Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube 9783031308857, 9783031308864

Table of contents :
Preface
Contents
1 Topology
1.1 Basic Concepts for Metric Spaces
1.2 Compactness for Metric Spaces
1.3 Compact Topological Spaces
1.4 Product of Topological Spaces
1.5 Completeness
1.6 Separation Properties
1.7 Open and Closed Subsets
2 Hilbert Spaces
2.1 Vector Spaces
2.2 Basis of Vector Space
2.3 Normed Spaces
2.4 Banach Spaces
2.5 Inner Product Spaces
2.6 Hilbert Spaces
2.7 Orthogonal Complements
2.8 Separable Hilbert Spaces
3 Measure Theory
3.1 Outer Measures
3.2 Measures in Abstract Spaces
3.3 Measures in Metric Spaces
3.4 Metric Outer Measures
3.5 Lebesgue Measures
3.6 Hausdorff Measures
4 Extension of Isometries
4.1 Basic Concepts and Remarks
4.2 First-Order Generalized Span
4.3 First-Order Extension of Isometries
4.4 Second-Order Generalized Span
4.5 Basic Cylinders and Basic Intervals
4.6 Second-Order Extension of Isometries
4.7 Extension of Isometries to the Entire Space
5 History of Ulam's Conjecture
5.1 Historical Background
5.2 Basic Definitions
5.3 Mycielski's Partial Solution
5.4 Fickett's Partial Solution
5.5 Jung and Kim's Partial Solution
6 Ulam's Conjecture
6.1 Basic Definitions
6.2 Cylinders
6.3 Elementary Volumes
6.4 Construction of Invariant Measure
6.5 Efficient Coverings
6.6 Ulam's Conjecture on Invariance of Measure
Bibliography
Index

Citation preview

Frontiers in Mathematics

Soon-Mo Jung

Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube

Frontiers in Mathematics Advisory Editors William Y. C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Spr¨oßig, TU Bergakademie Freiberg, Freiberg, Germany

This series is designed to be a repository for up-to-date research results which have been prepared for a wider audience. Graduates and postgraduates as well as scientists will benefit from the latest developments at the research frontiers in mathematics and at the “frontiers” between mathematics and other fields like computer science, physics, biology, economics, finance, etc. All volumes are online available at SpringerLink.

Soon-Mo Jung

Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube

Soon-Mo Jung Mathematics Section Hongik University (Sejong Campus) Sejong, Korea (Republic of)

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-031-30885-7 ISBN 978-3-031-30886-4 (eBook) https://doi.org/10.1007/978-3-031-30886-4 Mathematics Subject Classification: 28A35, 28A78, 46B04, 46C05, 46G12 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkh¨auser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface Ulam’s conjecture, which claims the invariance of the standard product probability measure defined on the Borel .σ-algebra of the Hilbert cube, was probably made in the late 1930s before the outbreak of World War II. However, Ulam’s conjecture did not become known to the world until the following book published by Stanislaw M. Ulam in 1960: • S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960. In fact, Ulam conjectured the invariance of measures defined on the compact metric space: Let X be a compact metric space. Does there exist a finitely additive measure .μ defined for at least all the Borel subsets of X, such that .μ(X) = 1, .μ(p) = 0 for all points p of X, and such that congruent subsets of X have equal measure? In the 1970s, J. Mycielski rephrased Ulam’s conjecture using modern mathematical terms: The standard product probability measure on the Borel .σ-algebra of the Hilbert cube is invariant. The above statement is now widely known as Ulam’s conjecture. He also succeeded in proving Ulam’s conjecture under the assumption that the sets involved in the conjecture are open. A few years later, J. W. Fickett, who closely analyzed Mycielski’s work, was able to partially prove Ulam’s conjecture by transforming the .δ-isometry to construct an exact isometry and combining it with the method developed by Mycielski. In fact, he proved Ulam’s conjecture for metrics defined by any sequence of positive real numbers that decreases to 0 very quickly. More than 30 years later, S.-M. Jung and E. Kim partially proved Ulam’s conjecture by improving Fickett’s method and constructing a Hausdorff measure using the concept of entropy. Their result significantly improved Fickett’s result. v

vi

PREFACE

In 2020, S.-M. Jung combined various methods to prove Ulam’s conjecture. Among them, he developed and applied a method for extending the domain of local isometries, and was able to completely prove that Ulam’s conjecture is true. This book is structured to give the reader an overview of the whole process of solving Ulam’s conjecture, with details of the proof process explained in detail. Moreover, since this book is written in an integrated and self-contained manner, it is easy for readers to understand the contents and to avoid the trouble of searching for literature on their own. This book consists of six chapters, and we will now briefly describe the content of each chapter. • Chapter 1 briefly introduces the basic concepts and theorems of general topology that are essential to understanding the subject of this book. All lemmas and theorems that are necessary to prove the theorems presented in this chapter are introduced so that the reader does not have to reference other literature when reading this book. • In Chap. 2, we briefly introduce basic concepts and theorems in vector space, normed space, Banach space, inner product space, and Hilbert space that are essential to prove Ulam’s conjecture. • The concept of measures is a generalization and formalization of length, area, volume, and other common notions such as mass and probability of events. Although these concepts may appear different at first glance, they have many similarities and can often be unified by the concept of measure. In Chap. 3, we briefly present the basic concepts and theorems of general measure theory that are essential for proving Ulam’s conjecture. • In Chap. 4, we define the first- and second-order generalized spans and the index set, examine their properties, and apply them to the study of the extension of isometries. To this end, we develop a theory that extends the domain of local isometries to the generalized spans. We also prove that under the axiom of choice, the domain of a local isometry can be extended to any real Hilbert space, where the domain of a local isometry need not be a convex body or an open set. • In 1974 and 1977, J. Mycielski published the first papers dealing with Ulam’s conjecture. In fact, he proved Ulam’s conjecture positive with the additional assumption that the relevant sets are open. In 1982, J. W. Fickett succeeded in partially proving Ulam’s conjecture by proving the following statement in a different way than Mycielski: If the .ai ’s decrease to 0 very rapidly, then any two Borel subsets of the Hilbert cube, which are .da -isometric, have the same

PREFACE

vii

standard product probability measure. Around 40 years later, in 2018, S.-M. Jung and E. Kim jointly studied Ulam’s conjecture and improved Fickett’s result by partially proving Ulam’s conjecture. In Chap. 5, we introduce the historical process of solving Ulam’s conjecture by presenting a summary of these papers. In some cases, the reader may skip reading this chapter. • The conjecture of Ulam states that the standard product probability measure on the Hilbert cube is invariant under the induced metric .da when the sequence .a = {ai }i∈N of positive real numbers satisfies the condition ∞  . a2i < ∞. In Chap. 6, we completely prove that Ulam’s conjecture is true i=1

under the axiom of choice. The author would like to express his sincere thanks to Professor Dr. Dietmar Kahnert for suggesting and guiding this topic as a subject of his doctoral thesis. The author cannot fail to express his sincere thanks to Mr. Doyun Nam of Seoul National University, who has improved the quality of this book by carefully reading the first manuscript of this book and pointing out various points for improvement. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2020R1F1A1A01049560). Sejong, Republic of Korea February 2023

Soon-Mo Jung

Contents Preface

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1 Topology 1.1 Basic Concepts for Metric Spaces 1.2 Compactness for Metric Spaces . . 1.3 Compact Topological Spaces . . . 1.4 Product of Topological Spaces . . 1.5 Completeness . . . . . . . . . . . 1.6 Separation Properties . . . . . . . 1.7 Open and Closed Subsets . . . . . 2 Hilbert Spaces 2.1 Vector Spaces . . . . . . 2.2 Basis of Vector Space . . 2.3 Normed Spaces . . . . . 2.4 Banach Spaces . . . . . 2.5 Inner Product Spaces . . 2.6 Hilbert Spaces . . . . . . 2.7 Orthogonal Complements 2.8 Separable Hilbert Spaces

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3 Measure Theory 3.1 Outer Measures . . . . . . . 3.2 Measures in Abstract Spaces 3.3 Measures in Metric Spaces . 3.4 Metric Outer Measures . . . 3.5 Lebesgue Measures . . . . . 3.6 Hausdorff Measures . . . . .

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55 55 61 72 76 81 86

CONTENTS

x 4 Extension of Isometries 4.1 Basic Concepts and Remarks . . . . . . . . 4.2 First-Order Generalized Span . . . . . . . . 4.3 First-Order Extension of Isometries . . . . 4.4 Second-Order Generalized Span . . . . . . 4.5 Basic Cylinders and Basic Intervals . . . . 4.6 Second-Order Extension of Isometries . . . 4.7 Extension of Isometries to the Entire Space

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5 History of Ulam’s Conjecture 5.1 Historical Background . . . . . 5.2 Basic Definitions . . . . . . . . 5.3 Mycielski’s Partial Solution . . . 5.4 Fickett’s Partial Solution . . . . 5.5 Jung and Kim’s Partial Solution

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6 Ulam’s Conjecture 6.1 Basic Definitions . . . . . . . . . . . . . . . 6.2 Cylinders . . . . . . . . . . . . . . . . . . . 6.3 Elementary Volumes . . . . . . . . . . . . . 6.4 Construction of Invariant Measure . . . . . . 6.5 Efficient Coverings . . . . . . . . . . . . . . 6.6 Ulam’s Conjecture on Invariance of Measure

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Bibliography

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Index

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Chapter 1

Topology In this chapter, we will briefly introduce the basic concepts and theorems of general topology necessary to understand the subject matter of this book. Among many other literatures listed in the References section, we mainly refer to the book [13] by R. H. Kasriel and the book [19] by G. F. Simmons for this purpose. All the lemmas necessary to prove the theorems presented in this chapter are introduced so that the reader, if possible, does not have to refer to other books when reading this book.

1.1 Basic Concepts for Metric Spaces A metric space is a set along with a metric on the set, and the metric is a function that generalizes the notion of a distance between any two elements of a set. In other literature, elements of sets are often referred to as points, but this book will refer to them as elements instead of using the term points whenever possible. Definition 1.1. Assume that X is a set and .d : X × X → [0, ∞) is a function that satisfies (i) .d(x, y) = 0 if and only if .x = y

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(ii) .d(x, y) = d(y, x)

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(iii) .d(x, y) ≤ d(x, z) + d(z, y)

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for all .x, y, z ∈ X. Then the function d is said to be a metric on X, and .(X, d) is called a metric space. When it is clear what d is, we often simply write X instead of .(X, d). Let Y be a subset of X and let .d∗ be the restriction of d to .Y × Y . Then .(Y, d∗ ) is a metric space and it is called a subspace of .(X, d). In this case it is customary to write .(Y, d) instead of .(Y, d∗ ). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4 1

1

CHAPTER 1. TOPOLOGY

2

For each positive integer n, let .Rn be the n-dimensional space of all ordered n-tuples of real numbers. We define the function .de : Rn × Rn → [0, ∞) by 1/2  n  2 .de (x, y) = (xi − yi ) i=1

for all .x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ) ∈ Rn . Then .de satisfies all the conditions in Definition 1.1, i.e., it is a metric on .Rn which is called the Euclidean distance for .Rn . Any set X can be considered a metric space in a simple way. For example, we define the function .d : X × X → [0, ∞) by  0 (for x = y), .d(x, y) = 1 (for x =  y) for all .x, y ∈ X. It is easy to check that d is a metric on the set X. This metric is called the discrete metric. Let x be a fixed element of a metric space .(X, d) and let r be a positive real number. Then we define the open ball, with center x and radius r, by .Br (x) = {y ∈ X : d(y, x) < r}. Similarly, we define the closed ball, with center x and radius r, by .B r (x) = {y ∈ X : d(y, x) ≤ r}. Definition 1.2. Assume that K and U are subsets of a metric space .(X, d). (i) U is said to be open if and only if for every .x ∈ U , there exists a real number .ε > 0 such that .Bε (x) ⊂ U ;

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(ii) K is said to be closed if and only if its complement .X \ K is open.

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We will often write “U is open in X” and “K is closed in X” instead of “U is an open subset of X” and “K is a closed subset of X,” respectively. Lemma 1.3. Assume that x is an element of a metric space .(X, d) and r is a positive real number. (i) The open ball .Br (x) is open in X.

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(ii) The closed ball .B r (x) is closed in X.

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Proof. .(i) For any element .y ∈ Br (x), we see that .d(y, x) < r and we set .ε = r − d(y, x) > 0. Assume that z is an arbitrary element of .Bε (y). Then we have d(z, x) ≤ d(z, y) + d(y, x) < r − d(y, x) + d(y, x) = r,

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which implies that .z ∈ Br (x), i.e., .y ∈ Bε (y) ⊂ Br (x). Thus, according to Definition 1.2 .(i), .Br (x) is an open subset of X. .2 We encourage the reader to do his own proof of .(ii).

1.1. BASIC CONCEPTS FOR METRIC SPACES

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Definition 1.4. Assume that X is a set and .T is a collection of subsets of X. The collection .T is said to be a topology for X if .T satisfies the following axioms: (i) .∅ ∈ T and .X ∈ T .

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(ii) If .U1 ∈ T and .U2 ∈ T , then .U1 ∩ U2 ∈ T .

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(iii) If .U is an arbitrary sub-collection of .T , then .



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U ∈U

U ∈T.

Then .(X, T ) is called a topological space. When it is clear what .T is, we often simply write X instead of .(X, T ). Let Y be a subset of X and let .TY be the collection .{U ∩ Y : U ∈ T }. Then .TY is a topology for Y and it is said to be the relative topology for Y induced by .T . Moreover, .(Y, TY ) is said to be a subspace of .(X, T ). In particular, a subset U of X is called open in X if and only if .U ∈ T , and a subset K of X is called closed in X if and only if .X \ K ∈ T . Let .P(X) be the power set of a set X, i.e., the collection of all subsets of X. Then .P(X) is a topology for X and it is called the discrete topology for X. Let .T = {∅, X}. Then .T is also a topology for X and it is called the trivial topology or the indiscrete topology for X. In general, any set can have at least two obvious topologies: the discrete topology and the indiscrete topology. Let .(X, d) be a metric space and let .T (d) be the collection of all open subsets of X defined by Definition 1.2. It is then easy to prove that .(X, T (d)) is a topological space. So .T (d) is said to be the topology for X generated by d. Definition 1.5. (i) Let .(X, d) be a metric space and let .T (d) be the collection of all open subsets of X defined by Definition 1.2. The collection .T (d) is a topology for X and it is said to be the topology for X generated by d.

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(ii) A topological space .(X, T ) is said to be metrizable if there exists a metric d on X whose collection .T (d) of generated open sets is exactly the given topology .T .

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For this reason, we often treat the metric space .(X, d) as a topological space (X, T (d)). We note that the discrete topology is generated by the discrete metric but the indiscrete topology cannot be generated by any metric.

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Definition 1.6. Let K and U be subsets of a topological space X. (i) A point .x ∈ X is called an interior point of U if there exists an open subset V of X such that .x ∈ V ⊂ U .

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CHAPTER 1. TOPOLOGY

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(ii) A point .x ∈ X is called a limit point of K if every open set containing x intersects K in a point distinct from x.

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(iii) The set of all interior points of U is called the interior of U and is denoted by .U ◦ .

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(iv) The union of K and the set of all its limit points is called the closure of K and is denoted by .K.

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Theorem 1.7. Let S be a subset of a topological space X. (i) The interior of S is the union of all open subsets of X that are included in S, i.e.,  ◦ .S = U : U is an open subset of X included in S .

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(ii) The closure of S is the intersection of all closed subsets of X that include S, i.e.,

 K : K is a closed subset of X including S . .S =

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Let .N be the set of all positive integers and let .N0 = N ∪ {0}. A sequence {xi }i∈N of elements of a subset S of a topological space X is said to converge to an element x of X if for every open subset U of X containing x, there exists a positive integer N such that .xi ∈ U ∩ S for all .i ≥ N . The proof of the following theorem is left as an exercise for the reader.

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Theorem 1.8. The closure of a subset S of a topological space is the set of limits of all convergent sequences of elements of S. Definition 1.9. Let X be a metric space. (i) A subset D of X is called a dense subset of X (or dense in X) if .D = X.

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(ii) X is called separable if there exists a countable subset of X that is dense in X.

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The following characterization of denseness is easy to prove and its proof will be left as an exercise. Theorem 1.10. Assume that D is a subset of a metric space X. Then D is dense in X if and only if each nonempty open subset of X intersects D. The following theorem gives an important property corresponding to separability for metric spaces.

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Theorem 1.11. A metric space X is separable if and only if there exists a countable collection .U of open subsets of X such that each open subset of X can be expressed as the union of a sub-collection of .U. Proof. Assume that the metric space .(X, d) is separable. Then there exists a countable dense subset .{ci : i ∈ N} of X. Let .{ri : i ∈ N} be the set of all positive rational numbers. Moreover, we define a countable collection .U of open subsets of X by .U = {Brj (ci ) : i, j ∈ N}. We now assert that each open subset of X can be expressed as the union of a sub-collection of .U . Indeed, we will prove that if V is an open subset of X and .x ∈ V , then there exists a .U ∈ U such that .x ∈ U ⊂ V . Assume that V is an arbitrary open subset of X and .x ∈ V . Then there exists an .ε > 0 such that .Bε (x) ⊂ V . Since .{ci : i ∈ N} is dense in X, there exists a .ci that satisfies .d(ci , x) < 13 ε. Now we note that .B(2/3)ε (ci ) ⊂ V . Let .rj be a positive rational number satisfying 1 2 . ε < rj < 3 ε. Then we see that .x ∈ B(1/3)ε (ci ) ⊂ Brj (ci ) ⊂ B(2/3)ε (ci ) ⊂ V , 3 i.e., if we set .U = Brj (ci ), then .U ∈ U and .x ∈ U ⊂ V . Now, we assume that there exists a countable collection .U = {Ui : i ∈ N} of open subsets of X such that each open subset of X can be expressed as the union of a sub-collection of .U , where each .Ui is nonempty. For every .i ∈ N, we choose a .ci ∈ Ui and define .D = {ci : i ∈ N}. Assume that V is an arbitrary nonempty open subset of X and .x ∈ V . Then there exists a .Ui such that .x ∈ Ui ⊂ V . Moreover, .ci ∈ V . Hence, .V ∩ D = ∅. Thus, due to Theorem 1.10, D is a countable dense subset of X. Therefore, the metric space X is separable. .2 Now we will introduce the concept of covering of a set, which is a terminology commonly used in topology. If S is a subset of a topological space .(X, T ),then a covering of S is a collection .{Uλ : λ ∈ Λ} of subsets of X satisfying .S ⊂ Uλ , λ∈Λ

where .Λ is some index set. In this case, we say that the sets .Uλ ’s cover S. A covering .{Uλ : λ ∈ Λ} of S is called an open covering if each of its members is an open set, i.e., .Uλ ∈ T for all .λ ∈ Λ.  Let .U be a covering of S. If .V is a sub-collection of .U and .S ⊂ V , then .V is sometimes referred to as a sub-covering of S.

V ∈V

Theorem 1.12 (Lindel¨of). Let X be a separable metric space. If .U is an open covering of a subset S of X, then there exists a countable sub-collection of .U that also covers S. Proof. Assume that .U is an open covering of S. Due to Theorem 1.11, there exists a countable collection .V = {Vi : i ∈ N} of open subsets of X such that each open subset of X is the union of a sub-collection of .V.

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Let.J = {i ∈ N : Vi ⊂ U for at least one U ∈ U }. We will prove that S ⊂ Vi . For any .x ∈ S, there exists a .Ux ∈ U that includes x. Since the

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i∈J

open set .Ux is the union of a sub-collection  of .V, we can choose a .k ∈ N such that .x ∈ Vk ⊂ Ux . Hence, .k ∈ J and .x ∈ Vi . For any .j ∈ J, we choose an open i∈J   set .Uj ∈ U with .Vj ⊂ Uj . Now it holds that .S ⊂ Vi ⊂ Ui . Obviously, i∈J

{Ui : i ∈ J} is a countable sub-collection of .U that covers S.

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i∈J

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1.2 Compactness for Metric Spaces The Heine–Borel theorem states that each subset of Euclidean space .R is compact if and only if it is closed and bounded. In general topology, compactness is a concept that generalizes the concept of closed and bounded subsets of the Euclidean space .R to topological spaces. A sequence .{xi }i∈N in a topological space X is said to converge to an element x of X if for every open subset U of X containing x, there exists a positive integer N such that .xi ∈ U for all integers .i ≥ N . Definition 1.13. (i) A topological space X is called compact if every open covering of X contains a finite sub-collection that covers X.

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(ii) A topological space X is called sequentially compact if every sequence in X has a subsequence that converges to an element of X.

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(iii) A topological space X is said to have the Bolzano–Weierstrass property if every infinite subset of X has a limit point in X.

