proves that —

§2 8 . GENERALIZED JACOBIANS (2 8 .5 )

cp! = V ! a.e. in A0 . An analogous proof holds for q>| = V|, cp_|_ = VJ. a.e. in A0 . Thereby (ii) is proved.

*28.5.

A Characterization of The Generalized Jacobian

We shall use here the local index j(w), weA, defined in (1 4 .6 ). Let us denote by sgm x, x real, the number + 1, - 1, 0 according x > o , x < 0 , x = o . We can now complete the statement (28.2, i) as follows: (i)

Theorem. If (T, A) is any BY plane mapping from an admissible set A C E2 into E^ have 3 (w) = V* (w)sgm j(w) a.e. in A.

we

PROOF. Since V 1 = V* = = 3 = j = 0 in A - A0 we may restrict ourselves to A0 . Let J+, J”, JQ denote the subsets of A0 where j(w) > 0, j 0, =0; v ! = V [ > 0, V| = 0; V 1 = V* = = 0,respectively. The sets J , J”, JQ are B-measurable (1 4 .6 , ii) and I+, I", I are L-measurable (28.1, i). We have only to prove that |(I+ + I’)JQ I = 0, |I+J“|= 0, |I'J+ | = 0. Indeed, if this is proved, then a.e. in I+ + I” we have j + 0; a.e. in I+ we have j > 0, a.e. in I” we have j < 0 and the statement is proved. Let us prove that the set H = (I+ + ^as measupe zero. Let be a B-measurable set such that H1 C H C A°, |H - H1| = 0. Then, by ( 2 8 . i and ii) we have (H, ) /V' = (H, ) / o everywhere in H 1• Thus IHJ = o and hence 0 = |H1| =

k30

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

An analogous reasoning holds for (i) is proved.

I~J+. Thereby

*28.5 - Bibliographical Notes The definition of generalized Jacobian as it has been given in (28.1) is due to T. Rado and P. Reichelderfer [7] and supplants previous equivalent definitions of T. Rado [20, II] and L. Cesari [12]. The theorems (28.2, i) and (28.2, ii) had equiva­ lent formulations in T. Rado [20, II] and L. Cesari [12]. The part (a) of the proof of theorem (28.3, i) is exclusively due to T. Rado [19]. The remaining parts of the proof of (28.3, i) have had different equivalent formulations in the already quoted papers of T. Rado, P. V. Reichelderfer and L. Cesari. The statements in (28.4) and (28.5) are due to T. Rado [17, 20, III] and T. Rad6 and P. V. Reichelderfer [4 ].

§29.

FORMULAS FOR THE TRANSFORMATION OF AREAS AND DOUBLE INTEGRALS 29.1 . Formulas for The Transformation of Areas (i) THEOREM. If (T, A) is any c. plane mapping from an admissible set A of the w-plane E2, w = (u, v), into the p-plane E^, p = (x, y), if (T, A) is BV and AC, if J(w, T), weA, N(p; T, A), n(p; T, A), peE^, denote the generalized Jacobian (28.1) and the absolute and relative multiplicity functions (12.1 ) (12.10), then (A) / |3 (w, T )| = (Ep

(

I

N(p; T, A),

1) (A) /

3

(w, T ) = (E|) / n(p; T, A) ,

and also (A) / V 1 = (Ep / N, (2 )

(A)

f V; =

(Ep

f

N ,

(A) / VI = (E,p / N 1 .

PROOF. The function V is AC and hence (27.5, ii) we have (A) / V ! = V(A, T) and, since V(A, T) = W(A, T) = (E4,) / N, we have also (A) / V 1 = (E^) / N. By (28.2, ii) also the functions V+, V” are AC and hence an analogous reasoning holds for V+ and V”. Thus relations (2) are proved. Since 3 = V|_ - V|_, |3 | = V* a.e. in A (28.1 H28.2, and n = N+ - N” a.e. in E^ (12.10), relations (1) follows from (2). (ii)

THEOREM.

Under the conditions of (i), if

K

is any

§29*

TRANSFORMATION OF AREAS AND DOUBLE INTEGRALS (2 9 -2 )

B-measurable set K C B = T(A), then H is B-measurable and

if

431

H = T"^(K) C A,

(H) / J(w, T) = (K) / N(p; T, A), (3 ) (H) / 3 (w, T) = (K) J n(p; T, A) , and analogous relations hold for

V*, V S V*.

PROOF. The B-measurability of the set B = T(A) and H C A follows from (16.1, Note). Let K be open in B, then H is open in A, H is an admissible set (12.12, ii) and N(p; T, H) = N(p; T, A) in K, N(p; T, H) = 0 in E^ -K. By (i) above we have (H)

/ |3 (w)| = (E^)/ N(p; T, H) = (K) / N(p; T,

A).

Analogously for the second of the relations (3). Since 3 Is L-integrable in A, and n < N where N is integrable in E^, both relations (3) can beextended to every B-measurable set K C B by repeated operations of limit. An analogous proof holds for V 1, Vj, V^.

29.2. (I)

Some Remarks on L-measurability of c. Mappings

If (T, A) is any BV and AC plane mapping from any ad­ missible set A of the w-planeE2, w = (u, v), into the p-plane El, p = (x, y), if K CB = T(A) is any L-measurable set and H = T” (B) C A, thenthere are certain B-measurable sets HQ, H1 C A, KQ, K1 C B such that Hq C H C H1 C A, Kq C K C K1 C B, |K1 - KQ| = and V' (w) = 0 a.e. in H1 - HQ • Therefore,VJ = V I = a.e. in H, - HQ and (a) (HQ )/ V' = (H,) / V' (Kq ) / N = (K1) /N. Analogous equalities hold VJ., N+, Vi, N~ •

0, 0 = for

PROOF. By [S. Saks, I, p. 69] there are two sets KQ, Kj such that K0 C K CK|, KQ an F^-set, K{a G5-set, |K] - KQ | = 0. Hence KqB = Kq, KB = K, K, = KjB C Kj, ^ C B, KQ C K C ^ C K], |Kj - KQ | = 0, and K 1 as the intersection of two B-measurable sets, KJ and B, is also B-measurable. On the other hand, if HQ = T""1(KQ ), H^ = T”1(K^ ), by(28.2, ii) we have (HQ )/ V* = (KQ ) / N, (E, ) J V 1 = (K, ) J N, and hence (IL, - HQ )f V 1 = (K1 - KQ ) f N = 0. Since V* > 0 and0 < Vj, V^ < V 1 everywhere in A, we have V* = V[ = V* = 0 a.e. in H1 - HQ. Thus (oc) is a consequence of the equalities above.

1+32

CHAPTER VIII. ( ii)

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

Under th e c o n d itio n s o f ( I ) , and th e s e t

H = T - 1 (K ) C A

if

K C.B

th e r e a re c e r t a in B -m easurable s e ts Kq , K 1 C B such t h a t K0 C K C

i s any s e t

i s L -m e a s u ra b le , th e n HQ, H1 C A,

Hq C H C H1 C A,

K, C B, Hq =T " 1 (KQ), H, = T " 1 (K,

),

I - Hq | = 0 and N(p) = 0 a.e. in K1 - KQ. Therefore also N+ = N~ = n = o a.e. in - KQ and (p) (Hq ) / V* = (H) / V f = (H, ) / Y 1 = (Kq ) / N = (K1) / N. Analogous equalities hold for VJ., N+, V_[, N”. PROOF. By [S. Saks, I, p. 69] there are two sets H^, Hj such that H» C H C Hj, an Fff-set, Hj a Gg-set, |Hj - H^l = 0. Hence H^A = H^, HA = H, Hj 1 = Hj A C Hj,Hj» C A, H' C H C Hj 1 C A, |Hj * - H^|= 0, and HjT, as the intersection of two B-measurable sets Hj and A, is B-measurable. Let us prove that A - Hjf is an Fff-set. Indeed, A - Hj1 = A - AHj = A(E2 - Hj 1), where E2 - Hj 1 is an Fa-set and A is in any case an F0 (16.2, Note); hence E2 - Hj1, and A are bothlimits of increasijng sequences of compact sets, hence the same holds for A(E2 - Hj1), i.e., (E2 - Hj1)A is an Fg-set [or an F-set, or a G-set]. Let

M = T (A

- Hj ’ ), KQ = T (H £ ), HQ = T_ 1 (K0 ), K, = B

- M, R, = T -1 (K1 ).

By (16.2, Note) M and KQ are F0-sets, hence is B-measurable and both HQ, H1 are B-measurable in A. Let us prove that H ’ C HQ C H C C H1 C HjT C A, KQ C K C K] C B. If weH^ then p = T(w)eKQ, T*“1(p)eH0 and since weT_1(p), also weHQ. Thus H^ C HQ. On the other hand, H^ C H, KQ = T(H^ ) y K = T(H) and hence KQ C K.This implies T”1(Ko ) C T-1(K), i.e., HQ C H. If peK, then T”1(p) CH C Hj1, T~1(p)(A - Hj !) = 0, and finally p is not in M, i.e., peK^ .This proves that K C K] and this implies T“1(K) C T”1 (K^ ), i.e., H C H1• In addition, ifwg H^ then p = T(w)eK1 and thus p is not in M, w is not in A -Hj1 and weHj1. This proves that H^ C Hj1. All stated inclusions are proved and hence we have also H1 - HQ C Hj1 - H^. By (i) we have (HQ ) / V 1 = (KQ ) / N, (H1 ) / V* = (K1 ) / N, and hence (K, - K0 ) / N = (H-, - Hq ) / V != 0; thus N = 0 a.e. in K, - KQ. (iii) Under the conditions of (i), if f(p), P^B, is a given function and F(w) = f [T(w)], weA, then if f(p)N(p) is L-measurable in B, then also F(w)V,(w) is L-measurable in A. An analogous statement holds for fN+, FVJ., and for fN”, FV[. PROOF. a nd

Let

B - J

J be the set of all points peB with are L-measurable. Since f = fN: N in

N(p) = 0. Then B - J, also f

J is

§2 9 . TRANSFORMATION OF AREAS AND DOUBLE INTEGRALS (2 9 .2 )

433

L-measurable In B - J. By [S. Saks, I, p. 7 5 ] there isa B-measurable set K C B - J such that f(p) is B-measurable in K and |(B - J) - K|= Let H = T~1(K). Then H is B-measurable and F(w) is B-measurable in H (16.2). Now we have (J) / N = 0, (B - J - K) / N = 0, and since B - K = J + (B - J - K), also (B - K) / N = 0. On the other hand, we have A - H = T~1(B - K), hence, by (i), (A - H) f V 1 = 0 and finally V ! = o a.e. in A - H. Therefore FV! is L-measurablein A. (iv)

Under the conditions function and F(w) = Is L-measurable in L-measurable in B. fN + , F V },

and f o r

of (i), if f(p), peB, is a given f[T(w)], weA, then, if F(w)V!(w) A, then also f(p)N(p) An analogous statement holds for

is

fN - , FV'_.

PROOF. Let I be the set of allpoints weA where Y !(w) = 0. Then I and A - I are L-measurable and, since F =FV1: V ! in A - I, also F is L-measurable in A - I. By [S. Saks, I, p. 72] there is an increasing sequence [R^] of compact sets C HQ+1, C A - I, such that F(w) is continuous on each and such that, ifH^ = lim Hw C A - I, then |A- I - H J =0. Let = T(Hn ). Each is compact and we shall prove that f(p) iscontinuous on each K^. Indeed, F(w) is -uniformly continuous on each and hence, given e > 0, there is a a > 0 such that |F(w) - F(w!)| < € for all w, w ^ H ^ |w - w*| < cr. Let pQ be any point of K^. Then, by (1 0 .4 , iv) there is a 6 = &(P0; Hq) > °> such that |T~1(p), T“1(p0 )| < a for all P6^ * Ip - P0 I < As a consequence there are two points w0eT~1(pQ ), weT~1(p), w, ^eH^, such that Iw - wQ | = (T~1(p), T~1(pQ )) < cr, hence |f(p) - f (P0 ^ = lF (w ) - F ( wQ )\ < € Thus f(p) is continuous at each point Pq6^ and, therefore, f(p) is continuous in Since R^ C Hn+1, C A, we have C ^ C B and = lim exists. We have also = T(Hqo). Consequently, is a OF -set, Koo C B, f(p) is B-measurable in 0K0 7 , and B - K oo is a 7 ’ ^ B-measurable set. Let H* = T~ (B - K^), and let us prove that H* C A - H00 . Indeed, if weH*, then p = T(w)eB - K00 , p is not in K 00 and in no set hence T~ (p)^ = 0 for all n and finally T”1(p)HQo = 0, i.e., T“1(p) C A - H^ and alsoweA - H^. This proves that H K A - Hw . As a consequence H* C (A - I - H^) +1, V !(w) = 0 a.e. in H*, and, by (29.1 , ii), 0 = (H*) / Y 1 = (B - K j / N. Hence N = 0,3fN = 0 a.e. in B - K 00. On the other hand, fN 7 is L-measurable in K00 and hence fN is L-measurable in B. *

*

^

NOTE. It is well known that if f(p) Is L-measurable in some L-measurable set M, then for every real a, b, a < 0 b, is L-measurable,

0.

434

CHAPTER VIII.

GEOMETRICAL PROPERTIES AMD THE SECOND THEOREM

bounded, and hence L-integrable in M, and the same happens for the charac­ teristic function 9 ab(p) of the subset of M where a < f < b. These notations shall be used in the statement below.

(v) If the functions f(p)G(p), G(p), peM, are L-measurable [L-integrable] in an L-measurable set M, and g(p), peM, is any other L-measurable function with I g (P)I < IGr(p)I in M, then all functions fab&> ^ab^' in M.

^ab^' ^ab^

are L “measurat>le

[L-integrable]

PROOF. If Mq is the subset of M where G = 0 , then MQ and M - MQ are L-measurable sets and f = fG: G is L-measurable in M - M . Hence all functions fab, epab, fabG, fabG, fg, fabg, 9 abg are L-measurable in M - Mq . Since those mentioned in (v) are all zero in MQ, they are L-measurable in M. If fG, G are L-integrable, then also |fG|,|G| are L-integrable and all the functions in (v) are L-integrable because of the inequalities: |fabG| < IfG|, Iq>abG| < |G|, |g| < |G|, |fg| < |fG|, ^^*ab^ ^

< \?s\> l (12.10) also N = N+ + N”, n = N+ - N~ a.e. in B. Therefore the L-integrability of F|3 | is equivalent to the L-integrability of FV1; the L-integrability of F 3 is a consequence of the L-integrability of FV|, FV^; the L-integrability of fn is a consequence of the L-integrability

§2 9.

TRANSFORMATION OF AREAS AND DOUBLE INTEGRALS (29-3)

of fN+, fN“; (7 ) is equivalent to (4 ); (7) and (8 ) are consequences of (5) and (6 ). If f is bounded also F is bounded and viceversa. If f is bounded, - M < f < M, and if fN is L-measurable, then fN is L-integrable; by (29*2, iii), FV’ is L-measurable and, since |FV!| < MV!, FV 1 is also L-integrable. Analogously, if F is bounded and FV’ is L-measurable, then by (29*2, iv) also fN is L-integrable If fN Is L-integrable, then also |fN| = |f|N is L-integrable hence L-measurable; since, by (29.2, v), also fN+, fN” are L-measurable and |fN+ | = |f|N+, |fN”| = |f|N“, the functions fN+, fN~ areL-integrable. Analogously, if FV1 is L-integrable, then also |F1V 1, FV|, FV^ are L-integrable. Suppose that f is the characteristic function of a set K C B. Then also F(w) is the characteristic function of a set, namely of the set H = T~1 Suppose that either one of the func­ tions fN, FV! is known to be L-integrable, then both are L-measurable and L-integrable (29.2, iii, iv, and (b) above). By using the notations of (29.2, ii), let H , , KQ, K 1 be the B-measurable sets there defined, HQ = T"1(KQ ), = T“1(K^ ) HQ C H C H1, KQ C K C K1. By (29.2, ii) we have (HQ ) / V ! = (Kq ) / N and since F = 1 in HQ, f = 1 in KQ we have also ( a ) (H ) / FV! =(K ) / fN. On the other hand, we have |H1 - Hq | = 0 (29.2, ii) and F = 0 in A ; hence FV* = 0 a.e.in A - HQ and (p) (A - HQ ) / FV* = 0. Finally, N = 0 a.e. in K 1 - KQ (29-2, ii) and f = 0 in B - K 1; hence fN = 0 a.e. in B - K Q and (7) (B - KQ ) / fN = 0. By adding (a), (p), (7) we have (A) / FV! = (B) f fN; thus ( k ) is proved. By (c) above, both fN+, fN“ are L-measurable and L-integrable and the same reasoning above proves (5) and (6) under the same conditions. Suppose that f(p), p^B, takes only a finite set of distinct values m^, m2, ..., in n disjoint sets K 1, K2, ..., K^, = B, 1 < n < + °°. Then the function F(w) takes the same values m^ in the n disjoint sets = T"1(K^), = A. Suppose that either one of the functions fN, FV1 is known to be L-integrable then both are L-measurable and L-integrable. Let f^(p), peB, be the characteristic func­ tion of the set K^. Then = f in B and, by (29*2, v), each function f-^(p) has the property that f^N is L-measurable and L-integrable in B. If F^ = f^[T(w)], weA, i = 1, 2, ..., n, then F = Em^F^ and, by (d), (A) / F^VT = (B) / f^N. By multiplication by m^ and addition,

CHAPTER VIII.

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

we obtain (4 ). By (c) also fN+, fN~ are L-integrable and the same reasoning above proves (5) and (6 ) under the same conditions. Suppose that f(p) is anybounded function, - M < f < M, and that either one of thefunctions fN, FV 1 is known to be L-integrable. Then both fN, FV 1 are L-measurable and L-integrable (see (b) above). If we divide (- M, M) into n equal parts (a^, ai+1), i = 1, 2, ..., n, of lengths ai +1 - a± = 2 Mn""1, - M = a 1 < a2 i), (29*1, ii), (2 9 -3 , i) extend, to the case where A is any admissible set, previous theorems which in equivalent forms, have been proved by T. Rado and P. V. Reichelderfer [ b ] , and by L. Cesari [27], for the case A is a simple closed Jordan region. An extension of (29.3, i) analogous to (29.1, ii) can be easily formulated. The present theorems supplant also various previous more restrictive state­ ments as in W. H. Young [ b ] , G-. Andreoli and P. Nalli [1]. See also R. Caccioppoli [10]. §3 0 . THE SECOND THEOREM 30.1.

The Area Derivative

Let (T, A) be any c. mapping from an admissible set A of the w-plane E2, w = (u, v), into the p-space E^, p = (x, y, z), and let (Tp, A), r = 1, 2, 3, be the plane mappings which arethe projections of T on the coordinate planes E2r, r = l , 2, 3,(5 -4 ). If we consider the family Cit} of all simple pol. regions it C A (in par­ ticular the family {q} of all closed squares q C A), then the Lebesgue area L(*r, T) is an overadditive function in [it] ( 5 - ^ b , iii; 2 7 -0 - If L(A, T) < + 00 then L (it, T) is a BV function in{tt} (27.2). For every point weA0 let us consider all squares qe{q), weq, and let &(q) denote the diameter of q. By (2 7 -5 , i) the limit (1)

L'(w) = L'(w, T) =

lim 5 ----*- 0

|q.|

exists a.e. in A0. We shall define L'(w) = 0 everywhere in A - A0 as well as in all, points wsA° where the limit (1) does not exist, or is + oo. Thus L(A, T ) < + oo implies that the function L !(w) (defined every­ where in A, finite and non-negative) is L-integrable in A and, by (2 7 .5 , i) (2) We shall denote by

L(A, T) > (A) / L f(w). L f(w) the AREA DERIVATIVE of

(T, A).

If we suppose V(A, T) < +», then the function V(*, T),*e{*}, is a BV overadditive functionin {it} (9 -3 > Hi; 2 7 «1 ) and hencethe limit (3 )

V T(w) =V*(w, T) =

lim -- 0

5

P) |q I

exists a.e. in A°. ¥e shall define V !(w)= o everywhere in A - A° well as in all points weA° where the limit (3) does not exist, or is + 00. Thus V(A, T) < + 00 implies that thefunction V !(w) (defined

as

438

CHAPTER VIII.

everywhere in A, by (2 7 -5 , i),

GEOMETRICAL PROPERTIES AND THE SECOND THEOREM

finite and non-negative) is L-integrable in

(4 )

A

and,

V(A, T) > (A) / V* (w). (i)

If L(A, T) < + °°, then V 1(w) = L !(w) every­ where in A and V r, L r are L-integrable func­ tions in A.

This statement is a consequence of the equality

L = V

proved in (2 4 . 1 ).

NOTE. If we do not make use of the deep-seated equality L = V [proved in (2 4 . 1 )], then from the elementary inequality L > V proved in (9 *4 , ii) it follows only L ! > V ! a.e. in A.

From V(A, T ) < + 00 it follows V(A, Tp )< + 00, r = 2, 3 (9 -1 , i) and the mappings (T , A), r = 1, 2, 3, are BV (12.8, ii). For each mapping (Tp, A) we have already defined the derivatives V !, VJ., VJ and the Jacobian 3 = VJ. - V[ and, in order to avoid confusion, we shall denote by Vp (w), V^+(w), V^_(w), 3 r (w), weA, the derivatives and the Jacobian rela­ tive to (Tr, A). By (28.2, i) we know that either V^ = V^+, V^ = 0, or V*, = V^_, V^+ = 0 a.e. in A and that

Pp(w)|

(5 ) For each (6 )

weA

=

V£(w)

a.e. in

A.

let 3

(w) = [ ^ ( w ) +

(ii) Theorem. If in A.

322

(w) +

3 32

V(A, T) < + 00, then

(v)]5 . V T(w) =

3

(w) a.e.

PROOF. The proof which follows is divided into eight parts lettered (a) through (h). Ofthese only the elementary parts (a) and (b) are actually applied in the following chapters [(b) is not utilized in the parts (c) through (h) of the proof]. The present proof of the theorem above is com­ pletely independent of the deep-seated equality L = V proved in (2 4 .1). (a) For every weA° and square q with weq, q C A, by (21.5, i) we have V(q, T) > [zpV2(q, Tp )]2; hence | q | _1V (q , T ) > { 2 r [ i q l “ 1V (q , T ) ] 2} 2 .

As 5 ---- ► 0 we deduce V f(w) > [ZpV^2 (w)]2 a.e. in A°. By V ! = V^ = 0, r = 1, 2, 3, everywhere in A -A°, and by (5) we conclude V 1 > [Zp5p2]2. i.e., V' > 3 a.e. in A.

§3 0 . THE SECOND THEOREM (3 0 . 1 ) Let us prove the equality V* = 3 a.e. in A under the addition­ al hypothesis that the three mappings (Tp, A), r = 1 , 2, 3, are AC. Let € > 0 be any positive number. By (9 - 1 ) there is a. finite system S of non-overlapping closed simple pol. regions ir C A such that

V(A, T) - e < zv(jt, T) = zjzrv2 (*, Tr ) ] s where 2 is extended over all have also

jteS.

Since

v < V

(12.1) we

V(A, T) - e < z [z rV2 (it, Tr ) ] 2 . Since the mappings (T , A) are AC, r = 1 , 2, 3 , the functions V(*, Tp ), jte{jr}, are AC (2 7 -2 ) and hence, by (2 7 *5 , ii) and relation (5 ) above, we have V(it, Tp ) = («) for every

iteS.

JV£(w) = (#) J|3r(v)|,

r = 1, 2, 3,

By the inequality (2.1 0 , C ) we conclude

V(A, T) - e < Z { E r [ (,t) J l 3 r (w)l ] 2} 2 *

f
■2 2 ~ wel0 are points where V !(w), 3 (w) = +32 + are the regular derivatives of their integrals in A. Consequently, for almost all points wel , we can suppose that we have also Iq.n I_1 • (q.n ) / V 1 V'(w) - a,

| q | “ 1V (q , Tr ) < V«,(w) + a, r =

1, 2, 3 ; ( 5 )

lql_1 • (q.) / V ' < V '( w ) + a, f i r s t consequence i s t h a t (e )

IqI-1 • (q) / 3 > 3 (w ) - a. |£ * q | < ( l

| q I Q| > (1 - 2 ^ 1 ) | q | , A

- 2ij’ 1 ) _1z * | q l 0 l
a , by (10) we deduce a2

^

u ( j t t , T) < 62cr^,

*f€S,» and hence (11)

X

Since

n(q) < a|q|

u ( * f , T) < 62a .

we have also

U(q, T)
I

{ Z r[ Z

Iqlc] }

21i

lu(«', Tr )| - | q | a ] 2 }

where each expression in brackets has to be replaced by zero if negative. Since each qetq] is interior to some *eS and we have already defined the numbers Pp relative to each jt1 and the numbers ap relative to n, (** C q°,- q C *), we have, also by formula (12) above 0 > Z #{ Z r [ Z lPr lu(#*, T) - |q|a]2} 2 ir*Cq° = Z { Z r[ Z

lar lu(nS T) +

*-cq°

Z

(iPr l " IapI )u ( 1>

« 1c q °

,2 1 |

- IqlaJ }

> Z

{ Z r [l“plU(q, T) + mr ]2} * ,

where mr =

Z

^ Pr* "

T) ~

jt'Cq0 and where, as above, each expression in brackets has to be re­ placed by zero if negative. Therefore, the last relation can be written in the following form: a* > Z { Z

[l«r lU(q, T) +

|

,

where 0 < 0p < 1, and 0p = 0 if oop > 0; 0p = 1 and |orp |U + cjop > 0; 0p = - a>p~1|ap |U if a>p < 0, |ap |U + a>p < 0 . Since a 2 + oc2 + a 2 = 1, by (2.10 ,

if

oj

c ) we have .1 2

U(q, T) = { [ Z p l « r l u f } 2 = { Z r [l«p|U + V r

“ Vr]}

JL 2

,{Zp[l.p|U + V r ] f ; { ^ r [ V r n < { Z r [ lar |U + er“r] Y

and finally, also by (12.9, iii),

+ Z #r 1(JQp 1

< 0


^ U ( q , T) Z r K 1 = S v ( q , T) - £ Z r l 1> ***' v > ? OI> a 1 1 Hn€'-Hn^* Thus we have iT^w) - T2 [Hq(w)]| < n“ 1 for all we7^ where Hn(w)e7^ and this implies C. ~ c!_, i = 0, 1, ..., v (2.6). Thus (d) is proved and thereby also the statement (i) and the Note are proved.

is any one of

CHAPTER IX.

THE REPRESENTATION PROBLEM

(ii) If (T1, J1 )~ (T2, J2 )and both T 1 and T2 are light (10.8), then T 1 and T2 are Lebesgue equivalent, i.e., there exists a homeomorphism Hbetween J1 and J2 such that T 1 = T2H. PROOF. Both r and are the collections of thesingle points of J1 and J 2 ; thus the mapping t defined in (i) is a homeomorphism between J1 and J2 and T 1 = T2t.

3 1 .5

.

Some Further Examples

The conditions (a), (b), (c), (d) are far from being sufficient as the following examples prove. V. Let J1 = J2 = [0 < u, v < 1] and T 1: x = u, y = v, z = 0, (u, v)eJ1; T2: x =k u 2 - bu + 1, y = v, z =0, (u, v)eJ2. We have T 1(J1)= T2(J2 ) = Q = [0 < x, y < 1, z = 0]. Each point (x, °)eQ>x k 0, isthe image of one point (u, v)eJ1 and of two points (u, v)eJ2 . Conditions a, b are satisfied, c, d, are not. It is natural to say that T 1 covers Q once and T2 covers Q twice as a veil folded along the line x = 0, 0 < y < 1. T 1 and T2 are not F-equivalent. VI. Let J.J, J2 as above and T 1:x = u, y = 0, z = 0, (u, v)eJ1; T2: x = 2(w, J*), y = 0, z = 0, (u, v)eJ2, where (w, J*3 denotes the distance of w = (u, v) from the boundary J* of J2. Here T 1 is constant on each segment g^ = [u = t, o < v < 1], T2 is constant onthe boundary g£ = [(w, J*) = 2~1t] of each square concentric to J2 and T ^ g t )= T2 (g^.)= (t, 0, 0), 0 < t < 1. Hence t is defined by g^ = ^ s seSment < x < y = 0, z = 0], then T 1 (J1 ) = T2(Jg ) = s. IfC: (T1, J*), C f: (T2,J*), then C is the segment s counted twice, C ! is the single point (0, 0, 0).Thus conditions (a), (b), (c) are satisfied, (d) is not. T^ and T2 are not F-equivalent. VII. Let J1 =J2 = (u2 + v2 i = 2, where g| is still though in J2, I = 1, 2. Then t has all required properties and (c) holds. Finally both (T^ J*), (T2, J*) are the curve reduced to the single point ( 2 , 0, 0). Thus (a), (b), (c), (d) are satisfied. Nevertheless T^ and T2 are not F-equivalent, as we will prove formally in the following lines. Let us suppose, if possible, that there exists |T.j(w) - T2(H(w ))| < 1 for all .

H

such that

Let T 13, T'23 be the projectionsof T ^ T2 on the (x, y)plane, then also |T13(w) - T23(H(w ))| < 1 for all wej^ . Suppose, e.g., that H retains the positive orientation. Then let R = [3 • 2~1 < p < 2] be the region in bounded by J* and by the counter-clockwise oriented circumference 1 = [p = 3 *2~1], and let R f = H(R) C J2. Thus R r is the region bounded by J* and the curve 1 ! = H(l). Since each continuum k such that k C R, kJ* 4 0 ,diam T^(k) < 1, is mapped by H into a continuum k ! such that k ? C R 1, k !J* 4 0, diam T2(k!) < 2 we have k ’g ’ = 0 since [g], J*)T =2. Thus k•g^ = 0 and also T23(w) 4 (0, 0) for all weR1- J*. If L2: (T23, 1 !) and L: (T23, \ ) where x C R ! is any circumference [p = p , 1 < p 0 < 2 ] = \ C r !, we have 0(0, 0, Lg ) = 0(o, 0; L) = - 1 (8 .1 4 , iv). On the other hand, if C: (T13, 1 ), we have ||L2, C || < 1, {(0, 0); C) = 1 and hence 0(o, 0, Lg ) = 0(o, 0; C) = + 1, (8 -3 , i), a contradiction. Therefore no H exists such that 5(T^ T2H, ) < 1 and hence T 1 and T2 are not F-equivalent.

CHAPTER IX.

THE REPRESENTATION PROBLEM

NOTE. The extremely difficult problem of a necessary and sufficient (topological) condition in order that two mappings T-j, T2 are F-equivalent has been com­ pletely solved by J. W. T. Youngs [ ’\b] not only for c. mappings from plane closed Jordan regions, but for c. mappings from 2-manifolds.