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Now we will prove in the following series of theorems that the properties .(i), .(ii), and .(iii) are equivalent to one another for metric spaces. The symbol .{xi }i∈N is used to denote a sequence whose terms are .xi ’s, while the symbol .{xi : i ∈ N} is used to denote a set whose elements are .xi ’s. The use of these symbols will not cause any misunderstanding. Theorem 1.14. A metric space is sequentially compact if and only if it has the Bolzano–Weierstrass property. Proof. Let X be a metric space that is sequentially compact. We assert that each infinite subset A of X has a limit point in X. Since A is infinite, we can choose a sequence .{xi }i∈N of distinct elements of A. Due to the sequential compactness of X, the sequence .{xi }i∈N has a subsequence which converges to an element x of

1.2. COMPACTNESS FOR METRIC SPACES

7

X. Then x is a limit point of the set of the subsequence, which is a subset of A, i.e., x is a limit point of A. Hence, X has the Bolzano–Weierstrass property. We now assume that the metric space X has the Bolzano–Weierstrass property. Let .{xi }i∈N be an arbitrary sequence in X. If .{xi }i∈N has a term that repeats infinitely, then it has a constant subsequence, and this subsequence is obviously convergent. If no term of .{xi }i∈N repeats infinitely, then the set A of terms in this sequence is infinite. Since X has the Bolzano–Weierstrass property, the set A has a limit point x and it is not difficult to choose a subsequence from .{xi }i∈N that .2 converges to x. Theorem 1.15. Every compact metric space has the Bolzano–Weierstrass property. Proof. Let A be an arbitrary infinite subset of a compact metric space X. We assume that A has no limit point. Then every element of X is not a limit point of A, so every element of X is the center of an open ball that contains no element of A different from that center. The collection of all these open balls is an open covering of X, and by the compactness of X, there exists a finite sub-collection that covers X. Since A is contained in the set of all centers of open balls in this sub-collection, A is obviously finite. This contradicts the assumption that A is infinite. Therefore, we may conclude that A has a limit point, i.e., X has the Bolzano–Weierstrass property. .2 Definition 1.16. A metric space .(X, d) is called totally bounded if it has the following property: For any .ε > 0, there exists a finite set .Fε contained in X such that  .X = Bε (x), x∈Fε

where .Bε (x) denotes the open ball with center x and radius .ε. In this case, .Fε is called an .ε-net. Theorem 1.17. Every sequentially compact metric space is totally bounded. Proof. Let X be a sequentially compact metric space. Given .ε > 0, choose an element .x1 from X and construct the open ball .Bε (x1 ). If this open ball contains all elements of X, then the single-element set .{x1 } is an .ε-net. If there are elements outside of .Bε (x1 ), then let .x2 be one of those elements outside of .Bε (x1 ) and form the set .Bε (x1 ) ∪ Bε (x2 ). If this union contains all elements of X, then the twoelement set .{x1 , x2 } is an .ε-net. If we could continue like this infinitely, then the sequence .{x1 , x2 , . . .} would be a sequence without a convergent subsequence, contrary to the assumed sequential compactness of X. Therefore, a union of the form .Bε (x1 ) ∪ Bε (x2 ) ∪ · · · ∪

8

CHAPTER 1. TOPOLOGY

Bε (xn ) necessarily contains all elements of X. Then the finite set .{x1 , x2 , . . . , xn } is an .ε-net, so X is totally bounded. .2 Let .(X, d) be a metric space. A subset B of X is said to be bounded if there exists a positive number M such that .d(x, y) ≤ M for all .x, y ∈ B. The diameter of B is defined as ⎧ (for B = ∅), ⎨ 0 sup{d(x, y) : x, y ∈ B} (for a nonempty bounded set B), .d(B) = ⎩ ∞ (for an unbounded set B). Let .{Uλ : λ ∈ Λ} be an open covering of a metric space X. A positive real number a is called a Lebesgue number for the open covering .{Uλ : λ ∈ Λ} if every subset of X whose diameter is smaller than a is contained in at least one .Uλ . The following lemma is known as Lebesgue’s covering lemma. Lemma 1.18 (Lebesgue’s Covering Lemma). Let X be a sequentially compact metric space. Then every open covering of X has a Lebesgue number. Proof. Let X be a sequentially compact metric space and let .{Uλ : λ ∈ Λ} be an open covering of X. A subset of X is said to be big if it is not contained in any .Uλ . When there are no big sets, any positive real number serves as our Lebesgue number a. We may thus assume that there are big sets, and we define .a as the greatest lower bound of their diameters. It is obvious that .0 ≤ a ≤ ∞. We will prove that .a > 0. If .a = ∞, then any positive real number can be a, and if .a is a positive real number, then we can take .a = 12 a . Therefore, we assume that .a = 0 and derive a contradiction from this assumption. Since every big set has at least two elements, we conclude from .a = 0 that for every positive integer n there is a big set .Bn such that .0 < d(Bn ) < n1 , where .d(Bn ) denotes the diameter of .Bn . We now choose an element .xn of each .Bn . Since X is sequentially compact, the sequence .{xn }n∈N has a subsequence which converges to some element x of X. Then the element x belongs to at least one set .Uλ0 in the open covering, and since .Uλ0 is open, x is the center of some open ball .Br (x) contained in .Uλ0 . Let 1 .B(1/2)r (x) be the concentric open ball with radius . r. Since the subsequence of 2 .{xn }n∈N converges to x, there are infinitely many positive integers n for which .xn belongs to .B(1/2)r (x). Let .n0 be one of these positive integers, which is so large that . n10 < 12 r. Since .d(Bn0 ) < n10 < 12 r, we see that .Bn0 ⊂ Br (x) ⊂ Uλ0 . This contradicts the fact that .Bn0 is a big set, and we complete the proof. .2 Finally we will prove that sequential compactness implies compactness.

1.3. COMPACT TOPOLOGICAL SPACES

9

Theorem 1.19. Every sequentially compact metric space is compact. Proof. Let X be a sequentially compact metric space and let .{Uλ : λ ∈ Λ} be an open covering of X. According to Lemma 1.18, this open covering has a Lebesgue number .a > 0. We put .ε = 31 a and use Theorem 1.17 to find an .ε-net  .Fε = x 1 , x 2 , . . . , xn . For any .k ∈ {1, 2, . . . , n}, we have .d(Bε (xk )) ≤ 2ε = 23 a < a. By the definition of Lebesgue number, for every k, we may select a .Uλk such that .Bε (xk ) ⊂ Uλk . Since every element of X belongs to one of the .Bε (xk )’s, the collection .{Uλ1 , Uλ2 , . . . , Uλn } is a finite sub-covering of X. Therefore, X is compact. .2 In this way we proved that compactness, sequential compactness, and the Bolzano–Weierstrass property are equivalent for any metric space.

1.3 Compact Topological Spaces Let X be a topological space. We recall that a collection .{Uλ : λ ∈ Λ} of open subsets of X is called an open covering of X if every element of X belongs to at least one .Uλ . A sub-collection of an open covering that is itself an open covering is called an open sub-covering. Theorem 1.20. Let C be a compact subset of a topological space X. If C is included in a subset K of X, then C is a compact subset of K. Proof. Assume that .V is an arbitrary covering of C and each member of .V is open in K, where K is equipped with the relative topology. Then there exists an open subset U of X such that .V = U ∩ K for each .V ∈ V. Let .U be the collection of these open sets U ’s. Then .U is an open covering of C. Since C is a compact subset of X, there exists a finite sub-collection .{U1 , U2 , . . . , Un } that covers C. Thus, .{V1 , V2 , . . . , Vn } is a finite sub-collection of .V that obviously covers C, where .Vi = Ui ∩ K for all .i ∈ {1, 2, . . . , n}. Therefore, C is a compact subset of K. .2 Indeed, the reverse assertion of Theorem 1.20 also holds. More precisely, if C is a compact subset of K, where K is a subset of X equipped with the relative topology, then C is a compact subset of X. We note that each closed subset of a compact metric space is compact. In the following theorem, we will prove that this is generally true in all topological spaces. Theorem 1.21. Every closed subset K of any compact topological space X is compact.

CHAPTER 1. TOPOLOGY

10

Proof. Let .U = {Uλ : λ ∈ Λ} be an arbitrary open covering of K. Then .V := U ∪ {X \ K} is an open covering of X. Since X is compact, .V contains a finite sub-collection .{Uλ1 , Uλ2 , . . . , Uλn } ∪ {X \ K} that covers X, where each .λi ∈ Λ. Hence, we see that .K ⊂ Uλ1 ∪ Uλ2 ∪ · · · ∪ Uλn , which implies that .U contains a .2 finite sub-collection that covers K. Therefore, K is compact. The intuitive idea behind the notion of continuity of functions is to maintain the proximity of points. Expressing this idea as sentences in general topology, we get the following definition. Definition 1.22. Let X and Y be topological spaces and let .f : X → Y be a function. (i) The function f is called continuous at .x0 ∈ X if for each open subset V of Y with .f (x0 ) ∈ V , there exists an open subset U of X such that .x0 ∈ U and .f (U ) ⊂ V .

.

(ii) If f is continuous at each element of X, then f is said to be a continuous function.

.

(iii) The function f is a continuous function if and only if the pre-image of each open subset of Y is an open subset of X.

.

The equivalence of Definition 1.22 .(ii) and .(iii) is well known and is left as an exercise for the reader. The following theorem states that the images of compact sets under every continuous function are also compact. Theorem 1.23. Let X and Y be topological spaces. If .f : X → Y is a continuous function and K is a compact subset of X, then .f (K) is a compact subset of Y . Proof. Let .V = {Vλ : λ ∈ Λ} be an arbitrary open covering of .f (K), where each Vλ is an open subset of Y . Since       −1 f (K) ⊂ f −1 .K ⊂ f Vλ = f −1 (Vλ ),

.

λ∈Λ

λ∈Λ

the collection .U = {f −1 (Vλ ) : λ ∈ Λ} is an open covering of the compact set K. Hence, the compactness of K implies that the open covering .U of K contains a finite sub-covering .{f −1 (Vλ1 ), f −1 (Vλ2 ), . . . , f −1 (Vλn )} of K, where each .λi ∈ Λ, i.e., K⊂

n 

.

i=1

f −1 (Vλi ),

1.3. COMPACT TOPOLOGICAL SPACES

11

from which it follows that  n  n n      −1 .f (K) ⊂ f f (Vλi ) = f f −1 (Vλi ) ⊂ Vλ i . i=1

i=1

i=1

Therefore, each open covering .V of .f (K) contains a finite open sub-covering {Vλ1 , Vλ2 , . . . , Vλn }. That is, .f (K) is a compact subset of Y . .2

.

Proving that a topological space is compact by referring directly to the definition is sometimes difficult. The following theorem gives an equivalent form of compactness that is often easier to apply. Theorem 1.24. A topological space is compact if and only if every collection of closed sets with empty intersection has a finite sub-collection with empty intersection. Proof. We assume that X is a compact topological space and .{Kλ : λ ∈ Λ} is an arbitrary collection of closed sets with empty intersection. Then, we have

. Kλ = ∅. λ∈Λ

Thus, we get X=X\



.



Kλ =

λ∈Λ

(X \ Kλ ) =

λ∈Λ



Uλ ,

λ∈Λ

where we set .Uλ = X \ Kλ for all .λ ∈ Λ. Hence, .{Uλ : λ ∈ Λ} is an open covering of X. Then, by Definition 1.13 .(i), the open covering .{Uλ : λ ∈ Λ} of X contains a finite sub-covering .{Uλ1 , Uλ2 , . . . , Uλn }, i.e., X⊂

n 

.

U λi .

i=1

Therefore, n

.

i=1

Kλi =

n

i=1

(X \ Uλi ) = X \

n 

Uλi = ∅,

i=1

i.e., the collection .{Kλ : λ ∈ Λ} has a finite sub-collection .{Kλ1 , Kλ2 , . . . , Kλn } with empty intersection. Conversely, we assume that X is a topological space and every collection of closed sets with empty intersection has a finite sub-collection with empty intersection. Let .{Uλ : λ ∈ Λ} be an arbitrary open covering of X. If we define

CHAPTER 1. TOPOLOGY

12

Kλ = X \ Uλ for all .λ ∈ Λ, then .{Kλ : λ ∈ Λ} is a collection of closed sets with empty intersection:

.

.

Kλ =

λ∈Λ



(X \ Uλ ) = X \

λ∈Λ



Uλ = ∅.

λ∈Λ

Thus, by our assumption, the collection .{Kλ : λ ∈ Λ} has a finite sub-collection {Kλ1 , Kλ2 , . . . , Kλn } with empty intersection. Hence, we have

.

X=X\

n

.

i=1

Kλi =

n 

n 

(X \ Kλi ) =

i=1

U λi .

i=1

That is, .{Uλ1 , Uλ2 , . . . , Uλn } is a finite sub-covering of X. Therefore, X is compact. .2

The statement “every collection of closed sets with empty intersection has a to the statement “for finite sub-collection with empty intersection” is equivalent  every collection .{Kλ : λ ∈ Λ} of closed sets, if . Kλ = ∅ then there exists a λ∈Λ

finite sub-collection .{Kλ1 , Kλ2 , . . . , Kλn } such that .

n 

i=1

Kλi = ∅,” which is again

equivalent to the statement “for every collection .{Kλ : λ ∈ Λ} of closed sets, if n   . Kλi = ∅ for any finite sub-collection .{Kλ1 , Kλ2 , . . . , Kλn } then . Kλ = ∅.” i=1

λ∈Λ

Indeed, a collection of subsets of a nonempty set is said to have the finite intersection property if every finite sub-collection has nonempty intersection. Hence, the statement “every collection of closed sets with empty intersection has a finite sub-collection with empty intersection” is equivalent to the statement “every collection of closed sets with the finite intersection property has nonempty intersection.” This concept allows us to express Theorem 1.24 as follows. Theorem 1.25. A topological space is compact if and only if every collection of closed sets with the finite intersection property has nonempty intersection. Definition 1.26. Let X be a topological space. (i) A collection of open subsets of X is called an open base for X if every open set is a union of some members of this collection. A collection of closed subsets of X is called a closed base for X if the collection of all complements of its members is an open base.

.

1.3. COMPACT TOPOLOGICAL SPACES

13

(ii) A collection of open subsets of X is called an open subbase for X if its finite intersections form an open base. A collection of closed subsets of X is called a closed subbase for X if the collection of all complements of its members is an open subbase.

.

By Theorem 1.11, every separable metric space has a countable open base. An open covering of a topological space, whose members are all in some given open base, is called a basic open covering. An open covering whose members are all in some open subbase is called a subbasic open covering. Theorem 1.27. Let X be a topological space. (i) X is compact if and only if every basic open covering has a finite subcovering.

.

(ii) X is compact if and only if every collection of basic closed sets with the finite intersection property has nonempty intersection.

.

Proof. .(i) We assume that every basic open covering of X has a finite sub-covering. Let .{Uλ : λ ∈ Λ} be an arbitrary open covering of X and let .{Vγ : γ ∈ Γ} be an open base for X. For each .λ ∈ Λ, .Uλ is the union of certain .Vγ ’s, and the collection of all these .Vγ ’s is a basic open covering of X. By our assumption, this collection of .Vγ ’s has a finite sub-covering. For any member of this finite sub-covering, we can choose a .Uλ that contains it. The resulting collection of .Uλ ’s is obviously a finite open sub-covering of X. Therefore, we conclude that X is compact. Now we assume that the topological space X is compact. Since each basic open covering is also an open covering of the compact topological space X, it has a finite sub-covering. .(ii) If X is compact, then it follows from Theorem 1.25 that every collection of closed sets with the finite intersection property has nonempty intersection. Let .{Bλ : λ ∈ Λ} be an arbitrary collection of basic closed sets with the finite intersection property. Since each basic closed set .Bλ is a closed set, according to the last argument, the collection .{Bλ : λ ∈ Λ} has nonempty intersection. Now we assume that every collection of basic closed sets with the finite intersection property has nonempty intersection. Let .{Bλ : λ ∈ Λ} be an arbitrary collection of basic closed sets with the finite intersection property. Then the collection .{Bλ : λ ∈ Λ} has nonempty intersection, i.e., . Bλ = ∅. Assume that λ∈Λ

{Kγ : γ ∈ Γ} is an arbitrary collection of closed sets with the finite intersection property. Then .{X \ Bλ : λ ∈ Λ} is an open base for X and .{X \ Kγ : γ ∈ Γ} is a collection of open subsets of X. For every .γ ∈ Γ, there exists a nonempty index

.

CHAPTER 1. TOPOLOGY

14 subset .Λγ of .Λ such that X \ Kγ =



.

X \ Bλ .

λ∈Λγ

Thus, we have  .

X \ Kγ ⊂

γ∈Γ



X \ Bλ ,

λ∈Λ

which implies that ∅ =



.

λ∈Λ

Bλ ⊂



Kγ .

γ∈Γ

Therefore, by Theorem 1.25, X is compact.

2

.

As the following theorem shows, we can partially extend the previous theorem to the theorem related to open subbase. Theorem 1.28. A topological space is compact if every subbasic open covering has a finite sub-covering, or equivalently, if every collection of subbasic closed sets with the finite intersection property has nonempty intersection. Proof. From Theorems 1.24 and 1.25, it is easy to see that the two sufficient conditions mentioned in this theorem for compactness are equivalent. Consider a closed subbase for the topological space, and assume that .{Bi }i is its generated closed base, i.e., the collection of all finite unions of its members. We assume that every collection of subbasic closed sets with the finite intersection property has nonempty intersection, and we prove from this assumption that every collection of .Bi ’s with the finite intersection property also has nonempty intersection. Due to Theorem 1.27, this is sufficient to prove our theorem. Let .{B j }j be a collection of .Bi ’s with the finite intersection property. We must show that . Bj = ∅. We use Zorn’s lemma to show that .{Bj }j is contained in j

some collection .{Bk }k of .Bi ’s which is maximal with respect to having the finite intersection property, in the sense that .{Bk }k has this property and any collection of .Bi ’s which properly contains .{Bk }k fails to have this property. The argument runs as follows. Consider the family of all collections of .Bi ’s which contain .{Bj }j and have the finite intersection property. This is a partially ordered set with respect to collection inclusion. If we consider a chain in this partially ordered set, the union of all collections in it is a collection of .Bi ’s which contains every member of the chain and has the finite intersection property, as we see from the fact that

1.3. COMPACT TOPOLOGICAL SPACES

15

every finite collection of its sets is contained in some member of the chain and that the member has the finite intersection property. We conclude that every chain in our partially ordered set has an upper bound, so Zorn’s lemma guarantees that the partially ordered set has a maximal element. This argument  yields theexistence of a collection .{Bk }k with the properties stated above. Since . Bk ⊂ Bj , it now j k  suffices to show that . Bk = ∅. k

Each .Bk is a finite union of members of our closed subbase, for example, .B1 = S1 ∪ S2 ∪ · · · ∪ Sn . It now suffices to show that at least one of the members .S1 , S2 , . . . , Sn belongs to the collection .{Bk }k . For if we obtain such a set for each .Bk , the resulting collection of subbasic closed sets will have the finite intersection property (since it is contained in .{Bk }k ), and therefore, by our hypothesis relating to the subbasic closed sets, it will  have nonempty intersection;  since this nonempty intersection will be a subset of . Bk , we shall know that . Bk is itself nonempty. k

k

We finish the proof by showing that at least one of the members .S1 , S2 , . . . , Sn does in fact belong to the collection .{Bk }k . We assume that each of these sets is not in this collection, and we deduce a contradiction from this assumption. Since .S1 is a subbasic closed set, it is also a basic closed set, and since it is not in the collection .{Bk }k , the collection .{Bk }k ∪ {S1 } is a collection of .Bi ’s which properly contains .{Bk }k . By the maximality property of .{Bk }k , the collection .{Bk }k ∪ {S1 } lacks the finite intersection property, so .S1 is disjoint from the intersection of some finite collection of .Bk ’s. If we do this for each of the sets .S1 , S2 , . . . , Sn , we see that .B1 —the union of these sets—is disjoint from the intersection of the total finite collection of all the .Bk ’s which arise in this way. This contradicts the finite intersection property for the collection .{Bk }k and completes the proof. .2

As we can guess from the complexity of the proof, the scope of this theorem is very wide. We will show the practicability of this theorem by applying the theorem to provide a simple proof of the classical Heine–Borel theorem. Theorem 1.29 (Heine–Borel). Every subset of the Euclidean space .R is compact if and only if it is closed and bounded. Proof. It is well known that if a subset of .R is compact, then it is closed and bounded. Therefore, we will prove that a subset of .R, which is closed and bounded, is compact. We note that a closed and bounded subset of .R is a closed subset of some closed interval .[a, b]. Thus, we will prove that .[a, b] is compact under the assumption that .a < b, without loss of generality. We note that the collection of all intervals of the form .[a, d) and .(c, b], where c and d are any real numbers such that

CHAPTER 1. TOPOLOGY

16

a < c < b and .a < d < b, is an open subbase for .[a, b]. Therefore, the collection of all .[a, c]’s and all .[d, b]’s is a closed subbase for .[a, b]. Let .S = {[a, ci ], [dj , b] : i, j ∈ N} be a collection of these subbasic closed sets with the finite intersection property. In view of Theorem 1.28, we will prove that the intersection of all members of .S is nonempty. Without loss of generality, we  ∅. If .S contains only intervals of the form .[a, ci ] or only intervals assume that .S = of the form .[dj , b], then the intersection clearly contains a or b. Hence, we assume that .S contains intervals of both forms. Now we set .d = sup{dj : j ∈ N} and we complete the proof by showing that .d ≤ ci for all i. If we assume that .ci0 < d for some .i0 , it then follows from the definition of d that there exists a .dj0 such that .ci0 < dj0 . Since .[a, ci0 ] ∩ [dj0 , b] = ∅, it contradicts the finite intersection property .2 for .S and completes the proof. .

In general, the Heine–Borel theorem holds in the Euclidean space .Rn . Theorem 1.30. Let n be a positive integer. Every subset of the Euclidean space Rn is compact if and only if it is closed and bounded.

.

The Heine–Borel theorem holds for all finite-dimensional normed spaces, but in general this is not the case. It is not difficult to find metric spaces or topological spaces for which the Heine–Borel theorem does not hold. In many metric spaces, such as incomplete metric spaces, the Heine–Borel theorem does not hold. Even in complete metric spaces, the Heine–Borel theorem may not hold. For example, the Heine–Borel theorem does not hold for any infinite-dimensional Banach space. Even more surprising is that the Heine–Borel theorem may not hold for the real line unless given the usual metric.

1.4 Product of Topological Spaces One of the ways to create a new topological space from existing topological spaces is to multiply the given spaces. Assume that .{X1 , X2 , . . . , Xn } is a collection of sets, where n is a positive n  integer. The Cartesian product . Xi is defined as the set of all finite sequences i=1

(x1 , x2 , . . . , xn ), where .xi ∈ Xi for all .i ∈ {1, 2, . . . , n}. We note that each n  .x ∈ Xi is a function defined on .{1, 2, . . . , n}. We can generalize the concept .

i=1

of Cartesian products to any indexed collections of sets as follows: Definition 1.31. Assume that .{Xλ : λ ∈ Λ} is an indexed collection of sets. The Cartesian product . Xλ of the collection .{Xλ : λ ∈ Λ} is defined as the λ∈Λ

1.4. PRODUCT OF TOPOLOGICAL SPACES

17

collection of all functions x defined on .Λ such that .x(λ)  ∈ Xλ . For any .λ ∈ Λ, Xλ is called the .λ-coordinate space, and for any .x ∈ Xλ , .x(λ), also written as

.

λ∈Λ

xλ , is called the .λ-coordinate of x.

.

 We note that if .Λ = ∅ and each .λ-coordinate space .Xλ is nonempty, then . Xλ is nonempty by the axiom of choice. λ∈Λ



Xλ be the Cartesian product of the collection .{Xλ : λ ∈  Λ}. For any .μ ∈ Λ, let .πμ : Xλ → Xμ be defined by .πμ (x) = xμ , where .xμ Definition 1.32. Let .

λ∈Λ

λ∈Λ

denotes the .μ-coordinate of x. The function .πμ is called the projection function of  . Xλ into the .μ-coordinate space .Xμ . λ∈Λ

Assume that .{(Xλ , Tλ ) : λ ∈ Λ} is an indexed collection of topological  spaces. In the following definition, we define a topology for the Cartesian product . Xλ . λ∈Λ

Definition 1.33. Assume that .{(Xλ , Tλ ) : λ ∈  Λ} is a collection of topological Xλ by spaces. We define the collection .S of subsets of . λ∈Λ

 S = πλ−1 (Uλ ) : λ ∈ Λ and Uλ ∈ Tλ . 

.

The topology .T for .X =



Xλ that has .S as a subbase is called the product

λ∈Λ

topology and the topological space .(X, T ) is called the product space. A base .B for the product topology given in Definition 1.33 is the collection of all sets of the form

. πλ−1 (Uλ ), λ∈Δ

where .Δ is a finite subset of the indexing set .Λ, and for each .λ ∈ Δ, .Uλ is open in Xλ . Moreover,  

Uλ (for λ ∈ Δ), −1 . πλ (Uλ ) = Vλ , where Vλ = Xλ (for λ ∈ Λ \ Δ).

.

λ∈Δ

λ∈Λ

We note that for each nonempty open subset U of . μ ∈ Λ \ Δ.

.



λ∈Λ

Xλ , .πμ (U ) = Xμ for all

CHAPTER 1. TOPOLOGY

18 Theorem 1.34. Let .



Xλ be a nonempty product space. For each .μ ∈ Λ, the

λ∈Λ

projection function πμ :



.