31.6.

F-distance

Let (T^ A1 ), (T2, A2 ) be any two continuous mappings, or surfaces S1: (T.j, A1 ), S2: (T2, A2 ) from the admissible sets A1, A2 of the w-plane into the same p-space E^ and let us suppose that A^ and A2 are homeomorphic, i.e., the class (H) of the homeomoirphisms between A^ and A2 is not empty. The non-negative number 5=

||T1, Tp || = Inf Sup|T-(w) - T p[H(w)]|, 1 ^ H€{H) W€A1 1 2

0 < 5 < + oo, -

is said to be the FRECHET DISTANCE of (T1, A1 ) to (T2, A2 ). If T 1 and T2 are oriented and we consider only the class fH)Q of the homeomorphisms between A^ and A2 which retain the positive orientation, then 5 is the Frechet distance for oriented mappings. The following statements are easily proved: (i) ||T1, T2 1| = 0 if and only if T, ~ T2; (ii) ||T, T|| = 0; (iii) ||Tn, T2|| = ||T2, TJI; (iv) ||T1, T3 1|< ||Tr T2|| + ||T2, T3||; (v) if T1 ~ T *, T2 ~ T^, then H^, T2 1| =||T], T|||. Given any two FrSchet surfaces S^, §2, that is, two maximal classes C^, C2 of F-equivalent mappings, let \\S^, 5 2|| = ||T.j, T2|| for every T^eC^ T2eC2. Because of (v) the number HS^, §2|| does not depend upon the choice of T 1 in C1 and T2 in C2 . The number , S2 || is said to be the F-distance between the F-surfaces S^, §2. Obviously, HS.J, S2|| = 0 if and only if 3 1 = S2,1)3 ,, S2 || = ||§2, Sj, l|S1, §3I | < H^, S2 || + ||§2, §3||. Given any two mappings (T^, A^ ), (T2, A2 ), let us denote by (T1p, A 1 ), (T2r> r = 2, 3, the plane mappings which are the projections of T ^ T2 on the coordinate planes. Then obviously ||T1p, T2p|| < ||T-j, T2|| for all r = 1 , 2 , 3 *In particular||T.j, T2| | = 0 implies ||Tlp, T2p|| = o, r = 1, 2, 3, (but not vice versa).

3 1 .7

(i)

. Invariance of Lebesgue Area with Respect to Frechet Equivalence

Let (T, A), (Tn, An ), n = 1, 2, ..., be c. mappings from homeomorphic admissible sets A, A^ into the same

§31.

FRECHET EQUIVALENCE (31-7)

p-space E^> such that ||Tn, T|| -- *■0 as n -- >»• Then (1 ) L(A, T) < lim L(A^, Tq )j (2 ) ¥(A, Tp )< lim W ^ , T^); (3 ) N(p; Tp,A) < 1 1 m N(p; Tm , An), as n --- > and p€E2p. Analogousstatements hold for V, W+ W", V+, V". PROOF. Let 5 n = ||T, Tn ||, n = 1, 2, ..., hence e>n -- >0 as n -- >°o. For each n let be a homeomorphism between A and An such that |T(w) - T^H^w)]! < 6n + n~1 for all weA. Then d[T, A] -- ► 0 as n -- >oo, and, by (6.3) and (5-10) we have LCA^, Tn ) = L (A,TnHn ^ L(A, T) < lim L(A, TnHn ). Therefore L(A, T) < lim L(A^, Tn ) as n ->00. An analogous proof holds for (2) and (3). (ii)

If (T1, A 1 ) ~ (T2, A2 ), thenL(A^ T 1 ) = L(Ag, T2 ), ¥(A1, T lp) = ¥(A2, T2p); N(p; Tlp, A1) = N(p; T2p, Ag ) for all peE2p and r = 1 , 2 , 3 * An analogous state­ ment holds for V, V+, V~, ¥+, ¥“.

PROOF. Let (Tn, A^) = (T2, A2 ), n = 2, 3, ..• . Then ||T1, Tn|| = 0, n = 2, 3 > •••> and, by (i) above, also L(A^, T 1 ) < lim L(AQ, T ) = L(A2, T2 ). Analogously we have L(A2, T2 ) < L(A^, T 1 ) and hence L(A^, T-j ) = L(A2, T2 ). An analogous proof holds for V, ¥, N.

NOTE 1. Given any Frechet surface 15, i.e., any maximal class Cof F-equivalent mappings (T, A), let L(S) = L(A, T) for every (T, A)eC. By (ii) the number L(S) does not depend upon the choice of the element TeC and is said to be the Lebesgue area of the FrSchet surface S.

NOTE 2. (T, A) ~ (T1, A 1) implies also V(A, T) = V(A!, T 1). Analogously for plane mappings (12.1 ) (T, A) ~ (T!, A f) im­ plies V(A, T) = V(A», T»), ¥(A, T) = ¥(A», T 1), N(p; T, A) = N(p; T', A 1) for every p, V+(A, T) = V+(A', T*), etc. . The proof is the same as for (i).

NOTE 3. The statement (2 3 *3 > ii) can be reinforced by the re­ quirement (T , A) ~ (T, A). The proof is the same. Indeed in (23.12) we have proved that THQn — frTQ uniformly in A where TQ = THQ . This implies T ~ TQ. As a further conse­ quence we have also that the inequality V(A, T ) < V(A, T) proved in (23.12) can be replaced by V(A, T ) = V(A, T).

CHAPTER IX.

458

3 1 »8

THE REPRESENTATION PROBLEM

. The Representation Problem

Given any c. mapping (T, A) from any admissible set A of the w-plane E2, w = (u, v), into E^, we know that the condition L = L(A, T) < + » does not imply that the functions x, y, z have first partial derivatives in some points of A0; hence the ordinary Jacobians J2, J^ of the couples (y, z), (z, x), (x, y) need not exist and neither the classical area integral JL l I = I (A, T) = (A0 ); [J2 + J2 + J2]2 dudv = (A0 ) / [EG - F Z f dudv. As we have proved the condition L(A, T ) < + oo implies that the generalized Jacobians 3 .j, 3 2, 3 ^ exist a.e. in A0, are L-integrable in A0, and hence the generalized area integral i. 3 = (A0 ) I [3 2 + 3 2 + 3 2 ] 2 dudv exists, is finite and we have L > 3 (the second theorem, 3 0 .3 , i, or L2 of 1 .4 ). We haveL = 3 if and only if the plane mappings (T , A), r = 1 , 2 , 3 , are AC.On the other hand, if the components x, y, z of T(w) have ordinary first partial derivatives a.e. in A then 3 p = Jp a.e. in A0, r = 1 , 2 , 3 , and the generalized area integral coincides with the classical area integral (but not necessarily with the Lebesgue area L ). The concepts of F-equivalence and of F-surface, in connection with the second theorem, lead naturally to the following main problems: I. Given any F-surface S with L ( S ) < + oo, is there at least one representation (T, A) of S such that L = 3 ; i.e., the Lebesgue area is given by the generalized area integral? II. Given any F-surface S with L(S) < + co, is there at least one representation of S where the functions x, y, z have first partial derivatives a.e. in A° and L = I; i.e., the Lebesgue area is given by the classical area integral? We shall see (3 7 *3 , ii) that both problems (representation problems) have an affinnative answer.

*31.9. If

Some Additional Observations and Bibliographical Notes , A2

are compact sets of

E2

and

(T^ A 1 ), (T2, A2 ) are

§31.

FRECHET EQUIVALENCE (31-9)

F-equivalent mappings, T 1 ~ T2 from A 1 and A2, respective­ ly, into E^, then we may consider the collections r(T.|, A 1 ), r(T2, A2 ) of maximal continua of constancy for T 1 in A 1 and for T2 In Ag, respectively, (1 0 .4 ). Letr2 be the corresponding hyperspaces, i.e., the collections r^, r2 con­ sidered as topological spaces (1 0 .4 ), and the decompositions T 1 = 1 ^ , T2 = l2m2 (10.8) of T 1 and T2 in products of monotone and light mappings. Thus m 1 , m2 are monotone, 1 ^, 1 2 are light, and m 1(A1 ) = m2 (A2 ) = Theorem (3 1 *4 , i) implies that r^, ?2 are homeomorphic and that 1 ^ , lg are Lebesgue equivalent. Indeed the mapping r from onto r of (3 1 *4 , i) can be thought of as the homeomorphism ^ _i ~ ~ _i t = m2xm1 from r1 onto r2, and then 1 2 = 1 ^? . In addition we have T2 = l2m2 = ll(T~1m2 ) = l^m^, where T"1m = m^ is a monotone mapping and = t ”1(f2 ) = T”*1m2 (A2 ) = m^(A2 ). We conclude that T ^ T2 admit of simul­ taneous decompositions T 1 = l^m^, T2 = 1^^, having the same hyperspace and the same light factor. If two mappings (T^, A^ ), (T2, Ag ) from compact sets .A 1, A2 admit of decompositions T 1 = lm^, T2 = lm2, where m 1, m2 are monotone, having the same hyperspace m 1(A1 ) = m2(A2 ) and the same light factor 1, then T ^ T2 are not necessarily equivalent. They are said to be Kerekjarto equivalent, or K-equivalent. For instance the two mappings of the example VTI are K-equivalent but not F-equivalent. The example VTI is due to J. W. T. Youngs [9]. Thus K-equivalence is a necessary but not sufficient condition for F-equivalence of c. mappings from compact sets. The ex­ tremely difficult problem of determining a necessary and sufficient condition for F-equivalence has been recently solved by J. W. T. Youngs [1 4 ] as we have already mentioned. J. W. T. Youngs has indeed formulated such a necessary and sufficient condition for the F-equivalence of c. mappings from 2-manifolds. On K-equivalence see also E. J. Mickle [7]. For further questions on F-equivalence and related questions, see the basic already quoted paper of J. W. T. Youngs [1 4 ], and also B. v. Kerekjarto [1], M. Frechet [5], C. B. Morrey [1, 2, 3], T. Rado [II], E. J. Mickle [4 ], E. J. Mickle and T. Rado [2, 3], P. V. Reichelderfer [4 ], W. R. Scott [2], J. W. T. Youngs [4 , 5, 6, 7, 8, 11, 12].

46o

CHAPTER IX.

§32.

THE REPRESENTATION PROBLEM

MEAN VALUE INTEGRALS OP L-INTEGRABLE FUNCTIONS 52.1.

Definition of Mean Value Integral

Let G be any open set of the w-plane E2, w = (u, v), let K be any compact set K C G, and let hQ = (K, G*} > 0 be the distance of K to the boundary G* of G [hQ be any number if G = E2 and thus G* is empty]. Let f(w), weG, be any single-valued real in G. For every point w = (u, v)eK and n > 2h~1 let q = q denote the square —1 nw _2 v < y < v + n 1 apea n completely f (n)(w) = n2 - (qnw)

ft

= n2 •

f

u+n- 1

u

L-integrable function for every integer q = [u < x < u + n~1, contained in G, and

f

v+n- 1 f(x,

y)

tody,

v

W€K.

The fimction f^n ^ = f^n ^(w), weK, n > 2h~1, is said to be the MEAN VALUE INTEGRAL of f of index n.

NOTE. The mean value integrals constitute a useful process of approximation in real functions theory. We shall discuss their properties in view of various appli­ cations to the representation problem. For applications of mean value integrals to surface area see for instance H. E. Bray [1], T . Rado [1 7 ], C . B. Morrey [1 ]; for applications to potential theory see G- T. Evans [I]. A brief account of mean value integrals is given in S. Saks [I, p. 178] and L. M. Graves [I, pp. 2 5 4 -2 5 9 ]* The present discussion is modeled partially (52.2, 3, 5, 7) on C. B . Morrey1s [1 ].

52.2. First Properties of Mean Value Integrals Given certain functions f, g, ..., all L-integrable in G, we shall denote by f ^ , g^n ^ ••• their mean value integrals on a common compact set K C G. (i)

If f, g are both L-integrable in G, and c is any constant, then (f + g)^n ^ = f^n ^ + g^n ^; (cf)(n) = c(f(n)).

(ii)

If lf(w) - f(wf)l < Mlw - w !I for all w, w TeG (M < + 00 a given constant), then also If(n)(w) - fTn)(w-)| < Mlw - w'| for all w, w TeK and the same constant M.

§3 2 . M E M VALUE INTEGRALS OP L-INTEGRABLE FUNCTIONS (3 2 .2 ) PROOF.

If

w = (u , v ) , w ' = 1 ( u I , v ' )

and

a = u ' - u,

b = v ' - v, th e n (a 2 + b 2 ) 2 = |w - w ' | , l^n w l = n~2 > and | f ( n ) (w ‘ ) - f ( n ) (w )| = n2 | ( q ) f l f ( x + a , y + b ) f ( x , y ) ] d x d y |; 1 hence | f ^ ( w ') - f ^ ( w ) |
o, there is a number -q > o such that (H) / |f | < n e —P for every measurable set H C G, |H| < t]. Let a = min[n , 4 “%n] and w = (u, v), w ! = (uT, v l)be any two points of K with |w - w !| < a. Then|u u !| < cr, |v - v T| < a q, q ! are the squares qnw, qn¥,, both of side-lengthn”1, then q,. q* are partially overlapping. The common part qq* is a rectangle whose area is > (n -1 cr) ; 2 hence |q - qq!|, |q! qqT|< n”2 - (n~1 -a)2 < 2an”1 < 2”\. Therefore |f(n)(w) - f (n)(w')| = In2 • (q - qq')"; f - n2 • (q* - qq') / f| “2 n2 (ne) = e for all w, w*€K, |w - w !|< cr. Thus the con­ tinuity of f(n ^ is proved. —

(iv)

If

f

(k ) /

is L-integrable in G, then i f ( n ) i < (g ) ;

if i.

PROOF. Let m > 3 be any integer and let a = m-1n"1. Let us divide the w-plane in the squares Q^. = [ia < u < (I + 1 )a, ja < v < (j + 1)a] of side-length a, (i, j = o, + 1, ... ). If w = (u, v) is anypoint of K, then for convenient r and s we have ra < u < (r +1)a, su < v 0, a^ > 0, i = 1, ..., N; |aa - ba | < a|aa“1 + b^”1| |a - b| for any a, b > o, or > o. |aQf- b a | < | a - b | Qr for any a, b > 0, 0 < a < 1.

PROOF. Of (Nm) < 0< b < for any < ata^-1 and 0 Hence

If m = max [a-, ..., a™] then (a1 + ... + a^)^ < J0£ C£ C£ \ !T(a1 + ... + a^) and (1 ) is proved. Suppose now a. Then aa - b^ = (a - b), where b < I < a, a > 0. In addition ia“1 < aa“1 if a >1, ba“1 If 0 < a < 1. In either case 0 < aa - ba < +ba~1](a - b) and (2) is proved. Suppose 0< a < 1 0, there is 5 >0 such that f(w) - e < f ( w f) < f(w) + € for all w, w f€Kp, |w - w 1| < 5 ;in particular for wgK, ^2n“1 < 5 . By integration in qnw we have f(w) - e < f^n ^(w) < f (w) + e for all n > 2 1/25 _1, weK. (ii)

If f is L-integrable in G, then f(n ) -- a.e. in K as n ------ >°°.

PROOF. This statement is a consequence of the well-known Lebesgue theorem for differentiation of L-integral functions [S. Saks, I, p. 115]. (iii)

If f is L-integrable in G, (K) / |f - f | -- > 0 as n -- «.