X λ → Xμ

λ∈Λ

is an open continuous surjection. Proof. By Definition 1.32, it is obvious that the projection function .πμ is a surjection. We know that if .Uμ is an open subset of .Xμ , then .πμ−1 (Uμ ) is a member of the defining open subbase for the product topology. It follows from this fact and Definition 1.26 that .πμ is continuous. Assume that .Δ is a finite subset of .Λ and .Uλ isan open subset of .Xλ for all .λ ∈ Δ. Then, by the remark above, the set .V = πλ−1 (Uλ ) is a member of a λ∈Δ

base .B for the product topology. Then .πμ (V ) = Uμ for .μ ∈ Δ or .πμ (V ) = Xμ for .μ ∈ Λ \ Δ. In either case, .πμ (V ) is open in .Xμ . Hence, .πμ is an open function. .2 We note that every coordinate space inherits many properties from the product space: every property that is invariant under continuous open surjections is inherited by every coordinate space from the product space. A topological space X is called a Hausdorff space or a .T2 -space if for any distinct elements .x, y ∈ X, there exist disjoint open subsets U and V of X such that .x ∈ U and .y ∈ V .   Theorem 1.35. Let . Xλ be a nonempty product space. Then . Xλ is a Hausλ∈Λ

λ∈Λ

dorff space if and only if each coordinate space .Xλ is a Hausdorff space.  Proof. If the product space . Xλ is a Hausdorff space, then every coordinate λ∈Λ

space .Xλ is a Hausdorff space according to Theorem 1.34 and the above remark. We now assume that the coordinate space .Xλ is a Hausdorff space  for all .λ ∈ Xλ . Then Λ. Assume that x and y are distinct elements of the product space . λ∈Λ

xλ0 = yλ0 for some .λ0 ∈ Λ. Since .Xλ0 is a Hausdorff space, there exists a pair of disjoint open subsets .Uλ0 and .Vλ0 of .Xλ0 such that .xλ0 ∈ Uλ0 and .yλ0 ∈ Vλ0 . Then .πλ−1 (Uλ0 ) and .πλ−1 (Vλ0 ) are disjoint open subsets such that .x ∈ πλ−1 (Uλ0 ) 0 0 0 −1 and .y ∈ πλ0 (Vλ0 ), which implies that the product space is a Hausdorff space. .2

.

Tychonoff’s theorem, often considered one of the most important results of general topology, states that the product of any collection of compact topological spaces is compact with respect to the product topology. Now we have everything we need to prove Tychonoff’s theorem.

1.5. COMPLETENESS

19

Theorem 1.36 (Tychonoff). The product space of any nonempty collection of compact topological spaces is compact with respect to the product topology. Proof. Assume that .{Xλ : λ ∈ Λ} is a nonempty collection of compact topological spaces and .X = Xλ is its product space. Let .{Fμ }μ be a nonempty collection λ∈Λ

of the defining closed subbase for the product topology for X. Then each .Fμ is of  the form .Fμ = Fλμ , where .Fμμ is a closed subset of .Xμ but .Fλμ = Xλ for all λ

λ=  μ. property, and We assume that the collection .{Fμ }μ has the finite intersection  by Theorem 1.28 we complete the proof by verifying that . Fμ = ∅.

.

μ

For a fixed .λ ∈ Λ, .{Fλμ }μ is a collection of closed subsets of .Xλ with the according to Thefinite intersection property. Since .Xλ is assumed to be compact,  orem 1.25, there exists an .xλ ∈ Xλ which belongs to . Fλμ . By applying this μ  .2 process to each .λ, we obtain an element .x = (xλ )λ∈Λ of . Fμ . μ

Indeed, Tychonoff’s theorem is more widely known in its generalized form:  Xλ be the nonempty product space of a nonempty Theorem 1.37. Let .X = λ∈Λ

collection .{Xλ : λ ∈ Λ} of topological spaces. Then X is compact if and only if each coordinate space .Xλ is compact.

1.5 Completeness From an intuitive point of view, a space is complete when there are no “missing points.” For example, the set of all rational numbers is not complete because, e.g., √ . 2 is “missing” in it, although we can construct a Cauchy sequence of rational numbers that converges to it. A metric space X is called complete if every Cauchy sequence in X has a limit that is also in X. Definition 1.38. Assume that .{ci }i∈N is a sequence in a metric space .(X, d). Then {ci }i∈N is said to be a Cauchy sequence in .(X, d) if .{ci }i∈N satisfies the condition: for every .ε > 0, there exists an integer .Nε > 0 such that if .m ≥ Nε and .n ≥ Nε , then .d(cm , cn ) < ε.

.

According to Cauchy criterion for convergence in .Rn , every sequence in .Rn converges if and only if it is a Cauchy sequence. Hence, we know that the collection

20

CHAPTER 1. TOPOLOGY

of Cauchy sequences in .Rn and the collection of convergent sequences in .Rn are the same. For all metric spaces, every convergent sequence is a Cauchy sequence. However, it is not true that all Cauchy sequences in every metric space necessarily converge. Since it is an important property that all Cauchy sequences converge, it is necessary to classify metric spaces with this property separately. Definition 1.39. A metric space .(X, d) is said to be complete if every Cauchy sequence in .(X, d) converges. The metric space .R of real numbers with the metric given by the absolute value is complete, as is the n-dimensional Euclidean space .Rn with the Euclidean distance. On the √ other hand, the metric space .Q of rational numbers is not complete because, e.g., . 2 is missing from it, although we can construct a Cauchy sequence of rational numbers that converges to it. Theorem 1.40. A metric space is compact if and only if it is complete and totally bounded. Proof. Assume that X is complete and totally bounded metric space. In view of Theorem 1.19, we will prove that X is compact by showing that every sequence has a convergent subsequence. Since X is complete, it suffices to show that every sequence in X has a Cauchy subsequence. Consider an arbitrary sequence  .S1 = x11 , x12 , x13 , . . . in X. Since X is assumed to be totally bounded, there exists a finite collection of open balls, each of radius . 12 , whose union equals X. Thus, we note that .S1 has a subsequence  .S2 = x21 , x22 , x23 , . . . whose terms are all lie in an open ball of radius . 21 . Similarly, another application of the total boundedness of X shows that .S2 has a subsequence  .S3 = x31 , x32 , x33 , . . . whose terms all lie in an open ball of radius . 31 . In this way we continue to form consecutive subsequences and define  .S = x11 , x22 , x33 , . . . . By this construction, S is clearly a Cauchy subsequence of .S1 . Hence, X is a sequentially compact metric space. According to Theorem 1.19, X is a compact metric space.

1.6. SEPARATION PROPERTIES

21

Let X be a compact metric space. In view of Theorems 1.14, 1.15, and 1.17, X is sequentially compact and totally bounded. Let .S1 = {x11 , x12 , x13 , . . .} be an arbitrary Cauchy sequence in X. Since X is sequentially compact, .S1 has a convergent subsequence. That is, the Cauchy sequence .S1 is convergent in X, .2 which implies that X is complete. We note that a closed subspace of a complete metric space is complete. If .(X, d) is a metric space and Y is a subset of X that is not closed, then there exists a sequence .{yi }i∈N in Y that converges to a point .x0 ∈ X \ Y . Thus, .{yi }i∈N is a Cauchy sequence in Y that does not converge in .(Y, d). Hence, .(Y, d) is not complete. These remarks are summarized in the following theorem. Theorem 1.41. If K is a complete subset of a metric space X, then K is closed. Moreover, if X is a complete metric space and K is a closed subset of X, then K is complete.

1.6 Separation Properties Let .F be the collection of all topologies for a nonempty set X. We may introduce the set inclusion “.⊂” for the collection .F, which is a partial order for .F, and the pair .(F, ⊂) is a partially ordered set. Then the discrete topology for X is the maximum element and the trivial topology is the minimum element of .F. In the following definition, we define a finite sub-collection .F0 of .F for a fixed nonempty set X such that • Each member of this finite sub-collection has the separation property for an intended purpose. • This finite sub-collection .F0 is a chain for the induced set inclusion. Definition 1.42. Let .(X, T ) be a topological space. (i) X is called a .T1 -space if for each pair of distinct elements .x, y ∈ X, there are open subsets U and V of X such that .x ∈ U , .y ∈ U , .y ∈ V , and .x ∈ V .

.

(ii) X is called a .T2 -space or a Hausdorff space if for each pair of distinct elements .x, y ∈ X, there exist open subsets U and V of X such that .x ∈ U , .y ∈ V , and .U ∩ V = ∅.

.

(iii) X is called a regular space if for any closed subset K of X and any .x ∈ X \ K, there exist open subsets U and V of X such that .K ⊂ U , .x ∈ V , and .U ∩ V = ∅.

.

CHAPTER 1. TOPOLOGY

22

(iv) X is called a .T3 -space if X is a .T1 -space and a regular space.

.

(v) X is called a normal space if for each pair of disjoint closed subsets H and K of X, there exist open subsets U and V of X such that .H ⊂ U , .K ⊂ V , and .U ∩ V = ∅.

.

(vi) X is called a .T4 -space if X is a .T1 -space and a normal space.

.

Let .(X, T ) be a .T1 -space and let x be an arbitrary element of X. According to Definition 1.42 .(i), for each element y of .X \ {x}, there exists an open  subset .Vy of X such that .y ∈ Vy but .x ∈ Vy . Then, it holds that .X \ {x} = Vy ∈ T . y=x  Hence, .{x} = X \ Vy is closed in X. That is, every set consisting of single y=x

element of the .T1 -space X is closed in X. Conversely, if each set of single element of X is closed in X, then X is a .T1 -space. Let .X = {a, b, c} be a set of three elements. Then we may verify that .T = {∅, {a}, {b}, {a, b}, X} is a topology for X. We now see that .X \ {c} = {a, b} ∈ T . Thus, .{c} is closed in .(X, T ). On the other hand, the sets .{a} and .{b} are not closed in .(X, T ) because .X \ {a} and .X \ {b} do not belong to .T . This example shows that there is also a topological space that is not a .T1 -space. We assert that in each Hausdorff space, the limits of sequences are uniquely determined: let .{xi }i∈N be a sequence in a Hausdorff space X that converges to x and y for some .x, y ∈ X. Assume that .x = y. According to Definition 1.42 .(ii), there exist open subsets U and V of X such that .x ∈ U , .y ∈ V , and .U ∩ V = ∅. Since the sequence .{xi }i∈N converges to x, there exists an .i0 ∈ N such that .xi ∈ U for all .i ≥ i0 . On the other hand, it follows from the disjointness of U and V that .xi ∈ V for all .i ≥ i0 , which is contrary to the assumption that .{xi }i∈N converges to y. Therefore, we conclude that .x = y. Assume that .(X, d) is a metric space and .T (d) is the topology for X generated by d. We now assert that the topological space .(X, T (d)) is a Hausdorff space: let x and y be distinct elements of X. Then we have .δ := d(x, y) > 0. If we define the open sets .U = B(1/2)δ (x) and .V = B(1/2)δ (y), then .x ∈ U , .y ∈ V , and .U ∩ V = ∅. Therefore, we conclude that .(X, T (d)) is a Hausdorff space. Not every compact subset of a topological space needs to be closed. However, as shown in the following theorem, all compact subsets of any Hausdorff space are closed subsets. Theorem 1.43. Every compact subset of a Hausdorff space X is a closed subset of X. Proof. We assume that K is a nonempty compact subset of a Hausdorff space X, without loss of generality. Let x be an arbitrary element of .X \ K. Since X is a

1.7. OPEN AND CLOSED SUBSETS

23

Hausdorff space, for every .y ∈ K, there exist disjoint open subsets .U (y) and .V (y) of X such that .x ∈ U (y) and .y ∈ V (y). Then the collection .{V (y) : y ∈ K} is an open covering of K. Since K is compact, there exists a finite sub-covering n n   .{V (y1 ), V (y2 ), . . . , V (yn )}. If we set .U = U (yi ) and .V = V (yi ), then i=1

i=1

U and V are disjoint open subsets of X satisfying .x ∈ U and .K ⊂ V . Since .x ∈ U ⊂ X \ K, it holds that .X \ K is open in X, and hence, K is closed in X. .2 The idea that was useful in proving Theorem 1.43 will also be used to prove the following theorem. Theorem 1.44. Assume that H and K are disjoint compact subsets of a Hausdorff space X. Then there exist disjoint open subsets U and V of X such that .H ⊂ U and .K ⊂ V . Proof. Let x be a fixed element of H and let y be an arbitrary element of K. Since X is a Hausdorff space, there are open subsets .Ux (y) and .Vx (y) of X such that .x ∈ Ux (y), .y ∈ Vx (y), and .Ux (y) ∩ Vx (y) = ∅. The collection .{Vx (y) : y ∈ K} is an open covering of the compact set K. Hence, there exists a finite sub-collection .{Vx (y1 ), Vx (y2 ), . . . , Vx (ym )} that also m m   covers K. We now define .Ux = Ux (yi ) and .Vx = Vx (yi ). Then, .Ux and .Vx i=1

i=1

are open subsets of X such that .x ∈ Ux , .K ⊂ Vx , and .Ux ∩ Vx = ∅. On the other hand, the collection .{Ux : x ∈ H} is an open covering of the compact set H. Thus, there exists a finite sub-collection .{Ux1 , Ux2 , . . . , Uxn } that n n   Uxi and .V = Vxi . Then, U and V are open also covers H. We define .U = i=1

i=1

subsets of X such that .H ⊂ U , .K ⊂ V , and .U ∩ V = ∅.

2

.

1.7 Open and Closed Subsets In Definition 1.4, we defined the relative topology for a subset of a topological space. We will show in the next lemma that we can make a similar claim for closed subsets. Lemma 1.45. Assume that Y is a subset of a topological space X and W is a subset of Y . (i) W is open in Y if and only if there exists an open subset U of X such that .W = U ∩ Y .

.

CHAPTER 1. TOPOLOGY

24

(ii) W is closed in Y if and only if there exists a closed subset K of X such that .W = K ∩ Y .

.

Proof. .(i) Since this statement is only the definition of the open set in the topological subspace, there is nothing to prove. .(ii) If W is closed in Y , then .Y \ W is open in Y . Thus, there exists an open subset U of X that satisfies .Y \ W = U ∩ Y . That is, .W = Y \ (Y \ W ) = Y \ (U ∩ Y ) = Y \ U = Y ∩ (X \ U ), where .X \ U is a closed subset of X. On the other hand, if there exists a closed subset K of X that satisfies .W = K ∩ Y , then we have .Y \ W = Y \ (K ∩ Y ) = Y \ K = Y ∩ (X \ K), where .X \ K is an open subset of X. Hence, .Y \ W is open in Y , i.e., W is closed in Y . .2 We will find that Lemma 1.45 is particularly useful for proving Theorem 1.46. Theorem 1.46. Assume that Y is a subset of a topological space .(X, T ) and W is a subset of Y . (i) Assume that Y is open in X. Then W is open in Y if and only if W is open in X.

.

(ii) Assume that Y is closed in X. Then W is closed in Y if and only if W is closed in X.

.

Proof. .(i) Assume that .Y ∈ T . According to Lemma 1.45, for any .W ∈ TY , there exists a .U ∈ T such that .W = U ∩ Y , i.e., W is open in X. On the other hand, assume that .W ∈ T . Then, .W = W ∩ Y ∈ TY , i.e., W is open in Y . .(ii) Assume that Y is closed in X and W is closed in Y . Due to Lemma 1.45, there exists a closed subset K of X that satisfies .W = K ∩ Y which is clearly closed in X. On the other hand, assume that W is closed in X. Then, due to Lemma 1.45, .W = W ∩ Y is closed in Y . .2 The following three theorems are directly quoted from [12]. The next theorem deals with a necessary and sufficient condition that an open subset of a closed subset of a topological space X becomes an open subset of X. Theorem 1.47. Assume that Y is a closed subset of a topological space X and W is an open subset of Y . Then W is open in X if and only if .W ⊂ Y ◦ . Proof. Assume that W is open in X. Since .Y ◦ is the union of all open subsets of X which are contained in Y , it follows that .W ⊂ Y ◦ . Since W is open in Y , Lemma 1.45 implies that there exists an open subset U of X satisfying .W = U ∩Y . If .W ⊂ Y ◦ , then W = W ∩ Y ◦ = U ∩ Y ∩ Y ◦ = U ∩ Y ◦.

.

1.7. OPEN AND CLOSED SUBSETS

25

Since U and .Y ◦ is open in X, we conclude that W is open in X.

.

2

The following theorem provides the necessary and sufficient condition that a closed subset of the open subspace of a topological space X becomes a closed subset of X. Theorem 1.48. Assume that Y is an open subset of a topological space X and W is a closed subset of Y . Then W is closed in X if and only if .W ⊂ Y . Proof. If W is closed in X, then it is obvious that .W = W ⊂ Y . On the other hand, since W is closed in Y , by Lemma 1.45, there exists a closed subset K of X satisfying .W = K ∩ Y . If .W ⊂ Y , then W = W ∩ W = K ∩ Y ∩ W = K ∩ W.

.

Since K and .W are closed in X, we conclude that W is closed in X.

2

.

Obviously, .U ∪ V is an open set whenever both U and V are open sets. Conversely, if .U ∪ V is open, under what conditions can we expect that both U and V are open? Theorem 1.49. Assume that U and V are arbitrary subsets of a topological space X which are mutually separated in X, i.e., .U ∩ V = U ∩ V = ∅. If .U ∪ V is open, then both U and V are open. Proof. Since U and V are mutually separated, it holds that .U ⊂ X\V . Since .U ∪V and .X \V are open, their intersection (U ∪ V ) ∩ (X \V ) = (U ∩ X \V ) ∪ (V ∩ X \V ) = U ∪ ∅ = U

.

is open. In the same way, we can prove that V is open.

2

.

Chapter 2

Hilbert Spaces In this chapter, we briefly introduce basic concepts and theorems in vector space, normed space, Banach space, and Hilbert space that are essential to prove Ulam’s conjecture, the main subject of this book. For this purpose, we mainly refer to the book [3] by L. Debnath and P. Mikusi´nski, among many other literatures. In the following chapters, we will see that some properties of the inner product are essential.

2.1 Vector Spaces Let .R be the field of all real numbers and let .C be the field of all complex numbers. Definition 2.1. Let .K denote either .R or .C. A nonempty set X is called a vector space over .K if there are two operations, called addition and scalar multiplication, such that the following conditions are satisfied: (i) .x + y = y + x for all .x, y ∈ X.

.

(ii) .(x + y) + z = x + (y + z) for all .x, y, z ∈ X.

.

(iii) For any .x, y ∈ X, there exists a .z ∈ X such that .x + z = y.

.

(iv) .α(βx) = (αβ)x for all .α, β ∈ K and any .x ∈ X.

.

(v) .(α + β)x = αx + βx for all .α, β ∈ K and any .x ∈ X.

.

(vi) .α(x + y) = αx + αy for all .α ∈ K and all .x, y ∈ X.

.

(vii) .1x = x for all .x ∈ X.

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4 2

27

CHAPTER 2. HILBERT SPACES

28

Elements of .K are called scalars and elements of X are called vectors. If .K = R, then X is called a real vector space, and if .K = C, then X is called a complex vector space. A subset Y of a vector space X over .K is called a vector subspace or a subspace if .αx + βy ∈ Y for all .α, β ∈ K and .x, y ∈ Y . We note that any subspace of a vector space is a vector space itself and each vector space is a subspace of itself. A subset Y of a vector space X is called a proper subspace of X if Y is a subspace of X and .Y = X. Throughout this book, we use the notation .{xi }i∈N or .{x1 , x2 , . . .} to denote the sequence whose ith term is .xi . Let X be the set of all sequences .{xi }i∈N of real numbers. We define the addition and scalar multiplication by .

{xi }i∈N + {yi }i∈N = {xi + yi }i∈N , α{xi }i∈N = {αxi }i∈N

for all .α ∈ R and .{xi }i∈N , {yi }i∈N ∈ X. Then X is a real vector space. The space of all bounded sequences of real numbers is a proper subspace of X. The space of all convergent sequences of real numbers is a proper subspace of the space of all bounded sequences.

2.2 Basis of Vector Space Let .x1 , x2 , . . . , xn be elements of a vector space X. An element x of X is called a linear combination of vectors .x1 , x2 , . . . , xn if there exist scalars .α1 , α2 , . . . , αn such that x = α1 x 1 + α2 x 2 + · · · + αn x n .

.

For example, each element of the n-dimensional Euclidean space .Rn is a linear combination of vectors .e1 , e2 , . . . , en , where we set .ei = (0, . . . , 0, 1, 0, . . . , 0) with 1 in the ith position. Definition 2.2. Let X be a vector space. (i) A finite collection .{x1 , x2 , . . . , xn } of elements of X is called linearly independent if .α1 = α2 = · · · = αn = 0 is the unique solution of the linear equation .α1 x1 + α2 x2 + · · · + αn xn = 0.

.

(ii) An infinite collection .A of elements of X is called linearly independent if every finite sub-collection of .A is linearly independent.

.

2.3. NORMED SPACES

29

(iii) A collection of elements of X is called linearly dependent if it is not linearly independent.

.

We note that a collection .A of elements of X is linearly independent if no member x of the collection .A is a linear combination of a finite number of members of .A different from x. Definition 2.3. Let A be a subset of a vector space X over .K. We denote by .spanA the set of all finite linear combinations of vectors from A, i.e.,  spanA = α1 x1 + α2 x2 + · · · + αn xn : n ∈ N, αi ∈ K, xi ∈ A  . for all i ∈ {1, 2, . . . , n} . Then .spanA is a vector subspace of X and it is called the space spanned by A. It is easy to see that .spanA is the smallest vector subspace of X containing A. Definition 2.4. Let B be a subset of a vector space X. Then B is called a basis of X if B is linearly independent and .spanB = X. In general, a vector space X has multiple bases, but the number of vectors in each basis is the same. In other words, if a basis of X has exactly n vectors, then every other basis also has exactly n vectors. In this case n is called the dimension of X and we write .dimX = n.

2.3 Normed Spaces The concept of the norm in the vector spaces is an abstract generalization of length in the Euclidean space. That is, the norm is axiomatically defined as a real-valued function that satisfies certain conditions. Definition 2.5. Let X be a vector space over .K. A function . ·  : X → R is called a norm if it satisfies the following conditions: (i) .x = 0 if and only if .x = 0.

.

(ii) .αx = |α|x for all .x ∈ X and .α ∈ K.

.

(iii) .x + y ≤ x + y for all .x, y ∈ X. In particular, this inequality is called the triangle inequality.

.

Since .x = 12 (x +  − x) ≥ 12 x + (−x) = 12 0 = 0, it holds that .x ≥ 0 for any element x of a vector space X with the norm . · .  The function . ·  : Rn → R defined by .x = x21 + x22 + · · · + x2n , for all n n .x = (x1 , x2 , . . . , xn ) ∈ R , is a norm on .R . This norm is called the Euclidean norm.