This statement is a particular case of (iv).

then

CHAPTER IX. (iv)

THE REPRESENTATION PROBLEM

If f is La-integrable in G, (K) / |f(n) - f|“ -> 0

then n ----*».

a > 1,

as

PROOF. By La-integrability of f in G and [E. J. McShane, I, p. 229, 4 2 .4 ] it follows that, given e > o, there is a de­ composition f = g + h, where g is continuous in G, h is La-integrab1 e in G and (G) / |h|a < e. By (32.2, i), state­ ment (1) above and (32.2, v), we have In = (K) / |f(n) - f|a =(K) ; |(g(n) + h (n)) < (K ) / [lg(n) - g| +

|h(n)| + |h|]a

< 3a (K) /

|g(n)


00. Thus I < 3 € for all nlarge enough. Thereby (iv) is proved.

NOTE. The statement (iv) holds also for 0 < provided f is supposed to be L-integrable.

a

< 1,

32.4. Further Theorems of Convergence Let be the Euclidean p-space, p = (x^, ..., x^. Given any set R C Em and any two constants 0 < 0 < 1 < oc, let C ^ de­ note the family of all functions F(p), p^R, satisfying the following condition |F(p) - F (p’)I < M (1 + |pI + Ip'I )a_P|p'

- plP

for all p,p'eRand a convenient constant M= M(F) < + 00 not /y ^ Q depending upon p and p f. Since (1 + |p| + IpM) > all Lipschitzian functions F of exponent P belong to each of the classes C ^ with oc > l. In the following, M denotes a con­ venient constant, not always the same. (i)

Let F(p), peR, be any function of the class C^, let fj_(w), weG, i = 1, ..., m, be m given functions La-integrable in the open set G of the w-plane E2, w = (u, v), and such that p = (f^ fm )£R for all weG, let

§3 2 . M E M VALUE INTEGRALS OF L-INTEGRABLE FUNCTIONS (32.4) fn

)

K C G be any closed bounded set, denote by f^ the mean value integrals of f^ and suppose that p (n) = (fjn)^ . f^n ))6R for all weK and n large enough. Then (K) / |F(f{n), ..., f^n)) - F(f1, ..., fm )| -- * 0 as

n

PROOF. Since |p' - p| = [^(x^ - x^)2] < Zjjx^ - X-jJ > Ip1 - p|P < MSj |x ! - x.|P, (l + |p| + |p'|)a"P < M(l + |p|“-e + lp'l“ P) < M + MS1 lxi|a P + Msi|x^|a P, we have |F(f(n), ..., f^n)) - F(f1, (1)

fn )| < Mz1|f^n) - f± |p

+ ME. .|fjn)|a-p|f;[n) - f± lP

H

+

+ MEiJ.|fJ.|a"p|f{n) - q l 3, where the sums are extended over all i,j = 1, 2, . m. Here p < a , and hence by (32-2, v) and (3 2 .3 * iv), we have (K) / |f-|“ < M,

(K) / |fjn)|a < M,

(2)

(K) / |fin) - f, |P — * 0 as

n -- > 00.

By Holder inequality [E. J* McShane, I, p. (K ) f

133]

| f j n ) | a _P | f [ n ) - f ± | P

we have


/ |fjn)|a] a

£

[(K) / |f^n) - ^

1“]

^

,

and by (2), the second member approaches zero as n -Analogously, for the expression (K) / If.Ia ~ ^ If|n ^ - f-J^. Therefore the integral on K of the expression at the left-hand member of (1 ) approaches zero as n -- >-00.

NOTE (1). Each polynomial belongs to the class

F(x.j, . ^•

xm ) of degree

or > 1

k66

CHAPTER IX.

THE REPRESENTATION PROBLEM

In order to prove this let us observe first that R = Em and that |F| < M (1 + |pI )Qr. Then observe that F(pl ) - F(p) = zA^(x^ - x^), where A^ denotes the first partial derivative of F with respect to x^ at a point p = [x^ + e (x^ x^)], 0 < e < 1 . Hence, Ip I 1. Indeed, if 0 < p < 1, then, by the statement (32.3, (3 )), we have iG(p’) - G(p)| = ||F(pl)|P - |F(p)|P | < |F(p1 ) - F(p)|P < M (1 + Ip| + lpfI) ^ ” 1 ^Ip' - plP- If P > 1, then, by (32.3, (2)), iG(p’) - G(p)| < M[ |F(p 1 )|3 ' "1 + |F(p)|P_1]|F(p!) - F(p)|. The last expression is < M (1 +|p| + |p» |)Qr( ^ - 1 )h-(q; - 1 )|pf p| , and, because of a(P - 1 ) +(oc - 1) = ap - 1, we have IG(p1 ) - G(p)| < M (1 + |p| + |pf|)aP_1 |pf - p|.

EXAMPLE I. If F = x2 + ... + x^, then oc = 2 and, if fi, i = 1, 2, ..., m, are L2-integrable functions in G, then (K) / z(f^n ^) 2 -- * (K) / Zf? as n -EXAMPLE II. 1 If F = [1 + x2 + y2]2, then m = 2, a = 2, p = 2 1 and in G, then ap = 1 . If f, g are L(K) / [ 1 + (f(n))2 + (g(n))2]i _ -* ( ) f [1 + f2 + g 2 ] i . k

EXAMPLE III. 1 If F = [AB - C2]2, A = x2 + x2 + x 2 , B = y2 + y2 + j 2 , C = x1y 1 + x2y2 + ^3y3, then m = 6, a = b , p = 2-1, ap = 2. If f^, g^, i = 1 , 2 , 3, are L2-integrable functions in G, and A, B, C, An, Bn, Cn, are the expressions above, with the functions f^, g^, or the1mean value integralsjL f^n ^, g(n), then (K) / (AnBn - C2 )2 -- > (K) f (AB - C2 )2 as n -- > 00.

32.5 » Mean Value Integrals of ACT Functions In the following, G denotes any open set in the w-plane E2 K any bounded closed set K C G. We have already defined in (3.2) the concepts of function f(w) = f(u, v), (u, v)eG,

and

§3 2 . MEAN VALUE INTEGRALS OF L-INTEGRABLE FUNCTIONS (3 2 .5 ) absolutely continuous in the sense of Tonelli (ACT) in the open set G(i)

If fis continuous in G» then the mean value / \ integrals f v of f have first partial de­ rivatives ( f ^ ) u,( f ^ ) v continuous in K.

PROOF. We have u+n - 1 f (n)(u,v )

= n2 J

u

v+n- 1

J

f(x, y) dxdy ,

v

and, by elementary direct computation, we have -1 v+n _ ( f ^ ) u = n2 + n"1, y) - f(u, v

j^f(u

J

u+n

j)

j

dy ,

-1

(f(n))v = n2 J u

f (x, v + n”1 ) - f (x, v )"j dx

j

and the continuity of both derivatives is trivial. (ii)

If f is continuous and ACT in G, then the mean value integrals of f have first partial derivatives continuous in K and (f (n))u = (fu )(n), (f(n))v = (fv )(n) in K, where (fu )^n ^> (fv ) ^ are the mean value integrals of the first partial deriva­ tives fu, fv of f.

PROOF.

For almost all f(u + n'1, y)

y, -

we have

f(u, y) =

J

u+n

-1

fu(x>y) ^

where w = (u, y)eK and, as we have seen in (3*2, i), fu is L-integrable in G. Hence, by (i) and integration with respect to y in (v, v + n”1), also

(f(n))u=n

2

J

v+n”1 dy

J

u+n”1 fu (x> y) dx = (f.u ;,(n) u

k6 7

CHAPTER IX.

468

THE REPRESENTATION PROBLEM

The same reasoning holds for

fv -

NOTE. As a consequence of (ii) the notation can be used for both (fu )^n ^ and (f^n ^)u> as f|n) for both (fv )(n) and (f(n))v -

(iii)

If f is continuous and ACT in G and the first partial derivatives f , f are La-integrable in G, or > 1 , then

( k ) ; i 4 n ) ia < ( g ) ; i f u i“ , (k) / if*n ) i“ < (g) i i f v r , (K) / |f1(1n) - f j ( + (K) / l 4 n) - fv l as

n

PROOF. This statement is a consequence of (32.2, v), (3 2 -3 * iv) and (ii) above.

32.6. (i)

An Application

If x(w), y(w), z(w), weG, are functions continuous and ACT in 2 G whose first partial derivatives are L -integrable in G, if E = xn + 4 + 4 ' ■^n' ^n' ^n are value integrals

G = 4 + 4 + 4 > P = xuxv + + zu V lf anal°g°us expressions relative to the mean x^n ^ J , z^n ^ of x, y, z, then 1 _ \2 as n (K) / (EnGn - F2 )2 -- » (K) / (eg -

This statement is a consequence of (32.5, ii) and (32.4, Example III). This result can be improved as follows. We shall say that two measurable functions f(w), g(w), weG, belong to conjugate classes in G if there are two numbers p, q > 1with p-1 + q_1 = 1 such that f is L^-integrable and g is L^-integrable in G. We shall say that f, g belong to conjugate classes even if one of them is bounded and the other one is L-integrable. If f, g belong to conjugate classes then fg is L-integrable in G and

§3 2 . MEAN VALUE INTEGRALS OF L-INTEGRABLE FUNCTIONS (3 2 .6 ) (K) / fngn ----------- » (K) / fg

as

The first part of this statement is well known. part let us observe that 00

/ |fngn - fgl < (K) / |fn - f||gn l + (K) < [(K) / |fn - f p

]1

P

+ [[(CK) K ) // IlflP f l ]l/P [(K) I

469

n ----*•».

For the second

IfI Ign - gl
f uniformly in K we have also (K) / -- > ( n ) r y m u (K) / I(f ) I as m -- »• 00. On the other hand, because of (32.5, ii), we have ( f ^ ) u = (fu ) ^ and by (32-2, v), also (K) / l(fin))u l“ = (K) / |[(fm )u l(n) |a < (K) / l(fm )u la < M. Consequently (K) J |(fu )(n)|a
oo

Here f is ACT in G and by G; hence (32.3, ii) (fu )^ l(fu )n |a -- > |fu la a.e. in p. 29] we deduce that l^u la (K) / lfu la
00

K,

(K)/ |(fm ) |a < M.

f . The statement (I) is proved.

Bibliographical Notes

Besides the books and papers already quoted in (32.1, Note), see R. G Helsel [1], R. G. Helsel and P* M. Young [1], E. J. Mickle [5].

472

CHAPTER IX.

THE REPRESENTATION PROBLEM

§3 3 - SOME PARTICULAR TYPES OF SURFACES 33.1.

Base And Non-degenerate Surfaces

In the present section we shall consider only continuous mappings (T, A) [i.e., surfaces S: (T, A)] from a closed simple Jordan region A of the w-plane E2 into E^. A mapping (T, A) [or the surface S], is said to be a BASE MAPPING [or a "base surface1’] provided (a) for every continuum g of the collec­ tion r(T, A) (1 0 .4 ) the open set E2 - g is connected; (p) no continuum ger(T, A) covers A [hence no ger may cover A*] . The mapping (T, A) [or the surface S] is said to be OPEN NON-DEGENERATE provided (7) for every continuum ger(T, A) the set A - g, open in A, is connected; (5 ) no continuum ger(T, A) covers A* [and hence no ger may cover A]; the mapping (T, A) [or surface S] is said to be CLOSED NON­ DEGENERATE provided (7) holds and (5 f) there is a ger(T, A) which covers A* but not A. By ( 3 1 i) and. (1 1 .2 , i; 1 0 .4 , v; 10.2, iv) it can be proved that the conditions above are all invariant for Frechet equivalence; hence a FrSchet surface S is said to be a base F-surface, or open non-degenerate, or closed non-degenerate, provided any representation (T, A) of S has the analogous property. Since A is a simple closed Jordan region, given any surface S: (T, A), the curve C: (T, A*), which is the image under T of the boundary A* of A, is said to be the boundary curve of S and denoted by C = 0 S. For any two surfaces S 1 : (T^ A 1 ), S2: (T2, A2 ) we have ||C.j, C2 1| < ||T-jj T 2 1| = ||S1, S2 1| and hence, if T 1 ~ T2, i.e., S 1 ~ S2, \\T ^, T 2 1| = 0 , then ||C1, C2|| = 0 , and C 1 ~ C2. Consequently If S denotes an F-surface and S: (T, A) anyrepresentation of S, then the curves C = 0 S belong to only one class of F-equivalent curves, that is, are representations of an F-curve C which is said to be the boundary of S, C = eS.

3 3 *2

. A First Property

(i)Any base surface S whose boundary curve Jordan curve is open non-degenerate.

C = eS

is

a

PROOFLetS: (T, A). Obviously condition (5 ) is satisfied. In order to prove (7) suppose, if possible,that there is a continuum ger(T, A) such that A - g is not connected, hence the sum of at least two components 7^9 7 2> ••• open in A. By (a), E2 - g is connected; hence, if w^,

§33 •

SOME PARTICULAR TYPES OF SURFACES (33-3)

= 1 , 2 , are any two points w^e/^, there is a simplearc 1 joining w 1 and w2, 1 C E2 - g. Necessarily, 1 ( E 2 “ A) 4 hence let wj, w^ be the points of A * 1 first encountered along 1 from w 1 and w2 . The two points wj, w^ are necessarily separated on A* by the closed set gA*, i.e., there are at least two points h ^ h2 egA* separating w 1 and w2 on A* and T(h1) = T(h2 ), a contradiction since C is a Jordan curve and T is not constant on neither arc h 1h2, h2 h 1 of J*. 1

We shall need later the following property of Jordan curves. (II)

PROOF.

For every Jordan curve C in E^ and e > o, there is a 5 = 5 (C, e) such that for any two points P-i> P2eC dividing C into two arcs oc2 each diameter > e, we have |p1 - p 2 1 > 5 .

of

This statement is a consequence of (1 0 -1 , ii). (iii)

For every Jordan curve C in E^, for every se­ quence Cn, n = 1 , 2 , ..., of continuous closed curves with ||C , C|| -* o as n -> °°, and € > 0 , there is a 5 = 5 (C, e) and an nQ = nQ (C, such that for all n > nQ and for all couples of points p^, P2€Cn dividing Cn into two subcurves Cm, C2 n eactl diameter > e, we have Ip-, - p2 I > 8 .

e)

This statement is a consequence of (ii).

33 -3

(i)

- Base Surfaces

If (T, A) is a base mapping, then for any e > o there exists 5 = b ( e , T) > 0 such that for any simply connected open set 7 C A° with diam T (7) > €, we have diam T (7*) > 5 .

PROOF. Suppose the statement is false. Then there is an € > 0 and a sequence [7R] of simply connected open sets 7n C A0 with diam T( 7 *)< n”1, diam T(7n ) > e , n = l , 2, ... . We will suppose n large enough in ~

1

—

1

o rd e r t h a t n (+ 0) = 0, and t h e r e i s a number tj > 0 su ch t h a t e, diam T ( 7* ) < n " 1 < 3~ 1e, 7* C 7 , th e re i s a p o in t Pn € T (? n ) su ch th a t {pn , T ( 7* ) ) > 3~ 1e and a p o in t wn e 7n su ch t h a t T (wn ) = Pn - I f cn i s th e c lo s e d c i r c l e o f c e n te r wn and r a d iu s tj, we h ave (T(cn ), T ( 7*)3 > 0 and h ence v n ecn ^ cn C 7 ,

CHAPTER IX.

THE REPRESENTATION PROBLEM

7n C A0 , 7 * C An - cn C A^ - c ° .

Let

w^

be any p o in t

of

7 *.

S in ce

wn , wn e-*-^ n = 1, 2, . . . , and A I s com pact, th e r e e x is t s a subsequence [rijjj] o f in te g e r s n such t h a t w ----- > wQ, w ' ----- > w ^, where wQ, w '

rn

m

are two points of A. For the sake of simplicity suppose [r^] = [1 , 2, ...]. If cQ is the circleof center wQ and radius r\/2, we have necessarily cQ C A, w^eA - c°, where A - c° is a closed connected set. On the other hand w^ ► w^ implies lim inf 7* + 0 as n ---- > where each 7* is a continuum 7n C A. Finally (10.2, i), k =lim sup 7* is a continuum, k C A. Since diam T (7*) < n~1, we have diam T(k) = 0 ; hence T is constant on k and k C g, ger(T, A). Since wn v wQ, tT(wn ), T(7*)} > we have {T(wQ ), T(k)} = (T(wQ ), T(g)) > 3 _1 e. As a consequence cQg = 0 , and g C A - cQ . Now let 0 Q be a circle of center wQ containing the whole set A in its interior. Thus 7* C CQ - cQ and 7* separates wQ from C*. Consequently, k C CQ - cQ, g C CQ - cQ, and g separates wQ from C* in E2 . Other­ wise by ( 11 .2, i) there would be a line 1 C E2 - g joining wQ and C* with 17^ + 0 for every n. If wnel7* and w is any point of accumula­ tion of the sequence fwn ], we have wek, wel,and finally weg, wegl, gl + 0 . Thus g separates E2, a contradiction, since T Is a base mapping. (ii) If

(T,

A)

Is a base mapping, then, given

e>

0,

th e re a re a > 0 and 5Q > 0 su ch t h a t , f o r any c . m apping (T^, A 1 ) w it h ||T.j, T || < a and f o r any open s im p ly co n n e cte d s e t 7 C A ° w it h diam T 1 (7 ) > e,

we have a ls o

diam T^ ( 7 * ) >

PROOF. Let 5 = &(T, 2~1 e) > 0 be the number defined by (i) and let 5 q = 2 ~15 , a = min [2 ”2 5 , 2“2 c]. Thus given (T^ A 1 ) and 7, there is a homeomorphism H between A and A 1 such that |T(w) - T 1 tH(w)]| < a < min [2 ~2 5 , 2 “2 e] for all weA. Thus if 7 ? = H we have 7* C A ° , 7f* = H_ 1 ( 7 * ) , and since diam T ( 7 ) > e, we have also —2 —1 diam T(7*) > e- 2(2 e) = 2 €. Consequently diam T(7!*) > 6 , and diam ^(7*) > 5 - 2 (2 _2 5 ) = 2 ~ 15 = 8 . (iii)

Statements (i) and (ii) hold also if (T, A) open non-degenerate.

is

PROOF. In the proof of (i), by the supposition that the statement is false, we have deduced the existence of a point wQ and of an element ger(T, A) separating wQ from E2 - A in E2. Now, since (T, A) is open non­ degenerate, g cannot separate A (condition 7) • Therefore A - g is a connected set, open in A, containing w . Suppose A* - g + °> if*

§33-

SOME PARTICULAR TYPES OF SURFACES (33-4)

475

possible. Then there would be a point weA*, w not in g; hence 5 = {w, g] > o and w would belong to A* - g together rith an arc c of A*, having w as an interior point. Here A* is a simple curve, hence w is accessible from both E2 - A and A°. If l 11 = ww1, I 1 = ww2,are any two arcs of diameter < 5 , 1 1T C E2 - A + (w), 1 * = A° + (w), then 1 ! C A - g and there is a further arc 1 C A - g joining w 2 to w . Thus the continuum 1 + l f+ 1 M is contained in E 2 -g and joins wQ to a point w 1 eEp - A, a contradiction. Thus we may deduce A* - g = 0 , i.e., g 3 A*, again a contradiction, because of (5 ). Thus (i) is proved also in the new hypothesis. (ii) follows as before. Thereby (iii) is proved. (iv) If S is any base Frechet surface, then given e > o there are a > 0 and 5 > 0 such that for any F-surface § 1 with 115^, S|| < cr, for any representa­ tion (T.j, A 1 ) of 3 .J, for any open simply connect­ ed set 7 C with diam T^ (7) > e, we have also diam T ^ 7 *) > 5 . In particular,if § 1 = S, this statement holds for all representations (T, A) of S. PROOF. Let us consider any representation (T, A) of S and determine and 5 as in (ii). Then for all S with ||S.j, S|| < cr and all repre­ sentations (T-j, A 1 ) of S 1 we have ||T1, T|| = H^, S|| < a. (v)

The statement (iv) holds also if non-degenerate F-surface.

S

a

is any open

The proof is analogous to the proof of (iv) where (iii) is used instead of (ii).

33 *4

(i)

• Non-degenerate Surfaces

If (T, A) is non-degenerate (open, or closed), then, given e > 0, there is a 6 = 5 (T, e) such that, for any continuum K C A such that A - k has at least two components 7 ^ 7 2 with diam T(7 ^) > e, i = 1 , 2 , we have diam T(k) > 5 .

PROOF. Suppose the statement false. sequence [k ] of continua kn C A, 7 1n> 7 2 n of A ~ kn' diam T(k^) < n~1, n =

Then there would be an € > 0 , a and, for any n, two components

suc]l that diam T ^7 in^ > 2, ... . We can suppose

1,

1

n

= 2> large enough in order

CHAPTER IX. —

1

—

THE REPRESENTATION PROBLEM

1

that n < 3 e* Let o>(5 ) be the modulus of continuity of (T, A) (1 0 .6 ) and let t\ > 0 be a number such that cd(t|) < 3 ”1 e« Since diam T(7 in) > €, diam ^(1^) < 3 ~1 e, there are two points i = 1 , 2 , such that fPjjy T(kn )} > 3 _1 e> and two points v;j_n€^in> i = 1 , 2 , such that T(w^) = p^. If c^, i = 1 , 2 , are the closed circles of centers w.in and radius t i we havein c. be 1 n k„ = 0 . Let w~ 3n any point of k^. Since w^eA, i = 1 ,2 , 3 , n = 1 , 2 , ..., and A is compact, there is a subsequence [r^] of integers n such that w. ►w ., i = 1 , 2 , 3 , as m -----►». For the sake of simplicity suppose [nm ] = [1 , 2 , ...]. Let c ± y i = 1 , 2 y be the circles of center w^ and radius r\. Because of w^n -- v e bave lim inf kn +0 , hence K = lim sup kn is a continuum, K C A. Since diam T(kn ) < n*"1, we have diam T(K) = 0 , i.e., T is constant on K and K C g, ger(T, A). Since (T(win), T(kn )} > 3 ” 1 e, I = 1,2, n = 1 , 2, ... we have {T(w± ), T(g)) > 3"1 e, and hence c^g = o, 1 = 1, 2 . Finally, since kn separates w ln and w2n in A, also g separates w 1 and w2 In A, (see an analogous proof in 3 3 -3 , i)> a contradiction since T is non-degenerate (condition 7 of 3 3 -1 )- The statement (i)is proved. (ii) If S: (T, A) is non-degenerate,then, given € > o, there are cr > o and 5 > 0 suchthat, for any c. mapping (T^ A 1 ) with ||T , T|| < cr and for any continuum k C A 1 such that for at least two components 7 ^, 72 of A 1 - k we have diam T 1 (7^) > e, i = 1 , 2 , we have also diam T 1 (k) > & • This statement can be deduced from (i) as (3 3 *3 , ii) was deduced from (3 3 -3 , i). (iii) If 3 is any non-degenerate F-surface (open, or closed), then given e > 0 , there are a > 0 , 5 > 0 such that for any F-surface S^ with ||S1, S|| < cr, for any representation (T^ A^ ) of S,, for any continuum k C A1, such that for at least two components 7 1, 72 of A 1 - k we have diam T (7^) > e, i = 1, 2, we have also diam T 1 (k) < 6 . This statement can be deduced from (ii) as (3 3 *3 , iv) was deduced from (3 3 .3 , ii)-

§34.

REPRESENTATION THEOREMS FOR NON-DEGENERATE SURFACES (3 4 . 2 ) * 3 3 -5

477

» Bibliographical Notes

Base, open non-degenerate and closed non-degenerate surfaces have been studied by C. B. Morrey [1, 2, 3] and E. J. McShane [4 ]. For the theorems of the present section see L. Cesari [17, 1 9 > 3 8 ] and L. C. Young [4 ].

§3 4 . REPRESENTATION THEOREMS FOR NON-DEGENERATE SURFACES 3 4 .1 .

The Dirichlet Integral

Let (T, A) be any c. mapping from a closed simple Jordan region the w-plane, w = (u, v), into the p-space E^, p = (x, y, z). The mapping (T, A) D-SURFACE] provided (1 )x, y, z are (2) the first exist a.e. Then, if E, F, G,

is said to be a D-MAPPING [i.e.,

S: (T, A)

a

= ( e g - F2)^ < ( e g ) ^ < 2 _ 1 (E + G ).

By definition we shall write J^ = J2 = J^ = E = G = F = 0 on I(T, A) be the classical area integral (31-8) I(T, A) = (A) ; («J2 + D(T, A)

of

BVT and ACT functions in A0; partial derivatives x^, yu, ..., z y (which in A0 ) are L2-integrable in A0 . , J2, J^ have the usual meaning (5 -5 ), wehave

( j 2 + J2 +

and let

A

J2

+ J2)r dudv = (A ) / (eg

A*.

Let

- F2)^ dudv,

be the socalled Dirichlet integral

D(T, A) = (A) / 2"1(E + G) dudv = = 2~1(A) /

( 4

+ y2 +

z2

+ x2 + y2 +

z2

) dudv-

If not needed, we shall not display some of the letters T, A. If S is any F-surface and (T, A) any D-mapplng, which is a representation of S, then we shall denote (T, A) as a D-representation of S. (i)

If S Is any F-surface possessing a D-representation (T, A), then L(S) = I(T, A) < D(T, A) < + 00.

This statement is a consequence of (5 -1 3 > i) and (3 1 -7 )*

3 4 .2

Let

A

.

Almost Conformal Representations

be any closed Jordan region of the w-plane,

w = (u, v).

In

478

CHAPTER IX.

THE REPRESENTATION PROBLEM

differential g e o m e t r y a c. mapping (T, A): x = x(u, v), y = y(u, v), z = z(u, v), (u, v)cA, is said to be CONPORMAL in A0 provided x, y. z are continuous in A0 together with their first partial derivatives xu9

A + 4 > G = 4 + 4 + 4>

yu' ***' zv

E =4 +

3X1(1 E = G, F = 0

everywhere in

p =^

A0, where

+ yuyv + zu V

A mapping (T, A) i,s said to be ALMOST CONFORMAL provided 0 ) x > Y> z are BVT and ACT in A0; (2) the first partial derivatives x^, •••, zy (which exist a.e. in A0 ) are L2-integrable in A°; (3) E = G, F = 0 a.e. in A°. Thus an almost conformal mapping is a D-mapping and we have, a.e. in A0 («J2 + J2 +

= (eg - F2 ) 2 = (bg)^ =

i(E + G)

and such a relation is trivial on A* where all J.j, ..., G are zero by definition. Thus we have (i)

For every almost conformal mapping (T, A) we have L(A, T) = I (A, T) = D(A, T ) < +00; i.e., L(A, T) = (A) / (EG- F2 )i dudv = 2 _1 (A) / (E + G) dudv.

The class of the F-surfaces S possessing an almost conformal representa­ tion is very large as we shall see in (34.6), nevertheless there exist F-surfaces S, even of finite Lebesgue area, which do not possess conformal representations, as the following example shows.

EXAMPLE. Let CQ : x = f^t), y = f2(t), z = fj(t), 0 < t < 1, be any simple continuous curve such that its Jordan length, as well as the length of each partial arc of CQ, is + » [e.g. CQ could be a simple curve whose subarcs have all positive 3-dimensional measure in E^ (Osgood curves), W. F. Osgood, 1]. Let A q be the square AQ = [0 < u, v < 1 ], and (TQ, AQ ) be the mapping(TQ, AQ ): x = f1(u), y = f2(u), z = f3(u), (u, v) e Aq. By ( 5 - 9 , vi) we have L(AQ, TQ ) = 0. Let A de­ note any closed simple Jordan region of the w-plane, C be the maximal class of all c. mappings (T, A) which are F-equivalent to (TQ, A ), and S the F-surface defined by C. Thus L(S) = 0 and also T(A) = TQ(A0 ) = [CQ]. Suppose, if possible, that there exists an element (T, A)eC which satisfies con­ dition (1 ). Then for almost all u and each maximal (open) segment s, intersection of A° with the straight line u = u, the functions x, y, z are BV on s. On the other

§3^.

REPRESENTATION THEOREMS FOR NON-DEGENERATE SURFACES (3^-3 )

^79

hand T(s) is a subcontinuum subcontinuum of [CQ], i.e., a subarc 7 of CQ and, ifif 77f: f: (T, (T, s), s), then then length length yy xx >> length length 77,, where where length 7 = + +«>«> ifif 77 is is aa proper proper subarc subarc of of CQ. CQ. This This implies implies that T isisconstant constant on on s. s. Since Since the the segments segments ss are are every­ every­ where dense in A, this implies that T is constant on all segments s of G parallel to the v-axis. The same proof, using the straight lines v = v, assures that T is constant on all segments s1 of A parallel to the u-axis, and because any two points of AA ,, can be canjoined be joined by a by finite a finite system system of of segments s, sT, T is constant on A0 and thus in A; i.e., T(A) and hence [CQ], is a single point, a contradiction.

3^*3» We shall denote by (i)

Q

Lemma on AA Lemma on Dirichlet Dirichlet Integral Integral

the closed unit square

Q = [0 < u < 1, 0 < v < 1].

Given any two positive numbers N, e there exists a positive number r\ depending only on N and e such that for any D-mapping (T, Q) with D(T, Q) < N, there exists a b, < b < e, and a finite subdivision of Q into rectangles whose side-lengths lie between 5 and and 25 25 and and such such that that the the image image of of each each side side of of such rectangles (not on Q*) is a rectifiable curve whose length is less than € [L. C. Young].

PROOF. Let D(T) be the D-integral relative to T,T, then then D(T) D(T)

F = V v + yuyv + w

Thus if S is the Frechet surface defined by the class of all c. mappings F-equivalent to (T, A), we shall say that (T, A) is a generalized con­ formal representation of S. If the conditions (1), (2), (3) are verified only in a set GA° with G C A, we shall say that (T, A) is generalized conformal in

G.

Since the functions x, y, z are continuous in A°, their partial Dini derivatives are measurable in A0 [S. Saks, I, p. 170]; hence condition 487

^88

CHAPTER X.

(1) implies that If

(T, A)

THE REPRESENTATION OP GENERAL SURFACES AND THE THIRD THEOREM x^, ..., zy

are measurable in

A0.

is generalized, conformal (in A0 ) then the ordinary Jacobians

J 1 = y u zv - ^ v V

J 2 = zux v - zv V

J3 = V v

[For the sake of simplicity we shall put AA*.] Thus the classical integral

J

- x vy u

e x ls t a -e - 111

A° -

= J2 = J^ = o everywhere in

I ( A , T ) = (A ) J * ( J 2 + J 2 + j| j^ d u d v = ( A ) J ( eg - P2j ?dudv

exists and, as we know by (28.3, i) and (30.3, i)> we have 0 < I(A, T) < L(A, T).