CHAPTER 2. HILBERT SPACES

30

Definition 2.6. A vector space with a norm is called a normed space. Several different norms can be defined on a vector space. For example, if we define .x1 = |x1 | + |x2 | + · · · + |xn | for all .x = (x1 , x2 , . . . , xn ) ∈ Rn , then this function . · 1 : Rn → R is a norm on the vector space .Rn . Similarly, we can see that the function .x∞ = max{|x1 |, |x2 |, . . . , |xn |} is also a norm on the vector space .Rn . This norm . · ∞ is called the sup-norm. Therefore, to define a normed space, we have to specify both the vector space and the norm. We denote a normed space as .(X,  · ), where X is a vector space and . ·  is a norm defined on X. However, if we are certain of the norm given in the norm space .(X,  · ), we can simply write that norm space as X. Definition 2.7. Let .(X,  · ) be a normed space. A sequence .{xi }i∈N of elements of X is said to converge to an .x ∈ X if for each .ε > 0 there exists a positive integer N such that .xi − x < ε for all integers .i ≥ N . In this case, we write . lim xi = x or simply .xi → x. i→∞

We note that .xi → x in X implies .xi − x → 0 in .R. As we see in the following remark, the convergence of sequences in the normed space has the basic properties of the convergence of sequences in .R. Remark 2.8. Let .(X,  · ) be a normed space over .K and let .{xi }i∈N and .{yi }i∈N be sequences of elements of X. (i) If .{xi }i∈N converges in X, then it has a unique limit.

.

(ii) If .xi → x and .λi → λ, where .x ∈ X and .λi , λ ∈ K, then .λi xi → λx.

.

(iii) If .xi → x and .yi → y for some .x, y ∈ X, then .xi + yi → x + y.

.

As we see in Definition 2.7, every norm on a vector space X induces a convergence in X. But the reverse does not hold. For example, let .C([0, 1]) be the set of all continuous complex-valued functions defined on .[0, 1]. Then it is known that .C([0, 1]) is a complex vector space and there is no norm inducing the pointwise convergence for this space. Definition 2.9. Let .(X,  · 1 ) and .(X,  · 2 ) be normed spaces over .K. The norms  · 1 and . · 2 are called equivalent if they induce the same convergence. More precisely, the norms . · 1 and . · 2 are equivalent if

.

xi − x1 → 0 if and only if xi − x2 → 0

.

for all sequences .{xi }i∈N in X and .x ∈ X.

2.3. NORMED SPACES

31

It is well known that any two norms on a finite-dimensional vector space are equivalent. The following theorem gives another useful criterion for the equivalence of norms. The condition in the theorem is often used as a definition of the equivalence of norms. Theorem 2.10. Let . · 1 and . · 2 be norms on a vector space X. Then the norms . · 1 and . · 2 are equivalent if and only if there exist positive real numbers .α and .β such that αx1 ≤ x2 ≤ βx1

.

(2.1)

for all .x ∈ X. Proof. Assume that the norms . · 1 and . · 2 are equivalent. Then, it follows from Definition 2.9 that xi − x1 → 0 if and only if xi − x2 → 0

.

(2.2)

for all sequences .{xi }i∈N in X and .x ∈ X. On the contrary, assume that for any .α > 0 there is an .x ∈ X such that .αx1 > x2 . Then there exists an .xi ∈ X that satisfies .

1 xi 1 > xi 2 i

for every .i ∈ N. We set 1 1 yi = √ xi i xi 2

.

for each .i ∈ N. It then holds that .yi 2 = √1i → 0 as .i → ∞. On the other hand, √ it holds that .yi 1 ≥ i → ∞ as .i → ∞, which is contrary to (2.2). Therefore, we conclude that there exists an .α > 0 such that .αx1 ≤ x2 for all .x ∈ X. In a similar way, we can prove the existence of the positive number .β. By Definition 2.9, it is obvious that the condition (2.1) implies the equivalence of the norms . · 1 and . · 2 . .2 Let .(X,  · ) be a normed space. If we define a metric by .d(x, y) = x − y, then .(X, d) becomes a metric space. The convergence induced by the norm is the same as the convergence induced by this metric. In view of Definition 1.5, the metric on X generates a topology for X. Now that we can see the normed space as a topological space, we will use the norm to define basic topological concepts. Since every normed space is a metric space, we would not get confused even if we use in the following definition the same symbols that we used for the metric space.

CHAPTER 2. HILBERT SPACES

32

Definition 2.11. Let x be an element of a normed space .(X,  · ) and let r be a positive real number. (i) We denote by .Br (x) the open ball, with center x and radius r, defined by .Br (x) = {y ∈ X : y − x < r}.

.

(ii) We denote by .B r (x) the closed ball, with center x and radius r, defined by .B r (x) = {y ∈ X : y − x ≤ r}.

.

(iii) Similarly, we use the symbol .Sr (x) to denote the sphere, with center x and radius r, defined by .Sr (x) = {y ∈ X : y − x = r}.

.

Since, as already mentioned, every normed space is a metric space and thus every normed space is a topological space, Lemma 1.3 also holds literally for every normed space. Lemma 2.12. Assume that x is an element of a normed space .(X,  · ) and r is a positive real number. (i) The open ball .Br (x) is open in X.

.

(ii) The closed ball .B r (x) is closed in X.

.

(iii) The sphere .Sr (x) is closed in X.

.

We recall that every normed space is a metric space. Theorem 2.13. Each compact subset of a normed space is closed and bounded. Proof. Let K be a compact subset of a normed space .(X,  · ). Assume that {xi }i∈N is a sequence of elements of K that converges to an element x of X. Since the compactness is equivalent to the sequential compactness for any metric space (see Sect. 1.2), the sequence .{xi }i∈N contains a subsequence .{xpi }i∈N that converges to some element y of K. Moreover, the subsequence .{xpi }i∈N converges to x. That is, .x = y and .x ∈ K, which implies that K is closed. Now we assume that K is not bounded. Then there exists a sequence .{xi }i∈N of elements of K such that .xi  ≥ i for each .i ∈ N. It is obvious that the sequence .{xi }i∈N does not contain a convergent subsequence. Hence, K is not sequentially compact, i.e., K is not compact, which leads to a contradiction. Therefore, K has to be bounded. .2 .

We note that a subset of a finite-dimensional normed space is compact if and only if it is closed and bounded. However, this is not generally the case.

2.4. BANACH SPACES

33

Example 2.14. We consider the normed space .C([0, 1]). The unit closed ball .B 1 (0) is a closed and bounded subset of .C([0, 1]), but it is not compact. To show this, we consider the sequence of functions defined by .xi (t) = ti for all .i ∈ N. Then .xi ∈ B 1 (0) for all .i ∈ N. Since the convergence in .C([0, 1]) is the uniform convergence, the sequence .{xi }i∈N does not have a convergent subsequence.

2.4 Banach Spaces We note that every Cauchy sequence of elements from .Kn converges and every absolutely convergent series of elements from .Kn converges. However, not all normed spaces have the above properties. We recall that a sequence .{xi }i∈N of vectors in a normed space .(X,  · ) is called a Cauchy sequence if for each .ε > 0 there exists a positive real number .Mε such that .xi − xj  < ε for all .i, j > Mε . If a sequence .{xi }i∈N of vectors in a normed space .(X,  · ) converges to a vector .x ∈ X, i.e., .xi − x → 0 as .i → ∞, then xi − xj  ≤ xi − x + xj − x → 0

.

as .i, j → ∞. This explanation implies that every convergent sequence of vectors in a normed space is a Cauchy sequence. However, the converse does not hold in general. For example, we denote by .P([0, 1]) the normed space of polynomials on .[0, 1], where the norm is given by .P  = max |P (x)|. We define 0≤x≤1

Pi (x) = 1 + x +

.

1 2 1 x + · · · + xi 2! i!

for all .i ∈ N. Then .{Pi }i∈N is a Cauchy sequence. However, it does not converge in .P([0, 1]) since its limit is not a polynomial. Lemma 2.15. If .{xi }i∈N is a Cauchy sequence in a normed space .(X,  · ), then the sequence converges. Proof. Since .|xi  − xj | ≤ xi − xj  for all .i, j ∈ N, the sequence .{xi }i∈N is a Cauchy sequence in .R. Hence, we conclude that .{xi }i∈N converges. .2 Lemma 2.15 implies that each Cauchy sequence in a normed space is bounded. Definition 2.16. Let .(X,  · ) be a normed space. (i) X is called complete if each Cauchy sequence in X converges to an element of X.

.

CHAPTER 2. HILBERT SPACES

34

(ii) If X is a complete normed space, then it is called a Banach space.

.

The n-dimensional Euclidean space .Rn is a typical Banach space. Definition 2.17. Let .(X,  · ) be a normed space. (i) A series .

∞ 

.

i=1

xi converges in X if there exists an element x of X such that  n      . xi − x → 0 as n → ∞.   i=1

In this case, we write .

∞ 

xi = x.

i=1

(ii) When .

∞ 

.

xi  < ∞, the series is called absolutely convergent.

i=1

In general, not all series converge even if they are absolutely convergent. The following theorem shows that completeness and absolute convergence of series are equivalent. Theorem 2.18. A normed space .(X, ·) is complete if and only if every absolutely convergent series in X converges in X. Proof. Assume that .(X,  · ) is a complete normed space and .xi ∈ X for all .i ∈ N ∞ ∞   such that . xi  < ∞, i.e., the series . xi converges absolutely. Let us define i=1

i=1

sn = x1 + x2 + · · · + xn

.

for all .n ∈ N. We prove that .{sn }n∈N is a Cauchy sequence. Assume that .ε is an arbitrary positive real number and N is a positive integer that satisfies ∞  .

xn  < ε.

n=N +1

Then, by triangle inequality, we get sm − sn  = xm+1 + xm+2 + · · · + xn  ≤

∞ 

.

xk  < ε

k=m+1

for all .m, n ∈ N with .n > m > N , which implies that .{sn }n∈N is a Cauchy sequence in X.

2.5. INNER PRODUCT SPACES

35

Since X is complete, the Cauchy sequence .{sn }n∈N converges in X, which ∞  implies that the series . xi converges in X. i=1

Now we assume that every absolutely convergent series converges and .{xi }i∈N is an arbitrary Cauchy sequence in X. Then, for any positive integer k, there exists a positive integer .pk such that .xi − xj  < 21k for all integers .i, j ≥ pk . We may assume that the sequence .{pk }k∈N is strictly increasing. ∞  Since the series . (xpk+1 − xpk ) is absolutely convergent, it is convergent by k=1

our assumption. Since xpk = xp1 + (xp2 − xp1 ) + · · · + (xpk − xpk−1 ) = xp1 +

k 

.

(xpi+1 − xpi ),

i=1

the sequence .{xpk }k∈N converges to an element x of X. Hence, it holds that xi − x ≤ xi − xpi  + xpi − x → 0

.

as .i → ∞, i.e., the Cauchy sequence .{xi }i∈N converges in X, which implies that X is complete. .2 Theorem 2.19. A closed subspace of a Banach space is a Banach space. Proof. Let K be a closed subspace of a Banach space X. Then K is a normed space which is closed in X. Due to Theorem 1.41, K is complete as a closed .2 subset of a complete space X. Therefore, K is a Banach space.

2.5 Inner Product Spaces An inner product space is a vector space over .K with an operation called an inner product. The inner product of two vectors x and y in the space is a scalar, denoted with angle brackets such as . x, y. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality of vectors. For this reason, the infinite-dimensional inner product spaces are widely used in functional analysis. We use the symbol .z to denote the complex conjugation of any complex number z. Definition 2.20. Let X be a vector space over .K. A function . ·, · : X × X → K is called an inner product on X if the function satisfies the conditions (i) . x, y = y, x

.

CHAPTER 2. HILBERT SPACES

36 (ii) . αx + βy, z = α x, z + β y, z

.

(iii) . x, x ≥ 0

.

(iv) . x, x = 0 if and only if .x = 0

.

for all .x, y, z ∈ X and .α, β ∈ K. A vector space with an inner product is called an inner product space or a pre-Hilbert space. It follows from Definition 2.20 .(i) that . x, x = x, x, which implies that

x, x is a real number for any .x ∈ X. Due to Definition 2.20 .(i) and .(ii), we have

.

x, αy + βz = αy + βz, x = α x, y + β x, z

.

for all .x, y, z ∈ X and .α, β ∈ K. Example 2.21. Let .Kn be the vector space of all ordered n-tuples .(x1 , x2 , . . . , xn ), where .xi ∈ K for each .i ∈ {1, 2, . . . , n}. We define a function . ·, · : Kn × Kn → K by

x, y =

n 

.

xi y i

i=1

for all .x = (x1 , x2 , . . . , xn ) and .y = (y1 , y2 , . . . , yn ). Then . ·, · is an inner product on the vector space .Kn and .(Kn , ·, ·) is an inner product space. Every inner product . ·, · on a vector space X naturally induces an associated norm through .x = x, x for all .x ∈ X. The reader is encouraged to prove, as an exercise, that the function defined above is indeed a norm. Therefore, every inner product space is a normed space. In general, the norm on an inner product space means the function defined by 

x, x. .x = Theorem 2.22 (Cauchy–Schwarz Inequality). Let .(X, ·, ·) be an inner product space. Then | x, y| ≤ xy

.

for all .x, y ∈ X. Proof. For any .x, y ∈ X, we set .A = x2 = x, x, .B = | x, y|, and .C = y2 = y, y. We note that B is a nonnegative real number. Then we can choose

2.5. INNER PRODUCT SPACES

37

a scalar .α ∈ K such that .|α| = 1 and .B = α y, x. For all .r ∈ R, it follows from Definition 2.20 that 0 ≤ x − rαy, x − rαy = x, x − rα y, x − rα x, y + r2 y, y = x, x − rα y, x − rα y, x + r2 y, y

.

= A − Br − Br + r2 y, y = A − 2Br + Cr2 . If .C = 0, it has to be .B = 0. (Otherwise, the above inequality does not hold for large .r > 0.) If .C > 0, we take .r = B C in the above inequality and we obtain 2 .B ≤ AC, which completes the proof. .2 It is a natural thing to ask whether every normed space is an inner product space. Unfortunately, the answer is negative. In the following theorem, we propose a condition for a norm on an inner product space, which is a necessary and sufficient condition for the normed space to be an inner product space.  We recall that the norm on an inner product space is defined by .x = x, x. Theorem 2.23 (Parallelogram Law). Let .(X, ·, ·) be an inner product space. Then x + y2 + x − y2 = 2x2 + 2y2

.

(2.3)

for all .x, y ∈ X. Proof. It holds that x + y2 = x + y, x + y = x, x + x, y + y, x + y, y

.

= x2 + x, y + y, x + y2 for all .x, y ∈ X. If we replace y by .−y in the above equation, then we get x − y2 = x2 − x, y − y, x + y2

.

for all .x, y ∈ X. We obtain the parallelogram law by adding the last two equations.

2

.

Let .(X,  · ) be a normed space over .K. For any norm that satisfies the parallelogram law (2.3), the inner product that generates the norm is unique as a consequence of polarization identity. In the case of .K = R, the polarization identity is

CHAPTER 2. HILBERT SPACES

38 given by

x, y =

.

1 x + y2 − x − y2 4

(2.4)

for all .x, y ∈ X. For the case of .K = C, the polarization identity is given by

x, y =

.

i 1 x + y2 − x − y2 + ix − y2 − ix + y2 4 4

(2.5)

for all .x, y ∈ X. So if the parallelogram law is satisfied in a normed space .(X,  · ), then the normed space is an inner product space that is correspondingly equipped with the inner product (2.4) or (2.5). Another necessary and sufficient condition for the existence of an inner product that induces the given norm is that the norm satisfies the Ptolemy inequality x − yz + y − zx ≥ z − xy.

.

More precisely, the Ptolemy inequality holds in every inner product space. Conversely, if the inequality holds in a real normed space, then the space has to be a real inner product space. We will now prove that the inner product and norm are continuous functions using Cauchy–Schwarz inequality and the triangle inequality. Theorem 2.24. Let .(X, ·, ·) be an inner product space over .K. For any fixed x, y ∈ X, the functions . x, · : X → K, . ·, y : X → K, and . ·  : X → R are .(uniformly.) continuous on X. .

Proof. By the Cauchy–Schwarz inequality, we have | x, y1  − x, y2 | = | x, y1 − y2 | ≤ xy1 − y2 

.

for all .y1 , y2 ∈ X and for a fixed .x ∈ X. That is, . x, · is a uniformly continuous function. Similarly, we can prove that if y is a fixed element of X, then . ·, y is a uniformly continuous function. Using the triangle inequality, we obtain x1  − x2  ≤ x1 − x2 ,

.

and if we interchange .x1 and .x2 in the above inequality, then we see that |x1  − x2 | ≤ x1 − x2 

.

for all .x1 , x2 ∈ X. Therefore, . ·  is also a uniformly continuous function.

2

.

2.5. INNER PRODUCT SPACES

39

We recall that a subset V of a vector space X is called a subspace of X if V is itself a vector space with the addition and scalar multiplication which are defined on X. In sentences relating to vector spaces, the term “subspace” always refers to the subspace mentioned above. Sometimes we write “linear subspace” instead of subspace. Theorem 2.25. If V is a subspace of an inner product space X, so is .V . Proof. Let x and y be arbitrary elements of .V and let .α be an arbitrary scalar. Then there exist sequences .{xi }i∈N and .{yi }i∈N in V which converge to x and y, respectively. Since V is a vector space, .{xi + yi }i∈N and .{αxi }i∈N are sequences in V which converge to .x + y and .αx, respectively. Therefore, .x + y ∈ V and .αx ∈ V , which implies that .V is a subspace of X. .2 One of the most important consequences of the inner product is the ability to define the orthogonality of vectors. This makes the theory of Hilbert spaces very different from the general theory of Banach spaces. Definition 2.26. Let X be an inner product space. Two vectors .x, y ∈ X are called orthogonal if . x, y = 0. In the case, we write .x ⊥ y. We note that if .x ⊥ y, then . x, y = 0, and hence, . y, x = x, y = 0 = 0, i.e., .y ⊥ x. Another example of the geometric character of the norm defined by an inner product is the Pythagorean theorem. The Pythagorean theorem is a fundamental relationship in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. By the following theorem, the Pythagorean theorem holds in every inner product space. Theorem 2.27 (Pythagorean Theorem). Let .(X, ·, ·) be an inner product space. If the norm . ·  on X is induced by the inner product . ·, ·, then the Pythagorean formula holds, i.e., x + y2 = x2 + y2

.

for any pair of orthogonal vectors in X. Proof. If x and y are orthogonal vectors, then . x, y = y, x = 0. Hence, we have x + y2 = x2 + x, y + y, x + y2 = x2 + y2

.

for all .x, y ∈ X.

2

.

CHAPTER 2. HILBERT SPACES

40

2.6 Hilbert Spaces A Hilbert space is a vector space endowed with an inner product that induces a distance function for which it is a complete metric space. Because Hilbert spaces allow the methods of linear algebra and calculus to be generalized from (finitedimensional) Euclidean spaces to possibly infinite-dimensional spaces, the Hilbert spaces are naturally and widely used in mathematics and physics. Definition 2.28. A Hilbert space is a complete inner product space. In the definition above, the completeness of an inner product space .(X, ·, ·) means the  completeness of the normed space .(X,  · ), where the norm is defined by .x = x, x for all .x ∈ X (see Sect. 2.4). There are many examples of Hilbert spaces. .Rn and .Cn are Hilbert spaces if n n   xi yi and . x, y = xi y i , they are equipped with the inner products . x, y = i=1

i=1

respectively. Another example of Hilbert spaces is .l2 , where .l2 is the space of all ∞  sequences .{xi }i∈N of complex numbers such that . |xi |2 < ∞ with the inner i=1

product defined by

x, y =

∞ 

.

xi y i .

i=1

We recall that two vectors in an inner product space are called orthonormal if they are orthogonal unit vectors. We say that a set of vectors forms an orthonormal set if all vectors in the set are mutually orthogonal and all have unit length. Definition 2.29. Let .(X, ·, ·) be an inner product space. (i) A collection S of nonzero vectors in X is called an orthogonal system if any two different vectors in S are orthogonal to each other.

.

(ii) If every vector in an orthogonal system S is a unit vector, i.e., .x = 1 for every .x ∈ S, then S is called an orthonormal system.

.

We note that in Definition  2.29 .(ii), the norm is induced by the inner product, i.e., it is defined by .x = x, x. Any orthogonal system of nonzero vectors can be normalized. If S is an orthogonal system, then the collection 

x  :x∈S .S = x is an orthonormal system.

2.6. HILBERT SPACES

41

Theorem 2.30. Every orthogonal system in an inner product space is linearly independent. Proof. Let S be an orthogonal system in an inner product space .(X, ·, ·) over .K. n  Assume that . αi xi = 0 for some .x1 , x2 , . . . , xn ∈ S and .α1 , α2 , . . . , αn ∈ K. i=1

Then we have 0=

n 

0, αi xi  =

i=1 .

=

n 

n 



i=1

n 

αj x j , αi x i

j=1

=

n 

αi xi , αi xi 

i=1

|αi |2 xi 2 ,

i=1

which implies that .αi = 0 for all .i ∈ {1, 2, . . . , n}. Therefore, .x1 , x2 , . . . , xn are linearly independent. .2 Definition 2.31. Any sequence of vectors in an inner product space is called an orthonormal sequence if they form an orthonormal system. For each .i ∈ N, we define .ei = (0, . . . , 0, 1, 0, . . .) with 1 in the ith position. Then the set .S = {e1 , e2 , . . .} is an orthonormal sequence in .l2 . We recall that .l2 ∞  is the space of all sequences .{xi }i∈N of complex numbers such that . |xi |2 < ∞ i=1

with the inner product defined by

x, y =

∞ 

.

xi y i .

i=1

According to Theorem 2.27, the Pythagorean theorem holds for every pair of orthogonal vectors in an inner product space. In the following theorem, we will generalize the Pythagorean theorem to a finite number of orthogonal vectors. Theorem 2.32 (Extended Pythagorean Theorem). If .(X, ·, ·) is an inner product space, then  n  n   2    . xi  = xi 2   i=1

i=1

for all orthogonal vectors .x1 , x2 , . . . , xn ∈ X.

CHAPTER 2. HILBERT SPACES

42

Proof. We will use mathematical induction. By Theorem 2.27, .x1 + x2 2 = x1 2 + x2 2 for any pair of orthogonal vectors .x1 and .x2 . Hence, our assertion is true for .n = 2. Now we assume that our assertion is true for .n − 1, i.e.,  n−1 2 n−1      . xi  = xi 2   i=1

i=1

for all orthogonal vectors .x1 , x2 , . . . , xn−1 ∈ X. Let .x1 , x2 , . . . , xn be orthogonal vectors in X. We set .x =

n−1 

xi and .y = xn .

i=1

Then .x ⊥ y. Thus we have

 n  n−1 n   2     . xi  = x + y2 = x2 + y2 = xi 2 + xn 2 = xi 2 ,   i=1

i=1

i=1

2

which proves our assertion.

.