NOTE. By comparison of the definitions it is clear that the above definition of generalized conformal mapping has been obtained by the definition of almost conformal mapping (3^.2) by replacing the condition that x, y, z are ACT in A° by the weaker condition (1 ) that x, y, z have first partial derivatives a.e. in A°.

35 .3

(i)

• A Lemma

There exists a c. plane mapping (♦, q) of a square q onto itself, monotone in q, identical on the boundary q* of q, constant on a countable set of disjoint closed squares q^ C q, i = 1, 2, ..., z|q^| = |q|, and there exists a sequence (*n, q) of quasi linear homeomorphisms of q onto itself, identical on q* such that d(4>n, , q) -- » 0 as n ---» 00.

NOTE. Since we shall use the mappings ♦ ,an infinite number of times as tools for convenient modifications of given arbitrary mappings (T, A), we prefer to give first (in ot) an elementary direct construction of particular mappings ♦ , * • In (p) we shall show how (i) can be deduced by gener­ al theorems of topology.

PROOF. ( a ) Let p = a-^a^a^ be any convex plane quadrilateral. Let d be thediameter of p, 0 be the center of gravity of the fourvertices (of the mass unity), let pQ = aja^a^a^ be the image of p under the similarity transformation with center 0 and ratio 1/n (n > 3 integer). Let us divide the sides of p into three equal parts, by means of the points b^, i = 1, 2, ..., 8, ordered on p* starting from a^ and in the same order

§35 • GENERALIZED CONFORMAL REPRESENTATIONS (35-3) as the points of a^. the 9 quadrilaterals

Let us indicate by

En

the subdivision of

= t>2a2b5a^,

= t>3b^a^a|

= *\a5b5a^,

P6 = b5b6a(a^

P? = b6a^b7a£,

P8 = b7b8a]a(

P5

p

into

P2 = .b^a^a*

Pi = ai V i ' V P3

489

It is elementary to prove that the 9 quadrilaterals *T1d, are all convex and have diameter -1 The diameter dQ of pQ is dn' Let q, q* be the squares q = [0 < a, 0 < q* = [0 < u, v < 1]. For each integer n > 3 define the transformation 4>n as follows. We perform the subdivision upon the square q, and let qQ, q1 , •••> Q.q t>e the nine squares so tained. Let the central square qQ be undivided and perform the sub­ division E* on each of the remaining squares q. If Q-i i 9 1 2 ip = o, 1, . 8 , are the nine squares so obtained from each q. , ilet the central square 2

q.

be undivided and perform

E^

1], we E^ ob­

+ o,

on the remaining

1

8 -squares. By repeating this procedure n times tain squares qQ, q^ , q. '^ii, ^n-l0 1a12° ^ -n , -n lengths •>3 , 3 Let us repeat the same procedure on

q*

we obtain in q cerof side ^i i ...i 12 n

by applying

E 2 instead of E,. n ^ We obtain in q ! a square q^ of diameter 22n~2 and certain convex quadrilaterals qj_ Q • ' 9i 1i 2 . . . i n-1o' qi i ...in of diameters qi,i. 12 zn .-no 4 ^-2 3 n - 4 -n2V 2 < 3 • 4-122n"2, 32 • 1T22V 2 •2*n 3n . k~n ■ 22, ( i 1# i 2, , 8). If in - 1 , 2, n is the maximum of > o as all these diameters we have n -y 00• n By dividing corresponding quadrilaterals q^, q^ into four triangles by means of the two diagonals we can define n linearly in each of such triangles in such a way that n makes each triangle of q ! correspond to the corresponding triangle of q. Let us observe that we have successively divided each side of q and the corresponding side of q T into three equal parts and each of these parts again into three equal parts and so on, and that is linear on each part. Therefore ^n is identical on the boundary q of q. We have now to prove that the sequence $n converges uniformly in q toward a mapping having the required properties.

>*•90 CHAPTER X.

THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM

Let us prove that for each integer r the four vertices p T of each quadrilateral q! . have a limit position p * as n -- »00 and that _ 2

1 ’

r

|p! - pQ| < rn . Indeed this is true for q^ whose four vertices p ! are at a distance < n”2 from the center of gravity pQ of the square q !. Therefore the same is true for each quadrilateral q! whose vertices are 1

either fixed points on q 1* or vertices of q^. Thus also the center of gravity g ! of the vertices p ! of each quadrilateral q! has also a _2 1 limit position g^ and |g! - g^l < n . The same holds for the points on the sides of qj dividing the same sides in three equal parts. Now for 1 _2 the vertices p 1 of the quadrilaterals q! we have Ip1 - g| < n , 1 —p and hence |p! - g | < 2n” , and so on. Byr repetition of this reason­ ing weprove the statement above. We provealso that $ converges on each of the squares q. . . and that, if is the limit mapping, 12” r 4> is constant on each of the same squares. We have also z|q. . . I= 1 2 * ’* n 9”1 + 8*9-2 + 82-9~^ + ... = 1. Since n is identical on q*, n con­ verges also on q* and the limit mapping 4> is identical on q*. Let w be anypoint of q. Thenfor each n, the point w belongs to some squares p = q. . . , 1 < r < n, of the subdivision relative to 1 2 *** r "" q and n. Then n (w) belongs to the corresponding quadrilateral p* = qJ . . of diameter < 5 and whose vertices are at a distance 1 ^ 2 * -‘^r n < n Thus from their limit position (as n -- >°°) which is < rn n (w) is contained in some circle cn whose center is one of these limit points and whose radius is &n + n~1. In addition, for any two integers n, n + h, we have pn 5 Pn+^ and cncn+h 4 °* As a consequence we have U n ^ ) " *n+h^w ^ - 25n + 25n+h + 2n~ + 2 + - ^5n + * This proves that, as n -- >°°, converges uniformly toward a c. mapping from —1 q onto q l, and d(4> , 0, q) < 2 5 n + 2n” . (P). The existence of a mapping 4> as in (i) canbe assured also as follows: Let us denote by K the collection of all disjoint squares q. . n - V *m±n u defined in (a) plus all remaining single points of q. Then K is an upper semicontinuous collection, whose elements are continua not separating q or the w-plane. Hence (10.4, Note) there exists a monotone mapping (f, q) from q into the unit circle c whose collection r(f, q) is K, and f is a homeomorphism between q* and c*. If h is any homeomorphism be­ tween q f and c coinciding with f on q*, then = h”1f has the required properties. Now the monotone mapping $ from the Jordan region q C E2 onto itself has the property that no element g of K = r(, q) separates q or E2 and hence can be approached by means of

§35-

GENERALIZED CONFORMAL REPRESENTATIONS (35-4)

491

homeomorphisms Fn (10.4 , vii), and, as above we can always suppose that each Fn is identical on the boundary of q. Finally since the two cells are here the two squares q, q 1 we can approach each Fn by means of quasi linear homeomorphisms (identical on the boundary) (6.1, ii).

3 5 -4

. A General Theorem

We shalldenote by (T, A): p = T(w), weA, (Tf, A): p = T !(w),weA, two c. mappingsfrom an admissible set Aof the w-plane E2, w = (u, v), into the p-space E y p = (x, y,z), or, with other notations: (T, A): x = x(u, v), (Tf, A): x = X(u, v),

y = y(u, v), y = Y(u, v),

z = z(u, v), z = Z(u, v),

(u, v)eA, (u, v)eA.

(i) Given any c. mapping (T, A) from an admissible set A C E2 into and any open set G C A0, there existsanother c. mapping (T!, A) such that (1 ) (TT, A) ~ (T, A); (2) (T1, A - G) = (T,A - G), in particular (T1, AA*) = (T, AA*); (3) T 1 is generalized conformal in G; OO Xu = Yu = ... = Z V = ° a.e. in G;

(5) *u = Yu = •••' Zv = zv at evei^ point (u, v)eA - G where xu, yu, •••, existPROOF. Obviously (2) and (5) are trivial if G = A = A0 . Let SQ,S ^ ..., ... be the subdivisions of E 0into squares Q of side lengths 2°, 2- 1,-22 ,..., obtained for each n bymeans of the straight lines u = 2-ni, v= 2~nj, i, j = 0, + 1, + 2, ..., and let Fn be the square Fn = [- 2n < u, v < 2n]. For each n let usconsider those closed squares QeSn> Q C GFn which are not contained In any square Q f€Sm > Q 1C GFm with 1 < m < n. If [Q3n is the finite collection of these squares Q, and [Q] = then [Q] is a countable collection of non-overlapping closed squares Q C G filling G. For each Qe[Q] let ^q (^) be the modulus of continuity (10.8) of T on the compact set Q C A; thus o>q (o +) = 0. For each Q we have Q C G where Q, is compact and G is open; hence the distance d^ = (Q, A - G) is positive. Let 5 ^ be a number such that o)q (Sq ) < minCd^, m"”1], where m is the integer for which QeSm . Let us divide eachsquare Qe[Q] into congruentsquares q of diameter < 6^. If [q] isthe new system of all squares q, we have

^ ( d i a m q) < ^q(5q) < dQ = where

q C Q, Qe[Qj.

^ A “ G}2 fop eveiT qe[q],

A"

Thus we have:

(a)

each qe[q]

lies in

G; (p)

b92

CHAPTER X.

THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM

cD^diam q) < {q, A - GV squares q fills G.

(7)

the countable system

Cq]

of non-overlapping

For each qe[q] there exists (3 5 -3 , i) a mapping of q onto itself : u = u(a, p), v = v(a', p), (a , p)eq, such that is continuous and monotone in q, identical on q*, and constant on a countable system [q!]q of disjoint closed squares q* C q such that z!Iq1I = Iql, where Z 1 is a sum extended over all q !e[qf]^. Let X(or, p) = x(a, p) if (a, $ ) e A - G; X(a, p) = x[u(a, p), v(a, p)] if (a, P)eq, qe[q]. Let us define Y(a, p), Z(a, p) analogously. Then a continuous single-valued mapping (T!, A) is defined on A and obviously (T!, A - G) = (T, A - G). Thus (2) is proved. Let Gq C G be the set of all open squares q* , q 1 e[q* ]^, qe [q] . We have G - GQ = [G - zq°] + z[q° - zfq |0], where z denotes a sum ranging over all qe[q] and Z1, as before, a sum ranging over all q !e[q!]q. Since G - Zq° is a countable set of segments we have |G - Zq°| = 0 and, on -.0 r t „ | 0 | ____________________________ „ mi , I r* n the other hand, lqw - z!q fU| = ^0 for every q. Thus IG — GQ II - Q Since X, Y, Z are constant on each square q 1, we have XQ = Yu = = 0 everywhere in GQ and hence a.e. in G. As a consequence E G = F = 0 a.e. in G, and (3) and ( b ) are proved. Let wQ = (a, p ) be any point of A - G (if any) where xu exists. Since Let w = (a + h, p ), h 4 0, be any w q€A - G, we have X(a, p) = x(a, p) be the incremental ratio Jh = [X(of + h, p) point of A and let x(or + h, p), = [x(a + h, X(a, P)]h'1• If weA - G, then X(a + h, p] G. If weG, then x(a, p)]h-1, J-. h -- »x„ ' “u as h -- > 0 , (or + h, 0 )eA weq for some qe[q] and we denote by w ! = (a + h 1, p) the point of inter­ is section of the segment wQw with q*. Then X(w* ) = x(w!), since identical on q*. On the other hand, qe[q], hence q C Q for some Q and |X(w) - X(w!)|, |x(w) - x(w!)I < ^(diam q) since maps q onto itself. Thus we have X(w) - X(w_)

-1

x (w) - x(wo ) h •1

x(w) - x(w*:

-1

-t

X(w!) - x(w)

= J1 + J2 + V

-1 a)Q (diam q) < -1 xu as h -- » 0 , while lj2 l> Here Jl . i r-1 fq., _ _ v < h -1^2 h (q, A - GT and, since wQeA - G, also |j, w0) iJ 3 l < h 0 . This as h J,^ * 0 && as uh -- 7 0, and u Therefore, J2> J3 -- 7 w *u implies that exists at the point (a, p) and that Xu = xu - The same and thereby (5 ) is proved. argument holds for Y.u ’ For each qe[q] (3 5 .3 , i) with

we consider now the q.l. homeomorphism in q as m -- » 00. For any m

* defined in qe[q] and integer

§35-

GENERALIZED CONFORMAL REPRESENTATIONS (35-5)

493

n

we shall fix m = m(q, n) In such a way that |T*(w) - Tm (w)| < n-* f o r all weq. P o p any integer n let us now define the q.l. homeomorphism of A onto itself by putting = in each square qe[q] with q C Q, QefQ]^, r = o, 1, ..., n; H^w) = w for all remaining points w of A. Thus we have |Tf(w) - TH^w)) =0, or < n ”1 for all weA. Thus T 1 ~ T and (1 ) is completely proved.

NOTE. Under the conditions of (i) above, itmay happen G C A C A , where A is an admissiblesubset of alarger admissible set AQ, and that T is defined in Aq and is continuous in A . Then also (T!, A) can be extended to AQ by putting T ! = T in AQ - A and (T!, A ) is con­ tinuous in A . The proof is the same.

3 5 -5

that

- Exceptional and Proper Sets

Given any c. mapping (T, A) from an admissible set A of the w-plane E2, w = (u, v), into the p-space E^> p =(x, y, z), a point weA is said to be an EXCEPTIONAL POINT for the mapping T provided there isan open circle U of center w such that L(UA, T)= 0. That is, the Lebesgue area of the mapping induced by T on the (admissible) set UA is zero. Anysubset I of A is said to be anEXCEPTIONAL SET provided I is open in A and all its points are exceptional. A set M C A is said to be a PROPER SET provided its complementary A - M is exceptional. An exceptional set I need not be maximal; if so I is said to be the maximal exceptional set for (T, A). It may happen that no point weA is exceptional; then (T, A) does not possess exceptional sets and M = A is aproper set. If L(A, T) = 0 then all points weA are exceptional and then I = A is the maximal exceptionalset, though every subset of A open in A is then an exceptional set. (i)

Given any c. mapping (T, A) from any admissible set A of the w-plane E2, w = (u, v), into the p-space E y p = (x, y, z), there is another c. mapping (T*, A) such that (1 ) (T», A) ~ (T, A); (2) (T1, AA*) = (T, AA*); ( 3 ) T* is generalized conformal in A0; (4 ) = Yu = ... = Zv = 0 a.e. In A0; (5 ) I (A, T !) = 0 where I is the classical area integral (31.8).

This statement is a corollary of (3 5 -4 , i) where we take

G = A0 .

k9k

CHAPTER X. (ii)

THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM

Given any c. mapping (T, A) from any admissible set A of the w-plane E2, w = (u, v), into the p-space E y p = (x, y, z), there is another c. mapping (Tf, A) such that (1 ) (T«, A) ~ (T, A); (2) T and T ! have the same maximal ex­ ceptional set I and (Tf, A - IA°) = (T, A - IA°); (3) T 1 is generalized conformal in IA°; OO ^ = Yu = ... = Zv = 0 a.e. in IA°; (5 ) = xu, . . Zv = zy at every point weA - IA° where x^, ..., zv exist.