The Bessel inequality gives us information about the coefficients of an element in an inner product space with respect to an orthonormal set. Theorem 2.33 (Bessel Inequality). Given .n ∈ N, assume that .{x1 , x2 , . . . , xn } is an orthonormal set of vectors in an inner product space .(X, ·, ·). Then  2 n n       .x −

x, xi xi  = x2 − | x, xi |2   i=1

i=1

and n  .

| x, xi |2 ≤ x2

i=1

for all .x ∈ X. Proof. By Theorem 2.32, we have  n 2 n n       . αi x i  = αi xi 2 = |αi |2   i=1

i=1

i=1

2.6. HILBERT SPACES

43

for all scalars .α1 , α2 , . . . , αn . Hence, we get  2 n      αi x i  x −   i=1

n n   = x− αi x i , x − αj x j i=1

j=1



.

= x − 2

x,

n 



−

αj x j

j=1

= x2 − = x2 −

n  j=1 n 

n 

αi x i , x

+

n  n 

i=1

αj x, xj  − | x, xi |2 +

i=1

n  i=1 n 

αi x, xi  +

αi αj xi , xj 

i=1 j=1 n 

αi α i

i=1

| x, xi  − αi |2 .

i=1

Putting .αi = x, xi  yields the Bessel equality. Finally, the Bessel inequality is an immediate consequence of the Bessel equality. .2 Remark 2.34. Let .{xi }i∈N be an orthonormal sequence of vectors in an inner product space .(X, ·, ·). (i) If we let .n → ∞ in the Bessel equality, then we get

.

∞  .

| x, xi |2 ≤ x2

i=1

for any .x ∈ X, which implies that the series .

∞ 

| x, xi |2 converges for any

i=1

x ∈ X. In other words, the sequence .{ x, xi }i∈N is an element of .l2 .

.

(ii) The expansion

.

x∼

.

∞ 

x, xi xi i=1

is called a generalized Fourier series of x. The scalar . x, xi  is called the generalized Fourier coefficient of x with respect to the orthonormal sequence .{xi }i∈N . In general, we do not know whether the generalized Fourier series of x converges. The following theorem states that the generalized Fourier series of x converges in any Hilbert space.

CHAPTER 2. HILBERT SPACES

44

Theorem 2.35. Let .{xi }i∈N be an orthonormal sequence in a Hilbert space .(X, ·, ·) ∞  and let .{αi }i∈N be a sequence of scalars. Then the series . αi xi converges if and only if

∞  .

i=1

|αi

|2

< ∞. In that case, it holds that

i=1

 ∞ 2 ∞      . αi x i  = |αi |2 .   i=1

i=1

Proof. By Theorem 2.32, it holds that  n 2 n      . αi x i  = |αi |2   i=m

for all integers m and n with .n > m > 0. We define .sn = If .

∞ 

(2.6)

i=m

n 

αi xi for each .n ∈ N.

i=1

|αi |2 < ∞, then it follows from (2.6) that .{sn }n∈N is a Cauchy sequence in

i=1

X. Since X is complete, the series . Now we define .tn =

n 

∞ 

αi xi converges.

i=1

|αi |2 for any .n ∈ N. If the series .

i=1

then it follows from (2.6) that .

∞ 

∞ 

αi xi converges,

i=1

|αi |2 converges because .{tn }n∈N is a Cauchy

i=1

sequence in .R. Putting .m = 1 and letting .n → ∞ in (2.6), we complete our proof.

2

.

By Remark 2.34 .(i) and Theorem 2.35, it holds that in any Hilbert space X the ∞  series . x, xi xi converges for each .x ∈ X. However, it should be noted that it i=1

may converge to an element different from x. Now we shall treat separately the inner product spaces in which the generalized Fourier series of x necessarily converges to x. Definition 2.36. Assume that .(X, ·, ·) is an inner product space. An orthonormal sequence .{xi }i∈N in X is called complete if x=

∞ 

.

x, xi xi

i=1

or equivalently

  n      . lim x −

x, xi xi  = 0 n→∞   i=1

2.6. HILBERT SPACES

45

for all .x ∈ X. An orthonormal set that forms a basis is called an orthonormal basis. Definition 2.37. Assume that .(X, ·, ·) is an inner product space. An orthonormal system .B in X is called an orthonormal basis if each element x of X has a unique representation x=

∞ 

.

αi x i ,

i=1

where the .αi ’s are scalars and .xi ’s are distinct elements of .B. Remark 2.38. Let .(X, ·, ·) be a Hilbert space over .K. (i) A complete orthonormal sequence .{xi }i∈N in X is an orthonormal basis in X.

.

(ii) If .{xi }i∈N is a complete orthonormal sequence in X, then the space spanned by .{xi }i∈N  n   .span(x1 , x2 , . . .) = αi xi : n ∈ N; α1 , α2 , . . . , αn ∈ K

.

i=1

is dense in X. Proof. .(i) It suffices to prove the uniqueness of the generalized Fourier series of x. If there are two generalized Fourier series of x x=

∞ 

.

αi xi and x =

i=1

∞ 

β i xi ,

i=1

then it follows from Theorem 2.35 that  2  ∞ 2 ∞ ∞ ∞           2 .0 = x − x =  αi x i − β i xi  =  (αi − βi )xi  = |αi − βi |2 ,     i=1

i=1

i=1

i=1

which implies that .αi = βi for every .i ∈ N. .(ii) We can easily prove this claim using Theorem 1.10. We encourage the .2 reader to try this proof as an exercise. Some important characterizations of complete orthonormal sequences in Hilbert spaces are given in the following theorems.

CHAPTER 2. HILBERT SPACES

46

Theorem 2.39. Let .(X, ·, ·) be a Hilbert space and let x be an arbitrary element of X. An orthonormal sequence .{xi }i∈N in X is complete if and only if . x, xi  = 0 for all .i ∈ N implies .x = 0. Proof. Assume that .{xi }i∈N is a complete orthonormal sequence in X. Then, each element x of X has the representation x=

∞ 

.

x, xi xi .

i=1

Hence, if . x, xi  = 0 for all .i ∈ N, then .x = 0. Conversely, assume that . x, xi  = 0 for all .i ∈ N implies .x = 0. For any .x ∈ X, let ∞  .y =

x, xi xi . i=1

By Remark 2.34 .(i) and Theorem 2.35, the sum y exists in X. Thus, we have

x − y, xn  = x, xn  − y, xn  ∞



x, xi xi , xn = x, xn  − .

= x, xn  −

i=1 ∞ 

x, xi  xi , xn ,

i=1

and since .{xi }i∈N is an orthonormal sequence in X, we obtain

x − y, xn  = x, xn  − x, xn  = 0

.

for all .n ∈ N. Thus, it follows from the assumption that .x − y = 0. Therefore, .x = ∞ 

x, xi xi for any .x ∈ X. According to Definition 2.36, the orthonormal y = i=1

sequence .{xi }i∈N is complete.

2

.

We remind here once again that the norm in the inner product space means the norm derived from the inner product. Theorem 2.40 (Parseval’s Formula). Let .(X, ·, ·) be a Hilbert space and let .{xi }i∈N be an orthonormal sequence in X. Then .{xi }i∈N is complete if and only if x2 =

∞ 

.

i=1

for all .x ∈ X.

| x, xi |2

2.7. ORTHOGONAL COMPLEMENTS

47

Proof. Let x be an arbitrary element of X. By Theorem 2.33, we have  2 n n       .x −

x, xi xi  = x2 − | x, xi |2   i=1

(2.7)

i=1

for any .n ∈ N. If .{xi }i∈N is a complete orthonormal sequence in X, then   n  x2 − . lim | x, xi |2 = 0, n→∞

i=1

i.e., x2 =

∞ 

.

| x, xi |2

i=1

for all .x ∈ X. Conversely, assume that the last equality holds for all .x ∈ X. Then the expression on the right of (2.7) converges to 0 as .n → ∞. Hence, it follows from (2.7) that  2 n      . lim x −

x, xi xi  = 0 n→∞   i=1

for all .x ∈ X. Finally, it follows from Definition 2.36 that .{xi }i∈N is complete. .2

2.7 Orthogonal Complements We remember that every inner product space is a normed space, so moreover it is a metric space. A subspace S of a Hilbert space X is an inner product space. If we additionally assume that S is a closed subspace of X, then S is a complete inner product space by Theorem 1.41. That is, S is itself a Hilbert space. In the following definition, we extend the concept of orthogonality defined in Definition 2.26. Definition 2.41. Let S be a nonempty subset of a Hilbert space .(X, ·, ·). An element x of X is said to be orthogonal to S if . x, y = 0 for all .y ∈ S. In this case, we write .x ⊥ S. The set of all elements of X which are orthogonal to S, denoted by .S ⊥ , is said to be the orthogonal complement of S. If an element x of a Hilbert space X satisfies the condition .x ⊥ y for all .y ∈ X, then .x = 0. Hence, .X ⊥ = {0}. Similarly, .{0}⊥ = X.

CHAPTER 2. HILBERT SPACES

48

Theorem 2.42. Let S be a subset of a Hilbert space X. Then the orthogonal complement .S ⊥ of S is a closed subspace of X. Proof. For all scalars .α, β and .x, y ∈ S ⊥ , it holds that

αx + βy, z = α x, z + β y, z = 0

.

for all .z ∈ S, i.e., .αx + βy ∈ S ⊥ for all scalars .α, β and .x, y ∈ S ⊥ . Thus, .S ⊥ is a vector subspace of X. Let .{xi }i∈N be an arbitrary sequence in .S ⊥ that converges to some element x of X. According to Theorem 2.24, . ·, y is a continuous function. Hence, we get

x, y =



.

 lim xi , y = lim xi , y = 0

i→∞

i→∞

for any .y ∈ S. That is, .x ∈ S ⊥ . This fact implies that .S ⊥ is closed.

2

.

Theorem 2.42 states that the orthogonal complement .S ⊥ is a Hilbert space for every subset S of a Hilbert space. Definition 2.43. A subset S of a vector space is called convex if .αx+(1−α)y ∈ S for all .x, y ∈ S and all real numbers .α with .0 < α < 1. We note that every vector subspace is a convex set. We now introduce a theorem about the minimization of the norm, which is of fundamental importance in approximation theory. Theorem 2.44. Let X be a Hilbert space. If S is a closed convex subset of X and x is an arbitrary element of X, then there exists a unique element .yx of S such that x − yx  = inf x − z.

.

z∈S

Proof. Assume that .{yi }i∈N is a sequence in S that satisfies .

lim x − yi  = inf x − z.

i→∞

z∈S

Since . 12 (yi + yj ) ∈ S, we have     1  (yi + yj ) . x − inf x − z  ≥ z∈S  2 for all .i, j ∈ N.

2.7. ORTHOGONAL COMPLEMENTS

49

Furthermore, it follows from Theorem 2.23 that  2 2      1 1 2 2    yi − yj  = 4x − (yi + yj ) + yi − yj  − 4x − (yi + yj )  2 2 = (x − yi ) + (x − yj )2 + (x − yi ) − (x − yj )2  2   1  (y − 4 x − + y ) i j   2 2     1 2 2  = 2 x − yi  + x − yj  − 4x − (yi + yj )  . 2

.

From the facts 2   2 2 .2 x − yi  + x − yj  → 4 inf x − z , z∈S

as .i, j → ∞, and

 2  2   1  ≤ −4 inf x − z ,  (y . − 4 x − + y ) i j   z∈S 2

it follows that .yi − yj 2 → 0 as .i, j → ∞. Hence, .{yi }i∈N is a Cauchy sequence in S. Since S is complete as a closed subset of a complete space X by Theorem 1.41, the limit . lim yi = yx exists and .yx ∈ S. Moreover, it follows from the continuity i→∞

of norm (Theorem 2.24) that     .x − yx  = x − lim yi  = lim x − yi  = inf x − z. i→∞

i→∞

z∈S

We now want to prove the uniqueness of .yx . Assume that there exists another element .yx of S that satisfies x − yx  = inf x − z.

.

z∈S

By the parallelogram law (Theorem 2.23), we have yx − yx 2 = (x − yx ) − (x − yx )2 = 2x − yx 2 + 2x − yx 2 − (x − yx ) + (x − yx )2  2 = 4 inf x − z − 2x − (yx + yx )2 .

.

z∈S

Since . 12 (yx + yx ) ∈ S, we obtain yx −

.

yx 2



= 4 inf x − z z∈S

2

2    1    − 4x − (yx + yx ) ≤ 0, 2

CHAPTER 2. HILBERT SPACES

50 which implies that .yx = yx .

2

.

The following theorem states that every Hilbert space is the direct sum of a closed subspace and its orthogonal complement. We denote by .α and .α the real part and the imaginary part of any complex number .α. Theorem 2.45. Let X be a Hilbert space over .K. If S is a closed subspace of X, then each element x of X has a unique decomposition in the form .x = y +z, where ⊥ .y ∈ S and .z ∈ S . Proof. We note that .S ⊥ is a closed subspace of X by Theorem 2.42. If .x ∈ S, then ⊥ .x = x + 0 is a unique decomposition, where .x ∈ S and .0 ∈ S . Assume now that .x ∈ S. Since S is a closed convex subset of X, it follows from Theorem 2.44 that there exists a unique element .yx of S such that x − yx  = inf x − z.

.

z∈S

Now we prove that .x = yx +(x−yx ) is the decomposition that satisfies .yx ∈ S and .x − yx ∈ S ⊥ . If .w ∈ S and .α ∈ K, then .yx + αw ∈ S and x − yx 2 ≤ x − (yx + αw)2 = x − yx 2 − 2α w, x − yx  + |α|2 w2 .

.

Hence, we have .

− 2α w, x − yx  + |α|2 w2 ≥ 0.

(2.8)

We assume that .α is a positive real number. Then we divide inequality (2.8) by α and let .α → 0 to get

.

 w, x − yx  ≤ 0.

.

(2.9)

Similarly, we replace .α in inequality (2.8) by .−iα (.α > 0) and divide the resulting inequality by .α, and we let .α → 0 to obtain  w, x − yx  ≤ 0.

.

(2.10)

Since S is a vector space, .w ∈ S implies .−w ∈ S. Hence, we can replace w with .−w in inequalities (2.9) and (2.10). Thus, it follows that . w, x − yx  =  w, x − yx  = 0, i.e., . w, x − yx  = 0 for any .w ∈ S, which implies that ⊥ .x − yx ∈ S . Finally, we prove the uniqueness of decomposition .x = y + z, where .y ∈ S and .z ∈ S ⊥ . We note that if .x = y  + z  , .y  ∈ S, and .z  ∈ S ⊥ , then .y − y  ∈ S and .z  − z ∈ S ⊥ . Since .y − y  = z  − z, it should hold that .y − y  = z  − z = 0. .2

2.8. SEPARABLE HILBERT SPACES

51

Remark 2.46. Let S be a closed subspace of a Hilbert space X. (i) Each element of X can be uniquely expressed as the sum of an element of S and an element of .S ⊥ . We express this symbolically simply as

.

X = S ⊕ S⊥

.

and we say that X is the direct sum of S and .S ⊥ . Moreover, the above equality is called an orthogonal decomposition of X. (ii) The union of a basis of S and a basis of .S ⊥ is a basis of X.

.

2.8 Separable Hilbert Spaces As we see in the following definition, a Hilbert space is said to be separable if it has a countable orthonormal basis. Definition 2.47. A Hilbert space X is called separable if there exists a complete orthonormal sequence in X. We recall that .l2 is the space of all sequences .{xi }i∈N of complex (or real) ∞  numbers such that . |xi |2 < ∞. For any .i ∈ N, we set .ei = (0, . . . , 0, 1, 0, . . .) i=1

with 1 in the ith position. Then we can easily prove that the orthonormal sequence 2 2 .{e1 , e2 , . . .} in .l is complete. Hence .l is a separable Hilbert space endowed with the inner product given by

x, y =

∞ 

.

xi y i

i=1

for all elements .x = {xi }i∈N and .y = {yi }i∈N of .l2 . In view of Definition 1.9, a metric space is said to be separable if it contains a countable dense subset. For this reason, the following theorem is often used as a definition of separability for the Hilbert space. Theorem 2.48. Each separable Hilbert space contains a countable dense subset. Proof. Let X be a separable Hilbert space. Since X is a separable Hilbert space, there exists a complete orthonormal sequence .{xi }i∈N in X. We define  S = (α1 + iβ1 )x1 + · · · + (αn + iβn )xn :  . α1 , . . . , αn , β1 , . . . , βn ∈ Q; n ∈ N ,

CHAPTER 2. HILBERT SPACES

52

where .Q denotes the set of all rational numbers. Then S is countable. We note that for all .x ∈ X,  n      .

x, xi xi − x → 0 as n → ∞.   i=1

Since .

n 

x, xi xi ∈ S, S is dense in X.

2

.

i=1

Theorem 2.49. Each orthogonal set in a separable Hilbert space is countable. Proof. Assume that S is an arbitrary orthogonal set in a separable Hilbert space X and .S  is the set of normalized vectors from S, i.e., 

1  z:z∈S . .S = z Since .S  is an orthonormal set, we have x − y2 = x − y, x − y .

= x, x − x, y − y, x + y, y =2

for all distinct .x, y√∈ S  , which implies that the distance between any two distinct elements of .S  is . 2. We now consider the collection of all open balls with radius . √12 and center at every points in .S  . It is obvious that no two of these open balls can have a common point. Since every dense subset of X has at least one element in each open ball and X has a countable dense subset, .S  has to be countable. Therefore, S is countable, .2 which completes the proof. Let .X1 and .X2 be vector spaces over .K. A mapping .L : X1 → X2 is called a linear mapping if .L(αx + βy) = αL(x) + βL(y) for all .x, y ∈ X1 and .α, β ∈ K. Definition 2.50. Let .X1 and .X2 be Hilbert spaces over the same scalar field. .X1 is called isomorphic to .X2 if there exists a one-to-one linear mapping T from .X1 onto .X2 such that

T (x), T (y) = x, y

.

for all .x, y ∈ X1 . In the case, the mapping T is called a Hilbert space isomorphism of .X1 onto .X2 .

2.8. SEPARABLE HILBERT SPACES

53

According to the following theorem, any separable infinite-dimensional Hilbert space is isometric to the space .l2 of all square-summable sequences. On the other hand, a separable n-dimensional Hilbert space is isometric to either .Cn or .Rn , depending on whether the scalar field .K is either .C or .R. Theorem 2.51. Let X be a separable Hilbert space over .K and let n be a positive integer. (i) If X is infinite-dimensional, then it is isomorphic to .l2 .

.

(ii) If X is n-dimensional, then it is isomorphic to .Kn .

.

Proof. Since X is a separable Hilbert space, there exists a complete orthonormal sequence .{xi }i∈N in X. .(i) First we consider the case where X is infinite-dimensional. Then it follows that .{xi }i∈N is an infinite sequence. We define a mapping T by  .T (x) = x, x1 , x, x2 , . . . , x, xi , . . . for all .x ∈ X. Then, due to Theorem 2.35, T is a one-to-one mapping from X onto .l2 . In addition, T is a linear mapping. Furthermore, considering the definition of inner product on .l2 and continuity of inner product, we have  

T (x), T (y) = ( x, x1 , x, x2 , . . .), ( y, x1 , y, x2 , . . .) ∞ 

x, xi  y, xi  = i=1 .

∞    x, y, xi xi = i=1

=

x,

∞ 

y, xi xi

i=1

= x, y for all .x, y ∈ X. Therefore, we conclude that T is a Hilbert space isomorphism from X onto .l2 . .(ii) In the same way, we can prove the second claim. The proof of this is left .2 to the reader. We can show that the isomorphism of Hilbert spaces is an equivalence relation. In a sense, there is only one real and one complex infinite-dimensional separable Hilbert space.

Chapter 3

Measure Theory The concept of a measure is a generalization and formalization of length, area, volume, and other common notions such as mass and probability of events. These seemingly different concepts have many similarities and can often be unified by the concept of measure. Measures are fundamental to probability theory and integration theory. In this chapter, we briefly present the basic concepts and theorems of general measure theory that are essential for the proof of Ulam’s conjecture, mainly with reference to C. A. Rogers’ book [18].

3.1 Outer Measures We remember that the power set of a set X is the collection of all subsets of X and we denote the power set of X with .P(X). In measure theory, an outer measure is a function defined on the power set of a given set with values in the extended real numbers that satisfy some additional conditions. Definition 3.1. Let .P(X) be the power set of a set X. A function .μ∗ : P(X) → [0, ∞] is called an outer measure on X if it satisfies the following conditions: (i) .μ∗ (∅) = 0.

.

(ii) If .A1 , A2 ∈ P(X) with .A1 ⊂ A2 , then .μ∗ (A1 ) ≤ μ∗ (A2 ).

.

(iii) If .{Ai }i∈N is a sequence in .P(X), then  ∞  ∞   ∗ .μ Ai ≤ μ∗ (Ai ).

.

i=1

i=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4 3

55

CHAPTER 3. MEASURE THEORY

56

For example, if .μ∗ (A) is the Lebesgue outer measure for any subset A of the n-dimensional Euclidean space .Rn , then .μ∗ is an outer measure on .Rn . Although outer measures are defined on the power set of the space, the class of all measurable sets defined below has special and useful properties. Definition 3.2. If .μ∗ is an outer measure on a set X, a subset E of X is called ∗ .μ -measurable if μ∗ (A ∪ B) = μ∗ (A) + μ∗ (B)

.

for all subsets A and B of X with .A ⊂ E and .B ⊂ X \ E. Remark 3.3. Let A, B, and E be subsets of a set X and let .μ∗ be an outer measure on X. (i) A and B are said to be separated by E if .A ⊂ E and .B ⊂ X \ E.

.

(ii) E is said to be .μ∗ -measurable if the outer measure .μ∗ is additive on sets that are separated by E.

.

Before proving the theorem that provides considerable information about the measurable sets, it is convenient to introduce the concept of .σ-algebras and to prove a simple lemma on .σ-algebras. Definition 3.4. A collection .A of subsets of a set X is said to be a .σ-algebra if it satisfies the following conditions: (i) .∅ ∈ A.

.

(ii) If .A ∈ A, then .X \ A ∈ A.

.

(iii) If .Ai ∈ A for all .i ∈ N, then .

∞ 

.

Ai ∈ A.

i=1

It follows from .(ii) and .(iii) that if .Ai ∈ A for all .i ∈ N, then  ∞  ∞   . Ai = X \ (X \ Ai ) ∈ A, i=1

i=1

which implies that every .σ-algebra is closed under the countable intersection. Now we prove a simple lemma giving some necessary and sufficient conditions specifying .σ-algebras. Lemma 3.5. If a collection .A of subsets of a set X satisfies the following conditions:

3.1. OUTER MEASURES

57

(i) .∅ ∈ A.

.

(ii) If .A ∈ A, then .X \ A ∈ A.

.

(iii) If .A1 , A2 ∈ A, then .A1 ∪ A2 ∈ A.

.

(iv) If .Ai ∈ A for all .i ∈ N and they are disjoint, then .

∞ 

.

Ai ∈ A,

i=1

then .A is a .σ-algebra. Proof. Let .{Ai }i∈N be an arbitrary sequence in .A. It follows from .(i) and .(iii) that i−1  .

Aj ∈ A

j=1

for any .i ∈ N, where we set .

0 

Aj = ∅. Furthermore, by .(ii), we have

j=1

X\

i−1 

.

Aj ∈ A

j=1

for all .i ∈ N. Moreover, it follows from .(ii) and .(iii) that     i−1  i−1   ∈A .Ai ∩ X\ Aj = X \ (X \ Ai ) ∪ Aj j=1

j=1

for any .i ∈ N. Since the above sets are disjoint sets of .A, by .(iv), we get    ∞ ∞ i−1    Ai ∩ X \ ∈ A, . Ai = Aj i=1

i=1

j=1

which implies that .A is a .σ-algebra.

2

.

We can now prove a theorem that gives considerable information about the measurable sets and the behavior of the outer measure on the measurable sets. Theorem 3.6. Let .μ∗ be an outer measure on a set X. (i) If .μ∗ (N ) = 0, then N is .μ∗ -measurable.

.

(ii) If A is .μ∗ -measurable, so is .X \ A.

.

CHAPTER 3. MEASURE THEORY

58

(iii) If .{Ai }i∈N is a sequence of .μ∗ -measurable sets, then .

∞ 

.

Ai and .

i=1

μ∗ -measurable.

∞ 

Ai are

i=1

.

(iv) If .{Ai }i∈N is a disjoint sequence of .μ∗ -measurable sets, then  ∞  ∞   ∗ .μ Ai = μ∗ (Ai ).

.

i=1

i=1

Proof. .(a) Assume that N is a subset of X with .μ∗ (N ) = 0 and that A and B are subsets of X with .A ⊂ N and .B ⊂ X \ N . Then, by Definition 3.1, we have μ∗ (B) ≤ μ∗ (A ∪ B) ≤ μ∗ (A) + μ∗ (B) ≤ μ∗ (N ) + μ∗ (B) = μ∗ (B).

.

Hence, it follows that μ∗ (A ∪ B) = μ∗ (A) + μ∗ (B),

.

which implies that N is .μ∗ -measurable, i.e., the statement in .(i) is true. ∗ .(b) Assume that A is .μ -measurable. Then μ∗ (B ∪ C) = μ∗ (B) + μ∗ (C)

.

for all subsets B and C of X with .B ⊂ A and .C ⊂ X \ A. Since .B ⊂ A and .C ⊂ X \ A if and only if .C ⊂ X \ A and .B ⊂ X \ (X \ A) = A, we conclude that .X \ A is .μ∗ -measurable, i.e., the statement .(ii) is true. ∗ .(c) Let .A1 and .A2 be .μ -measurable sets. Assume that B and C are arbitrary ∗ subsets of X with .μ (B) < ∞, .μ∗ (C) < ∞ and with B ⊂ A1 ∪ A2 and C ⊂ X \ (A1 ∪ A2 ).

.

We note that .B ∪ C = (B ∩ A1 ) ∪ ((B ∪ C) ∩ (X \ A1 )) and the sets .B ∩ A1 and .(B ∪ C) ∩ (X \ A1 ) are separated by the .μ∗ -measurable set .A1 . Thus, we get

∗ ∗ ∗ (B ∪ C) ∩ (X \ A1 ) . .μ (B ∪ C) = μ (B ∩ A1 ) + μ (3.1) On the other hand, we have

(B ∪ C) ∩ (X \ A1 ) = B ∩ (X \ A1 ) ∪ C

.

and the sets .B ∩(X \A1 ) and C are separated by the .μ∗ -measurable set .A2 . Hence, we obtain



∗ (B ∪ C) ∩ (X \ A1 ) = μ∗ B ∩ (X \ A1 ) + μ∗ C . (3.2) .μ

3.1. OUTER MEASURES

59

Finally, since .A1 is a .μ∗ -measurable set, we have

∗ ∗ B ∩ (X \ A1 ) = μ∗ (B). .μ (B ∩ A1 ) + μ

(3.3)

Therefore, it follows from (3.1), (3.2), and (3.3) that μ∗ (B ∪ C) = μ∗ (B) + μ∗ (C).

.

If one of the subsets B and C of X has an infinite .μ∗ -outer measure, then the above equality is obviously true, which implies that .A1 ∪ A2 is .μ∗ -measurable. We have thus proved .(iii) for the finite sequence of .μ∗ -measurable sets. ∗ .(d) Let .{Ai }i∈N be a disjoint sequence of .μ -measurable sets. We set ∞ 

A=

.

Ai

i=1

and let B and C be arbitrary subsets of X that satisfy the conditions B ⊂ A and C ⊂ X \ A.

.

By applying mathematical induction to .(c), we see that the set . measurable for any .n ∈ N. Since .C ⊂ X \ A ⊂ X \  ∗

μ (B ∪ C) ≥ μ

.

∗

B∩

n 

 Ai

 ∪C

=μ

n 

B∩

i=1

Ai , we have n 

 Ai

+ μ∗ (C).

i=1

and .μ∗ -measurable,

Since the sets .A1 , A2 , . . . , An are disjoint      n n−1   B∩ μ∗ B ∩ Ai = μ∗ Ai ∪ (B ∩ An ) i=1

 ∗

B∩

=μ



.

= μ∗ B ∩

Ai is .μ∗ -

i=1

i=1

 ∗

n 

i=1 n−1 



i=1 n−2 



+ μ∗ (B ∩ An )

Ai Ai

we get

+ μ∗ (B ∩ An−1 ) + μ∗ (B ∩ An )

i=1

= ··· = μ∗ (B ∩ A1 ) + μ∗ (B ∩ A2 ) + · · · + μ∗ (B ∩ An ). Then, it follows from (3.4) and the last equality that μ∗ (B ∪ C) ≥

n 

.

i=1

μ∗ (B ∩ Ai ) + μ∗ (C)

(3.4)

CHAPTER 3. MEASURE THEORY

60 for all .n ∈ N, which implies that ∗

μ (B ∪ C) ≥

.

∞ 

μ∗ (B ∩ Ai ) + μ∗ (C).

i=1

Thus, due to Definition 3.1 .(iii), we obtain μ∗ (B ∪ C) ≥

∞ 

μ∗ (B ∩ Ai ) + μ∗ (C)

i=1



≥ μ∗ B ∩

.



∞ 

Ai

+ μ∗ (C)

(3.5)

i=1 ∗

= μ∗ (B) + μ (C) ≥ μ∗ (B ∪ C). That is, .μ∗ (B ∪ C) = μ∗ (B) + μ∗ (C) for all subsets B and C of X with .B ⊂ A ∞ ∞   and .C ⊂ X \ A, where .A = Ai . In other words, . Ai is .μ∗ -measurable. i=1

i=1

On the other hand, assuming .C = ∅ in (3.5), we get ∗

μ (B) =

.

∞ 

μ∗ (B ∩ Ai )

(3.6)

i=1 ∞ 

for every subset B of .A =

Ai . Moreover, taking .B = A in the last equality, we

i=1

obtain

 μ

.

∗

∞  i=1

 Ai

=

∞ 

μ∗ (Ai ).

i=1

Therefore, we have proved .(iv) and also .(iii) for the disjoint sequence of .μ∗ measurable sets. ∗ .(e) According to .(a), the empty set .∅ is .μ -measurable. Moreover, using .(a), .(b), .(c), and .(d), together with Lemma 3.5, we conclude that the collection of all ∗ ∗ .μ -measurable sets in X is a .σ-algebra. Hence, the collection of all .μ -measurable subsets of X is closed under the operations of countable union and intersection. .2 This completes the proof of this theorem. Corollary 3.7. Let .μ∗ be an outer measure on a set X. Assume that E is an arbitrary subset of X and .{Ai }i∈N is a disjoint sequence of .μ∗ -measurable subsets of X. Then   ∞ ∞   ∗ E∩ .μ Ai = μ∗ (E ∩ Ai ). i=1

i=1

3.2. MEASURES IN ABSTRACT SPACES

61

Proof. Keeping in mind that .{Ai }i∈N is a disjoint sequence, we put .B = E ∩

∞ 

Ai

i=1

in (3.6) to obtain     ∞   ∞ ∞ ∞     ∗ ∗ E∩ .μ Ai = μ E∩ Aj ∩ Ai = μ∗ (E ∩ Ai ), i=1

i=1

j=1

i=1

2

which completes the proof.

.

3.2 Measures in Abstract Spaces At this point, it is convenient to introduce the concept of a measure defined on a .σ-algebra of sets, sometimes called a countably additive measure by some other mathematicians. Definition 3.8. Let X be a set and let .A be a .σ-algebra of subsets of X. A set function .μ : A → [0, ∞] is called a measure if it satisfies the conditions: (i) .μ(∅) = 0.

.

(ii) For any disjoint sequence .{Ai }i∈N in .A, it holds that  ∞  ∞   .μ Ai = μ(Ai ).

.

i=1

i=1

In view of this definition, we can rephrase Theorem 3.6 as follows: Theorem 3.9. Let X be a set and let .μ∗ be an outer measure on X. (i) The collection .M of all .μ∗ -measurable subsets of X is a .σ-algebra containing the null sets, where a subset N of X is called a null set if .μ∗ (N ) = 0.

.

(ii) The restriction of .μ∗ to .M is a measure on .M.

.

It is desirable to have a general method for constructing an outer measure from a pre-measure defined on any class of sets including .∅. Following M. E. Munroe [15], we will call this method Method I. Definition 3.10. Given a set X, let .C be some collection of subsets of X. A set function .τ : C → [0, ∞] is called a pre-measure if it satisfies the following conditions: (i) .∅ ∈ C.

.

(ii) .τ (∅) = 0.

.

CHAPTER 3. MEASURE THEORY

62

The assumption in Definition 3.10 that .∅ belongs to .C is useful in that finite unions can appear as special cases of infinite unions in the following theorem. Here we introduce Method I by M. E. Munroe (see [15, Theorem 11.3]). Theorem 3.11 (Munroe). Let .τ be a pre-measure defined on a collection .C of subsets of a set X. If the set function .μ∗ : P(X) → [0, ∞] is defined by  μ∗ (E) = inf

∞ 

.

τ (Ci ) : E ⊂

i=1



∞ 

Ci where Ci ∈ C for all i ∈ N

i=1

for all .E ∈ P(X), then .μ∗ is an outer measure on X. Proof. .(a) Since .0 ≤ τ (C) ≤ ∞ for any .C ∈ C, it holds that 0 ≤ μ∗ (E) ≤ ∞

.

for all subsets E of X. .(b) It holds that  ∗

μ (∅) = inf

∞ 

τ (Ci ) : ∅ ⊂

i=1 .

≤

∞ 

∞ 

Ci where Ci ∈ C for all i ∈ N

i=1

τ (∅)

i=1

= 0. Thus, we have .μ∗ (∅) = 0. .(c) If .E1 ⊂ E2 , then every covering of .E2 is also a covering of .E1 and thus μ∗ (E1 ) ≤ μ∗ (E2 ).

.

(d) Assume that .{Ei }i∈N is an arbitrary sequence of subsets of X. We assert

.

that  μ

.

∗

∞ 

 Ei

≤

i=1

.

i=1

μ∗ (Ei ).

i=1

Our assertion is obviously true when . ∞

∞ 

∞

μ∗ (Ei ) = ∞. Hence, we assume that

i=1

μ∗ (Ei )

< ∞.

Then, .μ∗ (Ei )

< ∞ for any .i ∈ N.

3.2. MEASURES IN ABSTRACT SPACES

63

Given any .ε > 0 and integer .i > 1, there exists a sequence .{Cij }j∈N in .C such that ∞ 

Ei ⊂

.

Cij and

j=1

∞ 

1 τ Cij ≤ μ∗ (Ei ) + i ε. 2 j=1

Assume that .{Di }i∈N is a sequence obtained by rearranging the double sequence {Cij }i,j∈N as a single sequence. Then, we see that

.

∞  .

Ei ⊂

i=1

and

 μ

∗

∞ 

 Ei

≤

i=1

∞ 

Di and Di ∈ C (for all i ∈ N)

i=1

∞ 

τ (Di ) =

i=1

.

=

∞ 

 ∞  ∞  ∞   1 μ∗ (Ei ) + i ε τ Cij ≤ 2 i=1 j=1

i=1

μ∗ (Ei ) + ε,

i=1

which proves our assertion, since we can choose .ε arbitrarily small.

2

.

Now we will show that any outer measure .μ∗ defined on X can be obtained by applying Method I to an appropriate pre-measure. Theorem 3.12. Let X be a set. Every outer measure .μ∗ on X can be constructed, by Method I, from the pre-measure .τ : P(X) → [0, ∞] that satisfies .τ (A) = μ∗ (A) for all subsets A of X. Proof. Since .∅ ∈ P(X) and .τ (∅) = μ∗ (∅) = 0, .τ is a pre-measure. Based on Theorem 3.11, we can construct an outer measure from .τ by applying Method I and we will call this outer measure .λ∗ . Then, for any subset A of X, we have  ∞ ∞   ∗ λ (A) = inf τ (Ci ) : A ⊂ Ci where Ci ∈ P(X) for all i ∈ N .

i=1

i=1

≤ τ (A) = μ∗ (A),

since A belongs to .P(X) and covers A. ∞  On the other hand, if .A ⊂ Ci and .Ci ∈ P(X) for all i, then  μ∗ (A) ≤ μ∗

i=1 ∞ 

.

i=1

Ci

 ≤

∞  i=1

μ∗ (Ci ) =

∞  i=1

τ (Ci ),

CHAPTER 3. MEASURE THEORY

64

which implies that  ∞ ∞   λ∗ (A) = inf τ (Ci ) : A ⊂ Ci where Ci ∈ P(X) for all i ∈ N .

i=1

i=1

∗

≥ μ (A). Therefore, we conclude that .λ∗ (A) = μ∗ (A) for all subsets A of X.

2

.

We will now explore the relation between a given outer measure .μ∗ and the measure .μ defined on the .σ-algebra of all .μ∗ -measurable sets. Theorem 3.13. Let X be a set. Assume that .μ is a measure defined on a .σ-algebra A of subsets of X. Then .μ is a pre-measure defined on .A. Let .λ∗ be the outer measure constructed from the pre-measure .μ by Method I. Then the restriction of ∗ ∗ .λ to the .σ-algebra of all .λ -measurable subsets of X is an extension of .μ. .

Proof. It is obvious that .μ is a pre-measure defined on .A. Thus, by Theorem 3.11, the set function .λ∗ constructed from .μ by Method I is an outer measure on X. Assume that E is any subset of X and .{Ai }i∈N is a sequence in .A with E⊂

∞ 

.

Ai .

i=1

Then, since .A is a .σ-algebra, the sets A=

∞ 

.

Ai and Bi = Ai \

i=1

i−1 

Aj (i ∈ N)

j=1

belong to the .σ-algebra .A. Since .μ is a measure on .A, we have  ∞  ∞ ∞    .μ(A) = μ Bi = μ(Bi ) ≤ μ(Ai ). i=1

i=1

i=1

Therefore, we get  ∞ ∞   λ∗ (E) = inf μ(Ai ) : E ⊂ Ai where Ai ∈ A for all i ∈ N .



i=1

i=1

 ≥ inf μ(A) : E ⊂ A and A ∈ A .

On the other hand, taking any set A in .A with .E ⊂ A and considering the covering of E by the sequence .{A, ∅, ∅, . . .}, we have   ∗ .λ (E) ≤ inf μ(A) : E ⊂ A and A ∈ A ,

3.2. MEASURES IN ABSTRACT SPACES

65

and thus, we conclude that   λ∗ (E) = inf μ(A) : E ⊂ A and A ∈ A .

.

(3.7)

We can choose a sequence .{Ai }i∈N in .A such that E ⊂ Ai and λ∗ (E) ≤ μ(Ai ) ≤ λ∗ (E) +

.

for all .i ∈ N. Then, the set .A =

∞ 

1 i

Ai belongs to .A and satisfies

i=1

E ⊂ A and λ∗ (E) = μ(A),

.

(3.8)

and the infimum in (3.7) is calculated explicitly. When E belongs to the .σ-algebra .A, we have   μ(E) = inf μ(E) : E ⊂ A where A ∈ A   ≤ inf μ(A) : E ⊂ A where A ∈ A . ≤ μ(E). Hence, it follows from (3.7) that λ∗ (E) = μ(E)

.

for all .E ∈ A. Therefore, .μ is the restriction of .λ∗ to the .σ-algebra .A. It remains to prove that each set of .A is .λ∗ -measurable. Assume that A is an arbitrary set of .A and that C and D are arbitrary subsets of X, which are separated by A, namely, C ⊂ A and D ⊂ X \ A.

.

Then, for any .i ∈ N, we can select a set .Bi from .A such that 1 C ∪ D ⊂ Bi and λ∗ (C ∪ D) ≤ μ(Bi ) ≤ λ∗ (C ∪ D) + . i

.

Since .C ⊂ A ∩ Bi and .D ⊂ (X \ A) ∩ Bi , we get

λ∗ (C) + λ∗ (D) ≤ μ(A ∩ Bi ) + μ (X \ A) ∩ Bi

= μ (A ∩ Bi ) ∪ ((X \ A) ∩ Bi ) .

= μ(Bi ) ≤ λ∗ (C ∪ D) +

1 i

CHAPTER 3. MEASURE THEORY

66

for all .i ∈ N, which implies that .λ∗ (C) + λ∗ (D) ≤ λ∗ (C ∪ D). Therefore, each .2 set A of .A is .λ∗ -measurable. If in Theorem 3.13 the measure .μ is the restriction of an outer measure .μ∗ to the collection of all .μ∗ -measurable sets, then the outer measure .λ∗ will generally not coincide with the original outer measure .μ∗ . But the two outer measures are equal if and only if .μ∗ is regular. Definition 3.14. Let X be a set. An outer measure .μ∗ on X is called regular if for each subset E of X there exists a .μ∗ -measurable subset A of X such that E ⊂ A and μ∗ (E) = μ∗ (A).

.

On the basis of this definition, we prove a corollary to Theorem 3.13. Corollary 3.15. The outer measure .λ∗ defined in Theorem 3.13 is regular. Proof. Since each set of .A is .λ∗ -measurable, it follows from (3.8) that for any subset E of X, there exists a set .A ∈ A such that E ⊂ A and λ∗ (E) = μ(A) = λ∗ (A),

.

which implies that the outer measure .λ∗ is regular.

2

.

Theorem 3.16. Let X be a set. Assume that .μ∗ is an outer measure on X and .μ is the restriction of .μ∗ to the .σ-algebra .M of all .μ∗ -measurable subsets of X. Then ∗ .μ is a pre-measure defined on .M, and the outer measure .λ constructed from the ∗ pre-measure .μ by Method I is regular. All .μ -measurable sets are .λ∗ -measurable and all .λ∗ -measurable sets E with .λ∗ (E) < ∞ are .μ∗ -measurable. Moreover, .λ∗ coincides with .μ∗ if and only if .μ∗ is regular. Proof. Due to Theorem 3.9, the set function .μ is a measure on the .σ-algebra .M of all .μ∗ -measurable sets. By Theorem 3.13, (3.8) and by Corollary 3.15, the outer measure .λ∗ is regular, coincides with .μ∗ and .μ on .M, and satisfies   λ∗ (E) = inf μ(E  ) : E ⊂ E  where E  ∈ M   . (3.9) = inf μ∗ (E  ) : E ⊂ E  where E  ∈ M for all subsets E of X. Moreover, due to Theorem 3.13, all .μ∗ -measurable sets are .λ∗ -measurable. Assume that E is any .λ∗ -measurable subset of X with .λ∗ (E) < ∞. According to (3.8), there exists a .μ∗ -measurable subset A of X such that E ⊂ A and λ∗ (E) = μ(A) = μ∗ (A).

.

(3.10)

3.2. MEASURES IN ABSTRACT SPACES

67

Hence, it follows from (3.10) that .λ∗ (A) = μ∗ (A) = λ∗ (E) < ∞. Since E is ∗ .λ -measurable, we have λ∗ (E) + λ∗ (A \ E) = λ∗ (A) = λ∗ (E) < ∞.

.

Hence, we get .λ∗ (A\E) = 0. Since .λ∗ coincides with .μ∗ on .M, we obtain .μ∗ (A\ E) = 0. Therefore, the null set .N = A \ E is .μ∗ -measurable by Theorem 3.6. Since A is .μ∗ -measurable, it holds that .E = A ∩ (X \ N ) is .μ∗ -measurable. ∗ .μ

Finally, since the outer measure .λ∗ is regular, .λ∗ and .μ∗ can only coincide if is regular. On the other hand, if .μ∗ is regular, then we have   μ∗ (E) = inf μ∗ (A) : E ⊂ A where A ∈ M

.

for all subsets E of X. Therefore, in view of (3.9), .λ∗ coincides with .μ∗ .

2

.

The following theorem gives two useful properties of measurable sets. Theorem 3.17. Let X be a set. Assume that .μ∗ is an outer measure on X and ∗ .{Ai }i∈N is a sequence of .μ -measurable subsets of X. (i) If .A1 ⊂ A2 ⊂ A3 ⊂ · · · and E is an arbitrary subset of X, then

.

 μ∗ E ∩

∞ 

.

 Ai

  = sup μ∗ (E ∩ Ai ) : i ∈ N .

i=1

(ii) If .A1 ⊃ A2 ⊃ A3 ⊃ · · · , E is an arbitrary subset of X, and .μ∗ (E ∩ Ai ) < ∞ for some .i ∈ N, then

.

 μ∗ E ∩

∞ 

.

i=1

 Ai

  = inf μ∗ (E ∩ Ai ) : i ∈ N .

CHAPTER 3. MEASURE THEORY

68

Proof. .(i) We set .A0 = ∅ and .Bi = Ai \ Ai−1 for all .i ∈ N. Then .{Bi }i∈N is a disjoint sequence of .μ∗ -measurable subsets of X. Due to Corollary 3.7, we have     ∞ ∞   ∗ ∗ μ E∩ Ai = μ E ∩ Bi i=1

i=1

=

∞ 

μ∗ (E ∩ Bi )

i=1 n 

= lim

.

n→∞

μ∗ (E ∩ Bi )

i=1



n 

= lim μ∗ E ∩ n→∞

 Bi

i=1

= lim μ∗ (E ∩ An ) n→∞   = sup μ∗ (E ∩ Ai ) : i ∈ N . (ii) Assume that .μ∗ (E ∩ An ) < ∞ for some .n ∈ N. We set .Bi = An \ An+i for all .i ∈ N. Then .{Bi }i∈N is a sequence of .μ∗ -measurable subsets of X with .

B1 ⊂ B2 ⊂ B 3 ⊂ · · · .

.

It then follows from .(i) that   ∞    ∗ E∩ .μ Bi = sup μ∗ (E ∩ Bi ) : i ∈ N . i=1

Thus, we have 

∞ 

μ∗ (E ∩ An ) \

.

 Aj

  = sup μ∗ (E ∩ (An \ An+i )) : i ∈ N .

j=n+1

Since .{Ai }i∈N is a non-increasing sequence, it follows that E∩

∞ 

.

Aj = (E ∩ An ) ∩

j=n+1

Aj , E ∩ An+i = (E ∩ An ) ∩ An+i .

j=n+1

Moreover, we see that  E ∩ An = .

∞ 

(E ∩ An ) ∩



∞ 

Aj

j=n+1





 ∪

∞ 

(E ∩ An ) \

j=n+1

= (E ∩ An ) ∩ An+i ∪ (E ∩ An ) \ An+i .

 Aj

(3.11)

3.2. MEASURES IN ABSTRACT SPACES

69

Since each .Ai is a .μ∗ -measurable subset of X, so is .