PROOF. Let I denote the maximal exceptional set for (T, A)and (T1, A) be the c. mapping which we obtain by applying (3 5 -4 , i) to (T, A) with G = IA°. Let I* be the maximal exceptional set for (TT, A). Obviously, we have only to prove that I 1 = I. Let us observe first that each point welA0 is an exceptional point for T; hence there isan open square h of center w such that L(h, T) = 0 . Since each square y, of the proof of (3 5 -4 , i) is a compact set of points of IA°, Q can be covered by a finite number of squares h. We can re­ quire, therefore, that each partial square q C Q, qe[q], is completely contained in at least one of the squares h above. We can even require, in the definition of the squares q, that not only each square q is con­ tained in at least one of the squares h, but also that each square q, concentric to q and side-length three times the side-length of q is also contained in at least one square h. Now let w be any pointweIA°. Then w is an interior point of a well determined simple pol. region c, which is either a square q, and then weq0, or the sum of two adjacent squares q, and then w is interior point of the segment intersection of their boundaries, or the sum of four adjacent squares q, and then w is their common vertex. In any case the whole region c is contained in the square q corresponding to any one of the largest of the namedsquares q, and q is contained in a square h, c C q C h. Consequently, L(c, T) < L(h, T) = 0 and L(c, T) = 0. On the other hand, (T, c) ~ (T!, c) (see last lines of 3 5 -4 ) and hence L(c, T 1) = L(c, T) =0. If U is any circle of center w with U C c, we have also L(U, T T) = 0 and thus wel1. Since weA0 we have weIfA° and thus we have proved that IA C I!A0. If wel - A0, set (5 . 1 ) and

then weA - A0 and thus weAA*. Since A is an admissible A cannotbe an open set, then either A is the finite sum

let

§ 3 5 - GENERALIZED CONFORMAL REPRESENTATIONS (3 5 -6 ) H

of disjoint closed closedJordan Jordan regions regions J, J,

open in

H.

In any case

there is is aa square square hh can suppose

h

(3 5 *^, i) we have

J*

and

w

components of

or or is is a subset a subsetA Aof of a set a setH, H, J

is a closed Jordan region and

such such that that

Ah Ah == Jh. Jh. Since Since

wel wel we we

L(Ah, L(Ah, T) T) == 00 and and then then Ah Ah CC I. I. containing

w

and contained in

Let TT == TT!! onon j.j. Let kk

ranges over all squares squares

kj = o,

where

of of center center ww

so small small that that

the maximal open arc of z

wej*,

1^95

be the the set be set kk

qe[q] qe[q]with with qh* qh* ++ o. o.Then Then k

together with

j,

h.

of squares

q, q,

By

k is is a continuum, a continuum, 7,

of the

Now - k. 7 Now is 7 the isthe countable countable sumsum

of non-overlapping squares squaresqe[q] qe[q] plus plus the the arc arc which are not in

jj bebe

h* ++ Zq Zq where where == h*

belongs to only one, say

J - k. k. Obviously Obviously 7 7C CA A- k.

Let Let

j

j, j, and and all all points points of of are on the boundaries

7* 7*

q*

or are or limit are points limit points of suchof such

points.In In anyany case caseT !T=! T= Ton on7* 7*andand thethe monotone mapping mapping defines

$$ of of

AA onto onto itself itself which which

TT 1, 1, maps maps also also

77 onto onto itself. itself. As As in in

the proof of (3 5 .^, i) we have U

L (7 > T) = L( 7 , T 1)-

Ah CC I, I,we we haveL( 7L( , T) and, finally, L( L(77,, TT1) 1) == 77 CC Ah have , 7T) = 0= 0and, finally,

Since

is aa circle circle of of center center w w and andradius radius small smallenough, enough, wewe have have

hence, Let

L(UA, T !) =

w

0

and, finally,

wel1.

00..

IfIf

UA UA CC 7 7and, and,

Thus we have proved that

I I C I*.

beany any point pointweA° weA° - I, - I,letleth h bebea asquare squarewith with h hC CA0,A0, and and

be the

set k == h* h* ++ zq, zq,

qh* + +0. 0. Then Then

kk

where where

zz ranges ranges over over all all squares squares

is is aa continuum, continuum, not not containing containing

longs, as an aninterior interior point, point, to to only only one, one, say say h - k,

and 7is is open, open,connected, connected,

qe[q]

w, w, and and hence hence w w be­ be­

7, 7,

of the the components components of of

77 CC h, h, 77 CC A0 A0.. The The set set

7

sum ofcount count ably ablymany many squares squares qe[q] qe[q] and and points points w w!eA !eA- -IA°. IA°. mapping

H

is isalso alsothe the The monotone

of (3 (53-^, 5 -^,i)i) maps maps 77 onto onto itself itself and and is is obviously obviously identical identical on on

7 *.

Hence

and

L( 7 , T) = L( 7 , T !)«

could take

k

withwith

i (by simply repeating the reasoning of (3 5 -^, i))-

(T, 7) ~ (T1, 7) h

7 C h,

also

points

w !€7 ,

Now suppose, if possible, that

small enough in order that L (7, T r) =

0

in particular

it is proved that

weA° - I T,

L(h, T f) =

L( 7 , T) =

and

w f = w, and thus

0.

0

w wel1. e l 1. Then we and hence, by

This would imply that all

belong to

I,

a contradiction.

Thus

A 0 - I C A° - I 1.

By combining the arguments of the two last paragraphs we can prove that also AA* - I C AA* - I f I = I1

and, finally,

A - I C A - I f,

i.e.,

I* C I.

Thereby

and (Ii) (ii) is proved.

3 35 *6 .

A Particular Monotone Mapping

Let a < b be any two real numbers and [x^] a countable set of distinct real numbers 0 < x. < 1 , i = 1 , 2, ... . Let v > 1 be any integer such that >

v -1 (x1 - x)/v and x ! - x < v ( y ! -y); i.e., •ty(y!) _ ^(y) < v(yr - y). This proves that t is Lipschitzian and, there­ fore, absolutely continuous (AC) in [a, b] . Let us prove that ^*(y) = v a.e. in h, and V(y) = o everywhere in or. The second part is obvious since \|r is constant on each Interval of a. In order to prove the first part let usobserve that V(y) exists a.e. in [a, b] since is AC and necessarily 0 < ^T(y) < v. For every o > 0 leth0 be the subset of h where t!(y) < v - a. Since f is AC, we have 1 = 'Kb) - ^(a) = /„ dy.Hence Oj 1f'(y) dy = (h) / ^'(y) dy = u[h + (h U - h )] / +’(y) 1 < |hal(v -

a) +

|h -

and finally |h I = 0. This proves that a set, say h ! C h, with Ih1I = IhI.

halv = 1 - or |ha I, ^f(y) = v

a.e. in h,

i.e., in

The function t: x = iKy), a < y < b, defines asingle-valuedcontinuous monotone mapping from [a, b] onto [o, 1], and each interval of a is mapped by t into a single point x = x^. Points y, y f not belonging

§35-

GENERALIZED CONFORMAL REPRESENTATIONS (3 5 -6 )

497

to the same interval of oc are mapped into different points x, x !. The inverse mapping t~1 is not single-valued in [0, 1], but is single­ valued in the set [o, 1] - £x^. Each set J C [a, b] is mapped by t onto a set I = t(J), I C [0, 1], and if |j| = 0 then also III = 0 since ^ is AC. For any L-integrable functionf(x), 0 < x < 1, we have also /1 f(x)dx = f 10 [ f^(y)]^!(y) dy, o a and hence / f(x) dx = v • (h) / f[^(y)] dy. In particular if f(x) o is the characteristic function of a set I C [o, 1] - Ex., then the set 1 J =t~ (I) is a subset of h, is measurable, and III = v|j|, since 1 = / 1 f(x) dx = v • (h) / fl>(y)] d y = v | j | . Thus III = o implies o IJl = 0. —

Finally, if f(x) Is any AC function in [o, 1], then also F(y) = f[i(y)] is continuous and AC in [a, b] since t(y) is Lipschitzian. In addition F f(y) = 0everywhere in cc and F f(y) = vff[i{r(y)] a.e. in h. Indeed, if I is the setof allpoints xe[o, 1] - Zx^, where f !does not exist, then III = 0 and, if J = t"1(I), J C h, we have also IJ"I = o. Now for every yeh* - J, let x = t(y), y 1 = y + k , x ! = x + A, A = ty(y + k) - t(y) and hence [F(yf) - F(y)]k“1 = [f(x + a) - f(x)]A“1 . Ak~1. Thus, as k -- » o, we have F !(y) = vf*[^(y)] everywhere in h* - J, i.e., a.e. in h, since lhfI= Ihl, |j| = 0. For every n, letus consider the disjoint intervals [cp(x^) cp(x^)] i = 1, 2, ..., n, and let us consider nintervals [a^, b^]such that a n, m = m(n), and let ^n (y)> a < y < b, be the q.l. function with = i = o, •••, m, which is linear in each interval [y^, i = 0, 1, ..., m - 1. Then *n (y) is strictly increasing in [a, b], maps each interval (a^, b^), i = 1, 2, ..., n, into an interval containing x^, ^n (aj_) < x^ < where the = sign may hold only if x^ = o and then = x ± < ^n ^ i ^ or xi = 1 and then ^n (aj_) < x^ = ^n^i^' have also " 'Ky) I < n”1 for all a < y < b. The last remark above implies, for instance, that if C : p = p ( x ) , o < x < i , is a given curve, and C 1: p = ptt(y)]* a < y < b, then C and C 1 are

498

CHAPTER X.

THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM

Fr§chet equivalent; i.e., as n -- > 00.

3 5 -7

C ~ C f. Indeed

dn = max|pO(y)] - p[^ (y)] I — > o

- The Two-Dimensional Extension of the Previous

Mapping

Let Q be the square Q = [o < u < 1, 0 < v < 1] in the uv-plane, let R be any rectangle R = [a < (• < b, c < T) < d] in the -plane, and let [u±], o < u± < 1, i = 1, 2, ..., [v.], o < v . < 1, j = 1, 2, ..., be any two countable sets each of distinct real numbers u^, vj. Let v be any integer such that v ~ 1 < b - a, v ~ 1 < d - c, and let e^ > 0, h, a, I = cp(u), o < u < 1, u = iU), a < | < b, be the numbers, the sets and the functions defined in (35*6) relative to the real numbers a, b, [u^]; let el > 0, h f, a 1, ti = cpf(v), 0 < v < 1, v = (“ n), c < "H < d, be the numbers, the sets and the functions relative to the real numbers c, d, [v -]. As we know, the equalities hold Ze . = b - a - v~1, — 1J Zet=d-c-v • Then the equations (H, R): u = iU), v = V(ri), (I, T])eR = [a, b; c, d], defines a continuous single-valued mapping from R onto Q. Let U, A

A

bethe open set of allpoints such that lea, r\eal ; thus is the sum of all the rectangles t\ )eR

. = f«P(u1 ) - €i < | < Ki), ^ ( t))] is continuous in R; if f has ordinary first partial derivatives fu, fy a.e. in Q, then F has ordinary first partial de­ rivatives F|, F a.e. in K and F|(l, tj ) = vfu l>(S), ^ !(t))], t]) = vfy [^U)> t!(r])] a.e. in K. This last statement is a conse­ quence of the previous one and of (35-6). As a consequence we have (k ) ;

f |(s ,

n) didri

= v2 • (k ) / f2 [y(i), **(»i)] asdri

= (Q) / provided

fu

is L2-integrable in Q.

=

dudv> Analogously for

and

fy .

For every n let us denote by^n (x), ^(y) the strictly increasing func­ tions defined at the end of (35-6). Then Hn :u = ^n (x), v = ^(y), (x, y )eR, is a homeomorphism from R onto Q, mapping R* onto Q*, and n disjoint rectangles [a^ < x < b^, at < y < bt], with a^ < cp(u^) < cp(u^) < b^, at < cpf(vj) - € t < cpf(vj ) < bt, onto rectangles [^(a± ) < u < y ( b ^ ^ 1(at) < v < ^(bt)] with ^(a^) = t(ai ), ^n (t'i ) = ^(bi), ^ ( a j ) = ^!(aj), ^(bj) = ^f(b t),and t(a± ) < u ± < ♦(b1 ), If* (al) < v. < ^!(bl) i, j = 1 , 2, ..., n. Finally we have l^n (x) - ^(x)l < n ”1, U^ty) - t(y) I < n for all (x, y)eR. This implies that if (T, Q): p = p(w), weQ, w = (u, v), p = (X, Y, Z), is any given mapping and (T1, R): p = p[H(w!)], w TeR, w ! =(I, i), then T 1 ~ T and hence L(T!, R) = L(T, Q). Indeed dR = max|p[Hn (w* )] p[H(wf)]l -- > 0 as n -- v °°. In addition T* is constant on each component r = [cp(u± ) - €± < x < cp(u± ), cp!(v. ) - et < y < cpT( v . )] of Q, i, j = 1, 2, ... If we suppose that (T, Q) is almost conformal (3 4 .2 ) thenE = G,F = 0, everywhere in a set Q,- N with INI = 0. Hence, if N* = H~1(NQ!), we have iNl = INQ,1I = 0, IN1I = 0. Finally X^(w!) = v2xQ(w), ..., Z ^ w 1) v2zy (w) for all w T = (R - B) - N I! = K - Nn, where w = H(w!)eQT, and |N! 1 I = 0.Thus E ! = G !, F ! = 0 everywhere in (R - B) -(N1 + N 11), i.e., a.e. in R - B = K. In addition

=

CHAPTER X.

500

THE REPRESENTATION OP GENERAL SURFACES AND THE THIRD THEOREM

L(R, T') = L(Q, T) = 2~1 • (Q) / (E + G) dudv = = 2~1 • (K) / (E1 + G' ) didrj.

5 5 -8

I.

- Examples of Representations

A SQUARE WITH A THREAD. Let S^ be the surface formed by the square c r = [ - i < x < 1 , - l < y < l , z = o ] of the xy-plane and by the segment X = [x = 0, y = 0, 0 < z < 1] of the z-axis issuing from the center of the square a. More precise­ ly, let S1 be the surface defined by the following mapping (T, Q) from the square Q = [- 1 < u < 1, - 1 < v < 1] of the uv-plane: x =

(2 1

u I - 1 )sgmu,

y =

(2 1

u I -1 ) v / | u | ,

z = 0

if

2"1 < |u| < 1, |v| < Iu|; x = (2 IvI - l)u/|vl,

y = (21vI - 1)sgmv, 2 -1 < Iv l < 1,

x = 0,

y = 0,

z = 0

if

lu l < I V|j

z = 1 - 2 max[|u|, |v|]

if

|v| < 2- 1

-1

Iu| < 2

In this representation the open square q = [- 2”1 < u < 2~1, - 2~1 < v < 2""1] is the maximal exceptional set; the closed annular region Q - q is a proper set. Such a representation is not "almost conformalM and not even "generalized conformal" in Q - q. Indeed, for instance, in the sector ,-1 < u < 1, Ivl < u, we have x = 2u - 1, y = (2u - 1)v/u, z = 0 and hence E = b + v2u”i+, G = b + u~2 - 4 u~1, F = 2vu-2 - vu~^, i.e., E i G, F i 0. Nevertheless the surface S1 has also the following representation (i.e., the following mapping (T.j, Q) is F-equivalent to (T, Q)): x = (21uI - 1)sgmu,

y = (21v| - 1)sgmv,

z = 0 if

2 1 < |u| < 1, 2 1 < |v | < 1; 2"1 x = 0,

y = (2Ivl - 1)sgmv,

z

0

if < Ivl

T1 2

0,

;

z = 0

y = (2|u| - 1)sgmu,

0,

< 1

1

< |u|
I . e . , 7 ! , 7 ! + 0, and s in c e no p o in t o f F ( 7 ! ) i s i n 7 M > we deduce t h a t a l l p o in t s o f 7 11 a re i n 7 ! , i . e . , 7 11 C 7 1. By 7 11 C 7 1, g M C 7 ^ we co n c lu d e t h a t 7 11 + F ( 7 , t ) C 7 * 1 + g M C 7 T . The same r e a s o n in g h o ld s b y e x c h a n g in g K f and K I ! , and ( 2 ) i s p ro v e d .

Since the sets 7 T are disjoint as well as the sets 7 !l, it follows that 7] C 7** C 7 x2 , or 7]1 C 7j C 7^*, with 7], 7 ^€ 1 ]> 7 { implies 7] = 7] 1 = 7 2>7 ]1 = 7 ]= 7 2!* If we denote by [70] the collection of all those 7 T, or 7 ! 1 which are not contained as a proper part of other sets 7 XX, or 7 X, then the sets 7Q are disjoint, and for every 7 = 7 or 7 = 7 !1, we have either 7 = 7 qj 7 Q€[7 0], op 7 contained as a proper part of 7 0 for some 7 0 €^7 o^' Therefore J - K = Z7 Q where z ranges over all 7^ [7 0]Obviously K is closed and also compact. Let us suppose, if possible, that K Is not a continuum. Then, by (36.1, i) there exists a line h C J separating J into two parts J^, J2 with J^K + 0 , + 0, hK = 0 . Let wQ be any point wQ€h. Then w0 e7 0 f°r some 70 ^[7 Q] and thus there is a maximal open arc on h which is in 7 Q - Such an arc must coincide with h otherwise its end point would be in K and hence Kh + 0 ,

7 ^le[7

§3 6 . A RETRACTION PROCESS FOR SURFACES (3 6 .3 )

511

a contradiction. Thus h C y Q where y Q = 71, or 70 = 7M « Say y Q = 7* and hence K fh = 0. Thus either K* C or K* C J2 . Say K 1 C J1 and then J2 C 7 1 = y Q> KJ2 = 0, a contradiction. Thus we have proved that K is a continuum. Obviously [ y Q ] = [7]^and since 70 = y \ or y Q = 71 1, and T is constant on all sets F(7!)and F(7IT), T is constant on F (7q ) and K satisfies condition P with respect to (T, J). Thereby (vi) is completely proved.

36.3. (i)

Lebesgue Area of a Retraction

Let (T, Q) be any c. mapping from a simple pol. reg. Q C E2 into E^ and (TQ, Q) any retraction of (T, Q). Then there are two sequences of q.l. mappings (Pn> Q), (Pno, Q), n = 1, 2, with Pn ->T, Pn o — a(Pn > — »L(T), a(Pno) — ^L(Tq ) as n -- > 00j such that, for each n, PnQ is a retrac­ tion of Pn, and a(PnQ) < a(Pn ).

PROOF. Let [7] denote the countable collection of all components 7 of Q - K, and by the subcollection of those 7with diam T(7) >(3n)~1 where n is any integer. Then,by (36.2, i), [7] is finite and M1 diam T(7 ) < (3n)~ for all y e [ y ] For each y e [ y ] n we consider the boundary F(7) of 7 in Q, and, since T is constant on F(7), we have F(7) C g for some continuum ger(T, Q). Let us denote by and [p]n the finite collections of all distinct ger(T, Q) with g )F(7); yeiy]^, and of all distinct points p = T(g)€E^. —1

—1

By (6.5, i), where € = (3n) there exists a number a , 0 < an < (3n), and a q.l. mapping from E^ into itself relative to the collection [p]n above. Let (Pn, Q), (Pno, Q), n = 1, 2, ..., be anytwo sequences of q.l. mappings such that, for each n, we have ( a ) d(P , T, Q), d(pno’ To’ Q) < 3 "1V a(Pn } * L(T) + n_1> a(Pno) ^ L(To ) + n ~' ' Let (P^, Q), (P^Q, Q) be the q.l. mappings = -rnPn, P^Q = TnPno. By (6.5, i) we have (P) d(P^, Pn, Q), cKP^ q * pno> Q) < (3n)_1. Let Sn be a regular finite subdivision of Q, into triangles t such that both P —1 and PnQ are linear on each teSn, and diam TQ(t) < 3 cr^, diam T(t) < 3 ~1 for all t€Sn . Wedenote by Rn the collection of all teSn with tK 4 0, and we denote by Rn also the set of points covered by all t€Rn As in (6.1, Note 2) we may distribute the triangles teR into classes by putting two triangles t !, t !I€Rn into the same class if and only if there Is in Rn a chain t ! = t2, ..., tm+1 = t 11 of triangles, any two consecutive having a side in common. Then each class covers a connected

512

CHAPTER X.

THE REPRESENTATION OF GENERAL SURFACES AND THE THIRD THEOREM

subset of Q which may not be a pol. region but, by a standard procedure as in (6.11, Note 2) we can define a slight refinement of Sn such that the various sets above are disjoint pol. regions whose boundaries do not contain points of K. Since each of these pol. regions must contain points of K, we conclude that there cannot be more than one of such regions. Thus it is not restrictive to assume that Rn is a pol. region, K C Rn C Q, F(Rn )K =0. As a consequence also Q, - Rn is the finite sum of pol. regions rni, i = 1, 2, ..., m, m = m(n), ^nj_K = 0, and the boundary F ( r n ^ ) of each rni in Q is a simple arc of one of the boundary curves of Rn - More exactly at most one of the regions rni is not simple and then of connectivity v = 1, and this occurs as Rr C Q°, rni is reS^on between q* and the exterior boundary curve r* of Rh and F(rn i ) = For each r ^ the arc is necessarily con­ tained in only one component 7 €[7]. We may add to Rn all regions rn^ (if any) with ^(r^) C 7, y e l y ] We shall denote by the new pol. region, K C Rn C R^ C Q. For each component r ^ of Q - R^, say pni' ^ = 1' 2' a set 7 e[7] •

ra,(n )> m*(n) < m(n),

the arc

F ( v n ^ )

is contained in

For each component r ^ of Q, - R^ the arc F(rnj_) a sum 0f* sides s of triangles t with tK 4 0. Thus for each wes we have wet with tK j= 0 and, since t contains points of F(7 ) and points of 7, t must contain at least one point w*€F(7).Therefore we have |T(w) - T(w')| < diam T(t) < |TQ (w) - T0 (w!)l < diam TQ (t) < for every wes. On the other hand, T and TQ are constant and equal on F(7 ) and T[F(7 )] = Tq [F(7 )] = petP]n • Thus |T(w) - p| < |TQ (w) - p| < 3 ”’1^n for all weF(rn i ). By (a) we have now |Pn (w) - T(w)| < 5~1crn, |Pno(w ) - TQ(w)| < ^Pn o ^

“ ^

< an

for all ^op a11

w€F(rn i ),

and hence

|Pn (w) - p|,

W€^ pni^‘ As a consecluence both

=

TnPn^w ^ Pn o ^ = TnPn o ^ are constant and equal on FCr^), say PJ!I[F(rni)] = PA o ^ pni^ = This assures that satisfies prop­ erty P with respect to both (P^, Q) and us denote by (P^1, Q), (P^qj Q) the (elementary) retractions of P^, P^Q with respect to R£. Then (7) atP^1) < a(P^) < a(Pn ), a(P^) < a(P^Q ) < a(PnQ). By (a), (p), and (7) we have now d(Pn, T, Q) < 3 _1 crn < (3n)_1, d(pn, Pn' ^ < (3n)"1, a(P^) < a(Pn ). Hence we have d(P^, T, Q) < n“1, P^ -- > T, L(T) < lim a(P}!L) as n -- > °° and, by (a) again, also a ^Pn^ -- > L (T) as n -- > °°. Analogously we have d(P^Q, TQ, Q) < n”1, pA o —

> V

a(pAo } —

>L(To )*

For each weK we have P^tw) = P^w), T(w) = TQ (w), |P” (w) - T (w)| = |P’(w) - T(w)| < n~1 . For each teS , with tK 4 0, diam T(t) < diam TQ (t)
7 € ^-7 ^n’ diam Pn T^Pni^ = diam PAo^rni^ = 0 anci' wf€F^rni^ then w ] eRn - K and also lP"(w) - Tq (w )| < |P-(w) - P ” (w')l + |P-(w') - TQ (w1)| + |T0 (w!) TQ (w)l < 0 + n"1 + 0 = n""1. Thus |P^*(w) - T (w)| < n~1 fop e v e r y weQ and also lT(w) - TQ(w)| < (3n)~1 fop evepy weR^. The same argument proves also that lpAc>(w )— T (w)I < n~1 fop evepy weQ. Thus we have (5 ) d i ? " ’ To ’ d ^Pno' To' ^ < n ~ ] ’ and consequently L(TQ ) < lim aCP^1), L(Tq ) < lim a^A^), as n -- * °°’ and ^ and lattep yields a(PAc>) -^ ( T q ) as n -> 00* us now PPOve that have also ( e ) a ( P ^ J ) ->L(TQ ) as n -► «>.

we

If (e) is not tpue then by L(TQ )< lim a(P^r) we have L(T ) < + 00 and L(Tq ) + n < a(PA* ) fop some r\ > 0 andinfinitely many n- Pop the same n let (P*, Q) denote the q.l. mapping defined by P*(w) = PA^V ^ f°p evepy weQ - R^, = pA o ^ = pAo^w ) foP evepy weR^. Since P^, PAq , coincide on F (pnj_); the mapping P^ is continuous in Q. On

i P A

the othep hand, " T(w)l = lpA (w) " T(w)l ^ ~ Pn l + ,Pn “ T 1 < 2 (3 n ) " 1 fop evepy wgQ - RA; |P*(w) - T(w)| < IPAo^ ~ pno^)l + lpn0(v ) “ TQ(w) | + |TQ (w) - T(w) | < (^n) ” 1 + (3 n ) ” 1 + (3 n ) ~ 1 = n ""1 fop and consequently evepy weRA* Thepefope d(P*, T, Q) -- > 0 as n -- » L(T) < lim a(P*). Now we [L(T) + n_1] + [- L(Tq ) L(T) < L(T) - r\, t) > 0 , hypothesis we have ppoved

have a (P*) = a^Pn^ ~ a ^Pn f^ + a^Pno^ t]] + [L(Tq ) + n"1] = L(T) + 2 n ~ 1 - r\, and hence a contpadiction, ifL(T) < + 00. In this pelation (e) and also (ii) is ppoved.

L(T) = + 00, then we may define P* as above and we have also ■X* L(T) = lim a(Pn ) as n > 00. On the othep hand, PAA is the t r a c t i o n of P* with pespect to---RA and, as we have seen above, a (pAA) -- > U T q ) . Thus (i) is ppoved also fop L(T) = +00. + 00 =

Let

-

If

re­

(T, J) be any c. mapping (T, J) fpom a simple Jopdan pegion J into and, f o r e v e r y continuum K C J having ppopepty P with pespect to (T, J), we shall considep the petpaction (TQ, J) of T with pespect to K and the c. mapping (T, J - K) defined by T on the set J - K, open In J, and hence admissible (5 * 1 )• We shall denote as usual by the collection of all components of J - K.

514

CHAPTER X. (ii)

THE REPRESENTATION OP GENERAL SURFACES AND THE THIRDTHEOREM

For every c. mapping

(T,J)

we have

L(J, T) = L(J, T0 ) + L(J - K, T) = = L(J, Tq ) + £L(r, T), where £ ranges over all L(J, T) > L(J, Tq ).

7e [7

Thus

PROOF- If H is any homeomorphism from J onto the unit square Q and (T', Q) = TH-1, K> = H(K), (T£, Q) = TQH"1, then by (6.5, D we have L(Q, T') = L(J, T), L(Q, ) = L(J, TQ ), L(Q - K 1, T') = L(J - K, T). On the other hand, it is obvious that K 1 is a subcontinuum of Q, possessing property P with respect to (Tl, Q) and (T^, Q) is the retraction of (T1, Q) with respect to K 1. For each n we shall consider two sequences (Pn, Q), (Pn0> n = 2 > •••> of q.l. mappings as in (i), where PnQ is the retraction of Pn with respect to a pol. region Rn, with dR = d(Pn, T 1, Q) -- > 0, d(PnQ, Q) -- >0, a(Pn, Q) -- >L(Q, T l), a(PnQ, Q) -- >L(Q, T^), and also Rn } Rn+-|jRn ^ Cq , Rn ^ K ! (cf. the proof of (i)above). There­ fore, if Fn = Q, -Rn, Fn is a figure (finite sum of disjoint pol. regions r), with Fn C Fn+1j C Q,- K f, and Fn t Q - K'. It isimmediate­ ly seen that we have also P° t (Q, - K r)°- Since d(Pn, T 1, Fn ) < d(Pn, T f, Q), we conclude that d(Pn, T !, Fn ) -> 0 , (Pr,Fr ) ---> (Tf, Q - K !), and hence L(Q - K 1, T T) < lim a(Pn, ^n ) n --------------------- * 00(5 -3 ; other hand, Pno = Pn on R , PnQ is constant on each region r, and hence

a(Pno> Q) = a(pno' Rn^ = a^Pn'Rn^*

F in a ll7> we have

L(Q, T£) + L(Q - K', T 1) < lim a(PnQ, Q) +

lim a(Pn, FR )=

= l i m ta(Pn0, Q) + a(PR, FR )] = (O = lim ta(Pn, Rn ) + a(Pn, Fn )J

=

= lim a(Pn, Q) = L(Q, T l). Now for each n and for each component r of Q, Rn, the twomappings Pn and PnQ are constant and equal on the polygonal line F(r); hence for each n and r,we have a well determined point p = p(n, r) = Pn [F(r)] = PnQ[F(r)]. Let (P*, Fn ), n = 1, 2, ..., be any sequence of q.l. mappings from the figures Fn above with (P*, F ) -- > (T!, Q, - K 1), a(P*, Pn ) -- »L(Q - K', T' )• Then d^ = (P*, T', Fn ) -- > 0 as n -- * °° by definition (5 *3 )* For each n and r, and for each weF(r) we have now Pn (w) = Pn [F(r)] = p(n, r), “ Pi < " T ’(w)l + |T’(w) - Pn (w) | < d* + d . Thus for all weF(r) the points p^(w) are in

5 •8 ).

§3 6 . A RETRACTION PROCESS FOR SURFACES (36.4)

515

the solid sphere S of center p and radius dn + d*. For each n and r weshall consider the mapping t = T(n, r) defined in (6.5, and Note 1) relativelyto the simple point p = p(n, r) and where € = 5(dR + Then a = dn + d* and we shall consider the q. 1 - mapping Fn ) where t is a different mapping for each of the components r of F . Since each sphere S is mapped into its center p by the corresponding mapping t, we can conclude that tP* is constant on each polygonal line F(r) and xP*[F(r)] = Pn [F(r)] = PnQ[F(r)] = p(n, r) for each n and r. We have also b(tP*, Fn ) < a(P*, Fn ), d(xP*, P*, FR ) < 3 (dn + d*); hence ^(tP*, Tf, FR ) < 3 (d-n + d*) +d* = 5dn + 4 d*. Finally, let us con­ sider the mapping (Pn, Q) definedby Pn = pn = pn0 in Rn> ?n = TPn in Fn - Then we have d(Pn* T l, Q) < 3 dn + (Pn, Q) ---- * (Tl, Q), and also L(Q, T 1)< lim a(Pn, Q) = (2)

lim[a(Pn, Fn ) + a(PR, Rr )]