∞ 

Aj . Hence, it holds that

j=n+1

 ∗

μ (E ∩ An ) = μ .

∗

E∩

∞ 



 +μ

Aj

∗

(E ∩ An ) \

j=n+1



= μ (E ∩ An+i ) + μ (E ∩ An ) \ An+i ∗

∞ 

 Aj

j=n+1

∗

for all .i ∈ N. Since .μ∗ (E ∩ An ) < ∞, so are the other measures in the above formulae. In addition, by (3.11), we get   ∞  μ∗ (E ∩ An ) − μ∗ E ∩ Aj .



j=n+1 ∗

 = sup μ (E ∩ An ) − μ (E ∩ An+i ) : i ∈ N . ∗

Therefore, since .{Ai }i∈N is a non-increasing sequence, we obtain     ∞ ∞   μ∗ E ∩ Ai = μ∗ E ∩ Aj .

i=1

which completes the proof.



j=n+1

 = inf μ (E ∩ An+i ) : i ∈ N   = inf μ∗ (E ∩ Ai ) : i ∈ N , ∗

2

.

In the following theorem, we will give a method of obtaining a new outer measure from a given collection of outer measures. Theorem 3.18. Let X be a set and let .Λ be an arbitrary index set. If .μ∗λ is an outer measure on X for all .λ ∈ Λ, then  ∗  ∗ .μ (E) = sup μλ (E) : λ ∈ Λ (for all subsets E of X) is an outer measure on X. Proof. It is easy to show that .0 ≤ μ∗ (E) ≤ ∞ for all subsets E of X. Moreover, it holds that  ∗  ∗ .μ (∅) = sup μλ (∅) : λ ∈ Λ = 0. Assume that .E1 and .E2 are arbitrary subsets of X with .E1 ⊂ E2 . Then we have  ∗    ∗ .μ (E1 ) = sup μλ (E1 ) : λ ∈ Λ ≤ sup μ∗λ (E2 ) : λ ∈ Λ = μ∗ (E2 ).

CHAPTER 3. MEASURE THEORY

70

Finally, we assume that .{Ei }i∈N is an arbitrary sequence of subsets of X. Then, for all .λ ∈ Λ, we have  ∞  ∞ ∞    ∗ .μλ Ei ≤ μ∗λ (Ei ) ≤ μ∗ (Ei ). i=1

i=1

i=1

Hence, we get  μ∗

∞ 

.

 Ei

i=1

≤

∞ 

μ∗ (Ei ),

i=1

2

which completes the proof.

.

We now generalize the concept of the regular outer measures. Definition 3.19. Let X be a set and let .R be a collection of subsets of X. An outer measure .μ∗ on X is called .R-regular if for each subset E of X there exists a set R in .R such that E ⊂ R and μ∗ (E) = μ∗ (R).

.

Comparing this definition with Definition 3.14, we note that the outer measure is regular if .μ∗ is .M-regular, where .M is the .σ-algebra of all .μ∗ -measurable subsets of X.

∗ .μ

Definition 3.20. Let X be a set and let .R be a collection of subsets of X. (i) We denote by .Rσ the collection of all countable set unions, i.e.,  ∞  .Rσ = Ri : Ri ∈ R for all i ∈ N .

.

i=1

(ii) We denote by .Rδ the collection of all countable set intersections, i.e.,  ∞  .Rδ = Ri : Ri ∈ R for all i ∈ N .

.

i=1

Theorem 3.21. Let X be a set and let .μ∗ be the outer measure constructed by Method I from a pre-measure .τ defined on a collection .C of subsets of X with ∗ .X ∈ C. Then .μ is .Cσδ -regular.

3.2. MEASURES IN ABSTRACT SPACES

71

Proof. Assume that E is an arbitrary subset of X. If .μ∗ (E) = ∞, then we have E ⊂ X and μ∗ (E) = μ∗ (X),

.

where .X ∈ C ⊂ Cσδ . Hence, from now on, we only need to treat each subset E of X with .μ∗ (E) < ∞. Since  ∞ ∞   ∗ .μ (E) = inf τ (Ci ) : E ⊂ Ci where Ci ∈ C for all i ∈ N , i=1

i=1

for any integer .j ∈ N, we can choose a sequence .{Ci,j }i∈N in .C such that E⊂

∞ 

.

Ci,j

i=1

∞ 

1 and τ Ci,j < μ∗ (E) + . j i=1

We write D=

∞ ∞  

.

Ci,j .

j=1 i=1

Then, it is obvious that .D ∈ Cσδ and .E ⊂ D. Furthermore, μ∗ (E) ≤ μ∗ (D)  ∞ ∞   τ (Ci ) : D ⊂ Ci where Ci ∈ C for all i ∈ N = inf i=1 .

i=1

∞ 

τ Ci,j ≤ i=1

< μ∗ (E) +

1 j

for every .j ∈ N, since .{Ci,j }i∈N is a sequence (in .C) covering D. Thus, we have μ∗ (E) = μ∗ (D).

.

Therefore, for any subset E of X, there exists a .D ∈ Cσδ such that .E ⊂ D and μ∗ (E) = μ∗ (D), which implies that the outer measure .μ∗ is .Cσδ -regular. .2

.

Definition 3.22. Let X be a topological space. The Borel .σ-algebra of subsets of X is the smallest .σ-algebra containing all open (or closed) subsets of X. Each set in the Borel .σ-algebra is called a Borel set.

CHAPTER 3. MEASURE THEORY

72

Corollary 3.23. Let X be a set and let .μ∗ be the outer measure constructed by Method I from a pre-measure .τ . (i) If the pre-measure .τ is defined on the collection .G of all open subsets of X, then .μ∗ is .Gδ -regular.

.

(ii) If the pre-measure .τ is defined on the Borel .σ-algebra .B, then .μ∗ is .B-regular .(Borel-regular.).

.

Proof. We note that .X ∈ G, .X ∈ B, and .Gσ = G and that the collection of Borel sets is closed under countable unions and under countable intersections. Then the .2 assertions of this corollary are direct consequences of Theorem 3.21.

3.3 Measures in Metric Spaces In this section, we introduce a second method called Method II to get an outer measure from a pre-measure. The names Method I and Method II go back to Munroe [15]. Let .(X, d) be a metric space. We recall that the diameter of a subset E of X is defined as ⎧ (for E = ∅), ⎨ 0 sup{d(x, y) : x, y ∈ E} (for a nonempty bounded set E), .d(E) = ⎩ ∞ (for an unbounded set E). Let .C be a collection of subsets of X. For any .δ > 0, we define   .Cδ = E ∈ C : d(E) < δ . We call the outer measure .μ∗ defined in the following theorem the outer measure constructed from the pre-measure .τ by Method II. Theorem 3.24 (Munroe). Let .(X, d) be a metric space and let .τ be a pre-measure defined on a collection .C of subsets of X. If the set function .μ∗ : P(X) → [0, ∞] is defined by  ∗  ∗ .μ (E) = sup μδ (E) : δ > 0 , (3.12) where ∗ .μδ (E)

 = inf

∞  i=1

τ (Ci ) : E ⊂

∞ 

Ci where Ci ∈ Cδ for all i ∈ N

i=1

for all .E ∈ P(X), then .μ∗ is an outer measure on X.

(3.13)

3.3. MEASURES IN METRIC SPACES

73

Proof. It is obvious that the restriction .τδ of .τ to .Cδ is also a pre-measure for any δ > 0. Then we have  ∞ ∞   ∗ .μδ (E) = inf τδ (Ci ) : E ⊂ Ci where Ci ∈ Cδ for all i ∈ N

.

i=1

i=1

for all subsets E of X. Hence, it follows from Theorem 3.11 that .μ∗δ is the outer measure constructed by Method I from the pre-measure .τδ . By Theorem 3.18, the set function  ∗  ∗ .μ (E) = sup μδ (E) : δ > 0 (for all subsets E of X) 2

is an outer measure.

.

As .δ gets smaller, the size of the class of coverings over which the infimum (3.13) is taken also reduces. Therefore, for any fixed subset E of X, .μ∗δ (E) does not decrease when .δ decreases, and it is the small values of .δ that are relevant to the supremum (3.12). In fact, we could immediately replace the formula (3.12) with μ∗ (E) = lim μ∗δ (E).

.

δ→0+

(3.14)

Definition 3.25. Let A and B be disjoint nonempty subsets of a metric space (X, d). Then A and B are said to be positively separated if the distance   . inf d(x, y) : x ∈ A and y ∈ B

.

separating A and B is positive. The main advantages of the outer measures constructed by Method II over those constructed by Method I all stem from the following theorem, which shows that the outer measures by Method II are additive when considered on the union of a pair of positively separated sets. Theorem 3.26. Let .μ∗ be an outer measure on a metric space .(X, d) constructed from a pre-measure .τ by Method II. Then μ∗ (A ∪ B) = μ∗ (A) + μ∗ (B)

.

for all disjoint nonempty subsets A and B of X that are positively separated. Proof. Let A and B be disjoint nonempty subsets of X that are positively separated. Since .μ∗ is an outer measure on X, we have μ∗ (A ∪ B) ≤ μ∗ (A) + μ∗ (B).

.

CHAPTER 3. MEASURE THEORY

74 So it suffices to prove that

μ∗ (A ∪ B) ≥ μ∗ (A) + μ∗ (B).

.

Without loss of generality, we may assume that .μ∗ (A ∪ B) < ∞. Since A and B are disjoint nonempty subsets of X that are positively separated, we can choose a .δ > 0 such that   . inf d(x, y) : x ∈ A and y ∈ B ≥ δ. Let .ε > 0 be given. Assume that .δ1 , .δ2 , and .η are constants given by   1 .0 < δ1 < d(X), 0 < δ2 < d(X), and η = min δ1 , δ2 , δ . 2 We see that .μ∗ (A ∪ B) < ∞ and  ∞ ∞   ∗ .μ (A ∪ B) = sup inf τ (Ci ) : A ∪ B ⊂ Ci where Ci ∈ C for all i , >0

i=1

i=1

where the pre-measure .τ is assumed to be defined on a collection .C of subsets of X. Thus, we have  ∞ ∞   . inf τ (Ci ) : A ∪ B ⊂ Ci where Ci ∈ Cη for all i ∈ N ≤ μ∗ (A ∪ B), i=1

i=1

and we may choose a sequence .{Ci }i∈N from .Cη such that A∪B ⊂

∞ 

.

i=1

Ci and

∞ 

τ (Ci ) ≤ μ∗ (A ∪ B) + ε.

i=1

Now, we assume that there exists an .i ∈ N such that Ci ∩ A = ∅ and Ci ∩ B = ∅.

.

(3.15)

Then there would exist .a0 and .b0 with a0 ∈ Ci ∩ A and b0 ∈ Ci ∩ B

.

so that   1 1 1 d(a0 , b0 ) ≤ d(Ci ) ≤ η ≤ δ ≤ inf d(x, y) : x ∈ A and y ∈ B ≤ d(a0 , b0 ), 2 2 2

.

which would lead to a contradiction that .d(a0 , b0 ) = 0.

3.3. MEASURES IN METRIC SPACES

75

So, we conclude that there does not exist .i ∈ N satisfying the conditions in (3.15). For each .i ∈ N, we set   Ci (for Ci ∩ A = ∅), Ci (for Ci ∩ B = ∅), and Bi = .Ai = ∅ (for Ci ∩ A = ∅) ∅ (for Ci ∩ B = ∅). Then we get ∞ 

.

i=1 ∞ 

Ai ⊃ Bi ⊃

i=1

∞  i=1 ∞ 

 (Ci ∩ A) = (Ci ∩ B) =

i=1

∞ 

i=1  ∞ 

 ∩ A = A,

Ci 

∩ B = B.

Ci

i=1

Moreover, .{Ai }i∈N and .{Bi }i∈N are sequences of sets from .Cδ . Since the conditions of (3.15) are not met, it holds that either τ (Ai ) + τ (Bi ) = τ (∅) + τ (∅) = 0

.

or τ (Ai ) + τ (Bi ) = τ (Ci ) + τ (∅) = τ (Ci )

.

for all .i ∈ N. In either case, we have τ (Ai ) + τ (Bi ) ≤ τ (Ci )

.

for all .i ∈ N. Hence, we get ∞  .

τ (Ai ) +

i=1

∞ 

τ (Bi ) ≤

i=1

∞ 

τ (Ci ) ≤ μ∗ (A ∪ B) + ε.

i=1

Furthermore, it holds that Ai ∈ C, d(Ai ) < η ≤ δ1 , A ⊂ .

Bi ∈ C, d(Bi ) < η ≤ δ2 , B ⊂

∞  i=1 ∞ 

Ai , Bi .

i=1

Hence, we obtain μ∗δ1 (A) ≤

∞ 

.

i=1

τ (Ai ), μ∗δ2 (B) ≤

∞  i=1

τ (Bi ),

CHAPTER 3. MEASURE THEORY

76 and

μ∗δ1 (A) + μ∗δ2 (B) ≤ μ∗ (A ∪ B) + ε

.

for all .δ1 and .δ2 with .0 < δ1 < d(X) and .0 < δ2 < d(X). Thus, we have μ∗ (A) + μ∗ (B) ≤ μ∗ (A ∪ B) + ε

.

for any .ε > 0. Therefore, we conclude that μ∗ (A) + μ∗ (B) ≤ μ∗ (A ∪ B),

.

2

which completes the proof.

.

3.4 Metric Outer Measures The property of the outer measures constructed by Method II presented in Theorem 3.26 is so important that it can be used as a definition. Definition 3.27. An outer measure .μ∗ on a metric space X is called a metric outer measure if μ∗ (A ∪ B) = μ∗ (A) + μ∗ (B)

.

for any disjoint nonempty subsets A and B of X that are positively separated. According to Theorem 3.26, the outer measure on a metric space, constructed by Method II, is always a metric outer measure. Theorem 3.28. Let .μ∗ be a metric outer measure on a metric space .(X, d). Assume that .{Ai }i∈N is a sequence of subsets of X which satisfies A1 ⊂ A2 ⊂ A3 ⊂ · · · and A =

∞ 

.

Ai .

i=1

If the sets .Ai and .A \ Ai+1 are positively separated for each .i ∈ N, then  ∞     ∗ .μ Ai = sup μ∗ (Ai ) : i ∈ N . i=1

Proof. It is obvious that   μ∗ (A) ≥ sup μ∗ (Ai ) : i ∈ N .

.

3.4. METRIC OUTER MEASURES

77

So it is enough to prove that   μ∗ (A) ≤ sup μ∗ (Ai ) : i ∈ N .

.

  Without loss of generality, we assume that .sup μ∗ (Ai ) : i ∈ N < ∞. For every integer .i > 1, we set .D1 = A1 and .Di = Ai \ Ai−1 . If i and j are positive integers with .j > i + 1, then we have Di ⊂ Ai and Dj ⊂ A \ Aj−1 ⊂ A \ Ai+1 .

.

Since .Ai and .A \ Ai+1 are positively separated, the sets .Di and .Dj are positively separated whenever .j > i + 1. Furthermore, the sets n  .

D2i+m and D2n+m+2

i=1

are positively separated for all .n ∈ N and .m ∈ {0, −1}. Since .μ∗ is a metric outer measure, we can apply Theorem 3.26 inductively to obtain  μ

.

∗

n 

 D2i+m

=

i=1

n 

μ∗ (D2i+m )

i=1

for all .n ∈ N and .m ∈ {0, −1}. Now we see that   n   n   .A2n = D2i ∪ D2i−1 i=1

i=1

for all .n ∈ N. Hence, we have n  .

 ∗

μ (D2i+m ) = μ

∗

i=1

n 

 D2i+m

  ≤ μ∗ (A2n ) ≤ sup μ∗ (Ai ) : i ∈ N < ∞

i=1

for all .n ∈ N and .m ∈ {0, −1}. Hence, the series ∞  .

i=1

converge.

∗

μ (D2i ) and

∞  i=1

μ∗ (D2i−1 )

(3.16)

CHAPTER 3. MEASURE THEORY

78 Finally, it holds that



∗

μ (A) = μ

∗





∞ 

Ai

i=1

= μ∗ An ∪ .

≤ μ∗ (An ) +

∞  i=n+1 ∞ 

 Di (3.17) μ∗ (Di )

i=n+1 ∞    μ∗ (Di ) ≤ sup μ∗ (Ai ) : i ∈ N + i=n+1

for all .n ∈ N. Since both series in (3.16) converge, if we let .n → ∞ in (3.17), then we get  ∗  ∗ .μ (A) ≤ sup μ (Ai ) : i ∈ N , 2

which completes the proof.

.

Theorem 3.29. If .μ∗ is a metric outer measure on a metric space .(X, d), then all closed subsets of X are .μ∗ -measurable. Proof. Assume that K is an arbitrary closed subset of X and that A and .B are arbitrary subsets of X which are separated by K, i.e., A ⊂ K and B ⊂ X \ K.

.

Without loss of generality, we assume that both A and B are nonempty sets. We claim that B is expressed as the union of an increasing sequence .{Bi }i∈N of subsets of X, where .Bi and .B \ Bi+1 are positively separated and A and .Bi are positively separated for each .i ∈ N. We define .Bi by   1 .Bi = x ∈ B : inf d(x, y) > y∈K i for each .i ∈ N. Then we see that B1 ⊂ B2 ⊂ B3 ⊂ · · · ⊂ B.

.

Since .B ⊂ X \ K, if .b ∈ B, then .b ∈ K and b is not a limit point of K. Hence, b belongs to .Bi for all sufficiently large integers i. Therefore, it holds that B=

∞ 

.

i=1

Bi .

3.4. METRIC OUTER MEASURES

79

Since .A ⊂ K, A and .Bi are positively separated for all .i ∈ N. Now we assume that b and c are arbitrary elements of X with b ∈ Bi and c ∈ B \ Bi+1

.

(3.18)

for some .i ∈ N. Then, it follows from the definition of .Bi+1 that   1 . . inf d(c, y) : y ∈ K ≤ i+1 Hence, we can select an element .c of K such that 1  .d(c, c ) ≤ . i + 1/2 If we had d(b, c) ≤

.

1/2 i(i + 1/2)

for some .i ∈ N, then we would have   inf d(b, y) : y ∈ K ≤ d(b, c ) ≤ d(b, c) + d(c, c ) 1 1/2 . + ≤ i(i + 1/2) i + 1/2 1 = i for some .i ∈ N, a contradiction to the definition of .Bi and the assumption .b ∈ Bi . Therefore, for any pair b and c satisfying (3.18), it had to hold that d(b, c) >

.

1 i(2i + 1)

and .Bi and .B \ Bi+1 are positively separated for any .i ∈ N. Finally, we use the metric property of the outer measure and Theorem 3.28 to have   μ∗ (A ∪ B) ≥ sup μ∗ (A ∪ Bi ) : i ∈ N   = sup μ∗ (A) + μ∗ (Bi ) : i ∈ N   . = μ∗ (A) + sup μ∗ (Bi ) : i ∈ N = μ∗ (A) + μ∗ (B), which implies that every closed subset K of X is .μ∗ -measurable.

2

.

We recall that the Borel .σ-algebra of subsets of a topological space X is the smallest .σ-algebra containing all open (or closed) subsets of X and each set in the Borel .σ-algebra is called a Borel set.

CHAPTER 3. MEASURE THEORY

80

Theorem 3.30. If .μ∗ is a metric outer measure on a metric space X, then all Borel sets in X are .μ∗ -measurable. Proof. Let .M be the collection of all .μ∗ -measurable subsets of X. Then it follows from Theorems 3.9 and 3.29 that .M is .σ-algebra and .M contains any closed subset of X. Therefore, in view of Definition 3.22, we conclude that .M contains the Borel sets in X. .2 We now examine the regularity of the outer measures constructed by Method II from a pre-measure. Theorem 3.31. Let .μ∗ be the outer measure on a metric space .(X, d) constructed, by Method II, from a pre-measure .τ defined on a collection .C of subsets of X with ∗ .X ∈ C. Then .μ is .Cσδ -regular. Proof. Given any .δ > 0, let .μ∗δ be the outer measure on X constructed, by Method I, from the restriction .τδ of .τ to the sets C of .C with .d(C) < δ. Then it follows that μ∗ (E) = sup μ∗δ (E)

.

δ>0

for all subsets E of X. Due to Theorem 3.21, every outer measure .μ∗δ is .Cσδ -regular. Hence, if E is an arbitrary subset of X, then we can choose a .Cσδ -set .Ci (i.e., .Ci ∈ Cσδ ) such that E ⊂ Ci and μ∗1/i (E) = μ∗1/i (Ci )

.

for any .i ∈ N. Then, the set C=

∞ 

.

Ci

i=1

is a .Cσδ -set with .E ⊂ C. Moreover, for any .δ > 0, there exists an .i ∈ N satisfying .0
0, which implies that .μ∗ (C) ≤ μ∗ (E). Thus, we have C ∈ Cσδ , E ⊂ C, μ∗ (E) = μ∗ (C).

.

Therefore, .μ∗ is .Cσδ -regular.

2

.

Corollary 3.32. Let .μ∗ be the outer measure on a metric space X constructed, by Method II, from a pre-measure .τ .

3.5. LEBESGUE MEASURES

81

(i) If the pre-measure .τ is defined on the collection .G of all open subsets of X, then .μ∗ is .Gδ -regular.

.

(ii) If the pre-measure .τ is defined on the Borel .σ-algebra .B, then .μ∗ is .B-regular .(Borel-regular.).

.

In either case, the outer measure .μ∗ is regular. Proof. We note that .X ∈ G, .X ∈ B, and .Gσ = G and that the collection of Borel sets is closed under countable unions and under countable intersections. Then the assertions of this corollary are direct consequences of Theorem 3.31. The last .2 assertion is true by Theorems 3.26 and 3.30.

3.5 Lebesgue Measures In this section, we explore the special but most important Lebesgue measure using the general theory we have developed so far. Definition 3.33. Given an .n ∈ N, let .Rn be the n-dimensional Euclidean space. (i) Let .a1 , a2 , . . . , an and .b1 , b2 , . . . , bn be arbitrary real numbers with .ai ≤ bi for all .i ∈ {1, 2, . . . , n}. The set   .R(a, b) = (x1 , x2 , . . . , xn ) ∈ Rn : ai < xi < bi for any i ∈ {1, 2, . . . , n}

.

is said to be the open rectangle with corners a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ),

.

where we set .R(a, b) = ∅ if .ai = bi for some .i ∈ {1, 2, . . . , n}. We use the symbol .R to denote the collection of all such open rectangles. (ii) We define a set function .τ : R → [0, ∞] by

.

n

 τ R(a, b) = (bi − ai )

.

i=1

for all .R(a, b) ∈ R. Then .τ (R(a, b)) is the “elementary volume” of the open rectangle .R(a, b). Obviously, .τ is a pre-measure defined on .R. Remark 3.34. Even though we replace .R(a, b) in Definition 3.33 with the halfopen rectangle or the closed rectangle with corners a and b defined by   .R[a, b) = (x1 , x2 , . . . , xn ) ∈ Rn : ai ≤ xi < bi for any i ∈ {1, 2, . . . , n}

CHAPTER 3. MEASURE THEORY

82 or

  R[a, b] = (x1 , x2 , . . . , xn ) ∈ Rn : ai ≤ xi ≤ bi for any i ∈ {1, 2, . . . , n} ,

.

where .R[a, b) = ∅ if .ai = bi for some .i ∈ {1, 2, . . . , n}, and we define .R to be the collection of all such half-open rectangles or closed rectangles, the following theorems about the properties of Lebesgue measures are still true. Let .μ∗ and .ν ∗ be the outer measures constructed from the pre-measure .τ by Methods I and II, respectively. Then it is easy to verify that .μ∗ is the usual Lebesgue (outer) measure. Our experience so far makes us confident that the outer measure .ν ∗ would have many advantages over the outer measure .μ∗ . In fact, however, .μ∗ shares all the advantages of .ν ∗ , as the following theorem shows. Theorem 3.35. Assume that .μ∗ and .ν ∗ are the outer measures constructed from the pre-measure .τ by Methods I and II, respectively. Then .μ∗ = ν ∗ . Proof. By Theorems 3.11 and 3.24, it is obvious that .μ∗ (E) ≤ ν ∗ (E) for all subsets E of .Rn . We now claim that .ν ∗ (E) ≤ μ∗ (E). Assume that E is an arbitrary subset of .Rn with .μ∗ (E) < ∞, without loss of generality. For any .ε > 0, we can select a sequence .{Ri }i∈N in .R such that E⊂

∞ 

.

i=1

Ri and

∞ 

τ (Ri ) ≤ μ∗ (E) + ε.

(3.19)

i=1

Given any .δ > 0 and any .R(a, b) ∈ R, we can select a large integer N such that the .N n closed rectangles ai +

.

ri − 1 ri (bi − ai ) ≤ xi ≤ ai + (bi − ai ) (for i ∈ {1, 2, . . . , n}), N N

where .r1 , r2 , . . . , rn ∈ {1, 2, . . . , N }, and such that all the closed rectangles have diameter less than .δ. Then, for any sufficiently small .η > 0, the .N n open rectangles .R(r1 , r2 , . . . , rn ) given by ai +

.

ri − 1 ri (bi − ai ) − η < xi < ai + (bi − ai ) + η (for i ∈ {1, 2, . . . , n}) N N

cover the open rectangle .R(a, b), where .d(R(r1 , r2 , . . . , rn )) < δ for .r1 , r2 , . . . , rn ∈ {1, 2, . . . , N }, and have

.

N  N  .

r1 =1 r2 =1

N 



··· τ R(r1 , r2 , . . . , rn ) ≤ τ R(a, b) + O(η), rn =1

3.5. LEBESGUE MEASURES

83

where the O denotes Landau’s symbol (O notation). Applying this process to every open rectangle .Ri , we replace .Ri with a finite sub-collection .{Rij }j∈{1,2,...,j(i)} of open rectangles from .R of diameter less than .δ that satisfies 

j(i)

Ri ⊂

Rij and

.

j=1

j(i) 

τ (Rij ) ≤ τ (Ri ) +

j=1

1 ε. 2i

Then, by (3.19), the collection of all such open rectangles .Rij is a .δ-covering of E with j(i) ∞   .

τ (Rij ) ≤ μ∗ (E) + 2ε.

i=1 j=1

Hence, we have  ∞ ∞   . inf τ (Ri ) : E ⊂ Ri where Ri ∈ Rδ for all i ∈ N ≤ μ∗ (E) + 2ε, i=1

i=1

which implies that .ν ∗ (E) ≤ μ∗ (E) + 2ε. Therefore, since .ε is an arbitrary positive .2 real number, we conclude that .ν ∗ (E) ≤ μ∗ (E) for each subset E of .Rn . The following theorem seems obvious, but its proof is not so easy. In fact, this theorem was originally proved in the one-dimensional case by E. Borel [1]. Theorem 3.36. Assume that .μ∗ is the outer measure constructed from the premeasure .τ by Method I. Then .μ∗ (R) = τ (R) for all open rectangles R of .R. Proof. It follows from Theorem 3.11 that .μ∗ (R) ≤ τ (R) for all open rectangles R of .R. It remains to prove that .μ∗ (R) ≥ τ (R) for all open rectangles R of .R. We consider a closed rectangle I of the form I = [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ],

.

where .ai and .bi are real numbers satisfying .ai < bi for each .i ∈ {1, 2, . . . , n}. Assume that .{Rj }j∈N is an arbitrary sequence of open rectangles from .R with I⊂

∞ 

.

Rj .

j=1

We note that .{Rj }j∈N is an open covering of the compact subset I of .Rn . Hence, we can choose an .N ∈ N such that I⊂

N 

.

j=1

Rj .

CHAPTER 3. MEASURE THEORY

84

For any .j ∈ {1, 2, . . . , N }, let .I ◦ ∩ Rj be the open rectangle of the form I ◦ ∩ Rj = (a1,j , b1,j ) × (a2,j , b2,j ) × · · · × (an,j , bn,j ).

.

For all .i ∈ {1, 2, . . . , n}, let .ci,1 , ci,2 , . . . , ci,2N (.ci,1 = ai and .ci,2N = bi ) be a rearrangement of the numbers .ai,1 , bi,1 , ai,2 , bi,2 , . . . , ai,N , bi,N in nondecreasing order. For any finite sequence .{k(1), k(2), . . . , k(n)} with .k(i) ∈ {1, 2, . . . , 2N −1}, we define the open rectangle .R(k(1), k(2), . . . , k(n)) by R(k(1), k(2), . . . , k(n)) .

= (c1,k(1) , c1,k(1)+1 ) × (c2,k(2) , c2,k(2)+1 ) × · · · × (cn,k(n) , cn,k(n)+1 ).

When this open rectangle is nonempty, its midpoint is in .I ◦ and so is in one of the open rectangles .R1 , R2 , . . . , RN . Hence, by the choice of the numbers .ci,k(i) , the whole rectangle .R(k(1), k(2), . . . , k(n)) lies in one of the rectangles .I ◦ ∩ Rj . However, by the choice of the numbers .ci,k(i) , the elementary volume of each open rectangle .I ◦ ∩ Rj is the sum of the elementary volumes of the open rectangles .R(k(1), k(2), . . . , k(n)) that it contains. Thus we have ∞ 

τ (Rj ) ≥

j=1

N 

τ (Rj )

j=1

≥

N 

τ (I ◦ ∩ Rj )

j=1

=

N 



n 

ci,k(i)+1 − ci,k(i)

j=1 R(k(1),k(2),...,k(n))⊂I ◦ ∩Rj i=1

.

≥

= =

−1 2N −1 2N  

2N −1 

···

k(1)=1 k(2)=1 n 

n 

ci,k(i)+1 − ci,k(i)

k(n)=1 i=1

(ci,2N − ci,1 )

i=1 n 

(bi − ai ).

i=1

Therefore, we get μ∗ (I) ≥

n 

.

i=1

(bi − ai ).





3.5. LEBESGUE MEASURES

85

Finally, for any given nonempty open rectangle .I ◦ , we can construct in .I ◦ closed rectangles with elementary volume, and hence with Lebesgue outer measure, as close as we please to .τ (I ◦ ). So we have .μ∗ (I ◦ ) ≥ τ (I ◦ ). Therefore, we conclude that .μ∗ (R) = τ (R) for all open rectangles R of .R, which completes our proof. .2 Definition 3.37. Let .μ∗ be an outer measure on a vector space X. For any subset E and element p of X, we define   E+p= x+p:x∈E .

.

The outer measure .μ∗ is called translation-invariant if μ∗ (E + p) = μ∗ (E)

.

for all subsets E and elements p of X. We note that the Lebesgue measure is the unique Borel-regular, translationinvariant outer measure on the Euclidean space .Rn , which assigns unit measure to the unit n-cube .[0, 1]n . Let .(X1 , A1 ) and .(X2 , A2 ) be measurable spaces and let .μ1 and .μ2 be measures on them. We denote by .A1 ⊗ A2 the .σ-algebra on the Cartesian product .X1 × X2 generated by subsets of the form  .

 A1 × A2 : A1 ∈ A1 and A2 ∈ A2 .

We define the product measure .μ1 ⊗μ2 on the measurable space .(X1 ×X2 , A1 ⊗ A2 ) by .(μ1 ⊗ μ2 )(A1 × A2 ) = μ1 (A1 )μ2 (A2 ) for all .A1 ∈ A1 and .A2 ∈ A2 . The Lebesgue measure .μn is a measure defined in an n-dimensional Euclidean space. Unfortunately, since the Hilbert cube is an infinite-dimensional space, it is not possible to define the Lebesgue measure in the Hilbert cube. Instead, in the Hilbert cube we can define the standard product probability measure that most closely resembles the Lebesgue measure. Definition 3.38. If a measure .π defined on the Borel .σ-algebra of the Hilbert cube has the following properties, we call it the standard product probability measure: (i) .π(I ω ) = 1.

.

(ii) .π(B) =

∞ 

.

i=1

μ1 (Bi ) for all Borel subsets B of .I ω with .B =

∞  i=1

Bi , where

μ1 is the Lebesgue measure in .R and each .Bi is a Borel subset of .[0, 1].

.

86

CHAPTER 3. MEASURE THEORY

3.6 Hausdorff Measures The Hausdorff measure is a generalization of the traditional concepts of length, area, and volume to non-integer dimensions, such as the Hausdorff dimension of fractals. The Hausdorff measure is a kind of outer measure, named after F. Hausdorff, that assigns a number in .[0, ∞] to each subset of a metric space. Definition 3.39. We denote by .H the set of all functions .h : [0, ∞) → [0, ∞] with the following four properties: (i) .h(0) = 0.

.

(ii) .h(t) > 0 for any .t > 0.

.

(iii) h is monotonically increasing.

.

(iv) h is continuous on the right for all .t ≥ 0.

.

Every function h in .H is called a Hausdorff function. It is well known that every Hausdorff function induces a corresponding Hausdorff measure. Definition 3.40. Let .(X, d) be a metric space and .T (d) the topology for X generated by d, and let h be a Hausdorff function. We set .h(G) = h(d(G)) for all .G ∈ T (d). Then the outer measure constructed from the pre-measure h, defined on .T (d), by Method II is called the Hausdorff measure corresponding to the Hausdorff function h and is denoted by .μh . Theorem 3.24 confirms that the Hausdorff measure .μh is an outer measure on the metric space X as described in the definition above. Moreover, it immediately follows from Theorems 3.26 and 3.30 that all Borel sets are .μh -measurable. Theorem 3.41. Let .(X, d) be a metric space and let .μh be the Hausdorff measure corresponding to a Hausdorff function .h ∈ H. Then all Borel subsets of X are h .μ -measurable. For any .n ∈ N, the function .h : [0, ∞) → [0, ∞), defined by .h(t) = tn for all .t ≥ 0, is a Hausdorff function. Definition 3.42. Let .(X, d) be a metric space and let .h : [0, ∞) → [0, ∞) be defined by .h(t) = tn for all .t ≥ 0, where n is a positive integer. The Hausdorff measure corresponding to the Hausdorff function h is denoted by .μ(n) and it is called the n-dimensional Hausdorff measure or the .(n)-measure.

3.6. HAUSDORFF MEASURES

87

The one-dimensional Hausdorff measure of a simple curve in .Rn is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesguemeasurable subset of .R2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of length, area, and volume. Indeed, there are d-dimensional Hausdorff measures for every .d ≥ 0, where d need not necessarily be an integer. These measures are fundamental in the geometric measure theory. Now we will establish the relationship between .(n)-measure .μ(n) and Lebesgue measure .μ∗ in the Euclidean space .Rn . Theorem 3.43. For any .n ∈ N, there exists a real constant .κn with .0 < κn < ∞ such that μ(n) (E) = κn μ∗ (E)

.

for all subsets E of the Euclidean space .Rn . Proof. .(a) We consider the cube .C0 of all points .x = (x1 , x2 , . . . , xn ) of .Rn with 0 ≤ xi < 1 (for i ∈ {1, 2, . . . , n}). √ For any .δ > 0, we select an .N ∈ N with .N > 1δ n such that the cube .C0 is covered by the collection of .N n smaller cubes of all points .x = (x1 , x2 , . . . , xn ) of .Rn with .

.

ri − 1 ri ≤ xi < (for i ∈ {1, 2, . . . , n}), N N

where .r1 , r2 , . . . , rn ∈ {1, 2, . . . , N }. √ Since each of these smaller cubes has the diameter . N1 n which is less than .δ, it holds that  N N  N    1√ n √ n (n) .μ ··· n = n . δ (C0 ) ≤ N r1 =1 r2 =1

rn =1

Thus, we have μ(n) (C0 ) ≤

√

.

n

n .

.(b) We assume that .{Si : i ∈ N} is an arbitrary covering of .C0 . For every i ∈ N, we select a closed cube .Ci containing .Si with the edge of .Ci equal to twice the diameter of .Si . Then, we have

.

C0 ⊂

∞ 

.

i=1

Si ⊂

∞  i=1

Ci

CHAPTER 3. MEASURE THEORY

88 and

 ∗

∞ 

∗

1 = μ (C0 ) ≤ μ

.

 Ci

i=1

≤

∞ 

∗

μ (Ci ) =

i=1

∞ 

n 2d(Si ) .

i=1

Hence, we get ∞  .

d(Si )n ≥

i=1

1 . 2n

Since the inequality above is true for all coverings .{Si : i ∈ N} of .C0 , we conclude that μ(n) (C0 ) ≥

.

1 . 2n

(c) We now define

.

κn = μ(n) (C0 )

.

and we have .

√ n 1 ≤ κn ≤ n . n 2

Then, since .μ∗ (C0 ) = 1, we get μ(n) (C0 ) = κn μ∗ (C0 ).

.

Since .μ(n) and .μ∗ are invariant under translation and they are homogeneous of degree n under similarity transformations, it holds that μ(n) (C) = κn μ∗ (C)

.

(3.20)

for any cube C of all points .x = (x1 , x2 , . . . , xn ) of .Rn with ai ≤ xi < ai + s (for i ∈ {1, 2, . . . , n}),

.

where .ai ∈ R for all .i ∈ {1, 2, . . . , n} and s is an arbitrary real number with .0 < s < ∞. .(d) We now choose a special collection .C of all cubes of all points .x = (x1 , x2 , . . . , xn ) of .Rn with .

ri − 1 ri ≤ xi < k (for k ∈ N0 , ri ∈ Z, i ∈ {1, 2, . . . , n}). k 2 2

3.6. HAUSDORFF MEASURES

89

Each cube in .C has the property that if two cubes have a point in common, one is contained in the other. n lies in a cube C of .C .(e) Each point g of an arbitrary open subset G of .R that is contained in G and so in a cube C of .C which is maximal in the sense that it is a cube C of .C contained in G but not contained in any larger cube .C  of .C contained in G. Therefore, G is the union of the maximal cubes of .C contained in G. According to the property mentioned in .(d), these maximal cubes are obviously disjoint. Hence, we have G=

∞ 

.

Ci ,

i=1

where the .Ci ’s are disjoint cubes in .C. Since all Borel subsets of .Rn are measurable for both .μ(n) and .μ∗ by Theorems 3.26 and 3.30, it follows from (3.20) that μ

.

(n)

(G) =

∞ 

μ

(n)

(Ci ) = κn

i=1

∞ 

μ∗ (Ci ) = κn μ∗ (G)

(3.21)

i=1

for any open subset G of .Rn . n (n) (H) = ∞ = μ∗ (H), .(f ) Assume that H is an arbitrary .Gδ -set in .R . If .μ then μ(n) (H) = κn μ∗ (H).

.

If .μ(n) (H) < ∞, then we can cover H by a sequence .{Gi }i∈N of open subsets of .Rn with ∞  .

d(Gi )n < ∞.

i=1

As we did in .(b) (but using open cubes instead of closed cubes), we replace the open sets .Gi with open cubes .Ci of edge twice the diameter of .Gi and construct an open covering .{Ci : i ∈ N} of H such that  μ∗

∞ 

.

 Ci

< ∞.

i=1

On the other hand, when .μ∗ (H) < ∞, H can be covered by an open subset of .Rn with finite .μ∗ -measure. It follows from (3.21) that open sets of finite

CHAPTER 3. MEASURE THEORY

90

μ∗ -measure have finite .μ(n) -measure. Hence, when H is a .Gδ -set with either (n) (H) < ∞ or .μ∗ (H) < ∞, H can be expressed as .μ .

H=

∞ 

.

Gi ,

i=1

where .{Gi }i∈N is a non-increasing sequence of measurable open sets of finite measure for both .μ(n) and .μ∗ . Thus, by (3.21) and Theorem 3.17 .(ii), we have     (n) .μ (H) = inf μ(n) (Gi ) : i ∈ N = κn inf μ∗ (Gi ) : i ∈ N = κn μ∗ (H) for all .Gδ -sets H. (n) and .μ∗ are .G -regular outer measures. If .(g) According to Corollary 3.32, .μ δ n E is an arbitrary subset of .R , we can choose .Gδ -sets .H1 and .H2 such that E ⊂ H1 , μ(n) (E) = μ(n) (H1 ), E ⊂ H2 , μ∗ (E) = μ∗ (H2 ).

.

Therefore, we conclude that μ(n) (E) = μ(n) (H1 ∩ H2 ) = κn μ∗ (H1 ∩ H2 ) = κn μ∗ (E),

.

which completes the proof.

2

.

Perhaps the most apt description of the relationship between .μ(n) and .μ∗ is that the former assigns unit measure to the ball of unit diameter, while the latter assigns unit measure to the n-cube of unit edge. It is well known that .μ(n) (E) = κn μ∗ (E) for any subset E of the Euclidean space .Rn , where     2 n n √ .κn = Γ 1+ 2 π for all .n ∈ N.

Chapter 4

Extension of Isometries In this chapter, we define the first- and second-order generalized spans and the index set, examine their properties, and apply them to the study of the extension of isometries. To this end, we develop a theory that extends the domain of local isometries to the generalized spans, where we call an isometry defined in a subset of a Hilbert space a local isometry. In addition, we prove that the domain of a local isometry can be extended to any real Hilbert space, where the domain of a local isometry does not have to be a convex body or an open set. To this purpose, among many references, S.-M. Jung’s paper [9] is mainly cited.

4.1 Basic Concepts and Remarks Throughout this chapter, using the symbol .Rω , we represent an infinite-dimensional real vector space defined as   Rω = (x1 , x2 , . . .) : xi ∈ R for all i ∈ N .

.

From now on, we denote by .(Rω , T ) the product space .

∞ 

i=1

R, where .(R, TR ) is

the usual topological space. Since .(R, TR ) is a Hausdorff space, it follows from Theorem 1.35 that the product space .(Rω , T ) is also a Hausdorff space. ∞  Let .I ω = I be the Hilbert cube, where .I = [0, 1] is the unit closed interval. i=1

We denote by .(I ω , Tω ) the (topological) subspace of .(Rω , T ). Then, .Tω is the relative topology for .I ω induced by .T . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4 4

91

CHAPTER 4. EXTENSION OF ISOMETRIES

92

In this book, let .a = {ai }i∈N be a sequence of positive real numbers satisfying the condition ∞  .

a2i < ∞.

(4.1)

i=1

Using this sequence .a = {ai }i∈N , we define the metric on .I ω by  da (x, y) =

∞ 

.

1/2 a2i (xi − yi )2

(4.2)

i=1

for all .x = (x1 , x2 , . . .) ∈ I ω and .y = (y1 , y2 , . . .) ∈ I ω . Remark 4.1. It is to be noted that (i) .da is consistent with the topology .Tω and it is translation-invariant .(see [17].).

.

(ii) .(I ω , Tω ) is a Hausdorff space as a subspace of the Hausdorff space .(Rω , T ).

.

(iii) .(I ω , Tω ) is a compact subspace of .(Rω , T ) by Tychonoff’s theorem.

.

We define  Ma =

.

(x1 , x2 , . . .) ∈ Rω :

∞ 

a2i x2i < ∞ ,

i=1

where .a = {ai }i∈N is a sequence of positive real numbers that satisfies the condition (4.1). Then .Ma is a vector space over .R, and we can define an inner product .·, ·a on .Ma by x, ya =

∞ 

.

a2i xi yi

i=1

for all .x = (x1 , x2 , . . .) and .y = (y1 , y2 , . . .) of .Ma . This inner product induces the norm

.xa = x, xa for all .x ∈ Ma . Remark 4.2. .Ma is the set of all elements .x ∈ Rω satisfying .x2a < ∞, i.e.,   .Ma = (x1 , x2 , . . .) ∈ Rω : x2a < ∞ .

4.1. BASIC CONCEPTS AND REMARKS

93

In view of definition (4.2), the metric .da on .I ω can be extended to the metric on .Ma , i.e.,

.da (x, y) = x − y, x − ya for all .x, y ∈ Ma . Similarly as [10, Theorem 2.1] and [13, Theorem 70.4], we prove the following theorem. Theorem 4.3. Assume that the sequence .a = {ai }i∈N satisfies the condition (4.1). The inner product space .(Ma , ·, ·a ) is complete, i.e., it is a real Hilbert space. Proof. Assume that .{xi }i∈N is an arbitrary Cauchy sequence in .Ma , where we set xi = (xi,1 , xi,2 , xi,3 , . . .) ∈ Ma

.

for all .i ∈ N. .(a) Since .{xi }i∈N is a Cauchy sequence in .Ma , for any .ε > 0, there exists an .Nε ∈ N such that .da (xm , xn ) < ε for all integers .m, n ≥ Nε . If we fix a .j ∈ N, it follows from the above fact that 1/2  ∞    .aj xm,j − aj xn,j  ≤ a2 (xm,k − xn,k )2 = da (xm , xn ) < ε k

k=1

for all integers .m, n ≥ Nε , where .aj is a fixed positive real number as the jth term of the sequence .a = {ai }i∈N . This consideration implies that .{xi,j }i∈N is a Cauchy sequence in .R for each fixed .j ∈ N. Since .R is complete, there exists a real number .yj such that yj = lim xi,j

.

i→∞

(4.3)

for every .j ∈ N. We now set y = (y1 , y2 , y3 , . . .) ∈ Rω .

.

.(b) Since .{xi }i∈N is a Cauchy sequence in .Ma , for any .ε > 0, there exists an Nε ∈ N such that .da (xn , xn+p ) < ε for any .n, p ∈ N with .n ≥ Nε . Then, this fact implies that

.

∞  .

a2k (xn,k − xn+p,k )2 = da (xn , xn+p )2 < ε2

k=1

and hence, we have m  .

k=1

a2k (xn,k − xn+p,k )2 < ε2

(4.4)

CHAPTER 4. EXTENSION OF ISOMETRIES

94

for all .m, n, p ∈ N with .n ≥ Nε . By (4.3) and (4.4), we get m  .

a2k (xn,k − yk )2 = lim

p→∞

k=1

m 

a2k (xn,k − xn+p,k )2 ≤ ε2

k=1

for any .m, n ∈ N with .n ≥ Nε . Hence, we see that xn −

.

y2a

2

= da (xn , y) =

∞ 

a2k (xn,k − yk )2 ≤ ε2

(4.5)

k=1

for all integers .n ≥ Nε , i.e., we see by Remark 4.2 that .xn − y ∈ Ma for each integer .n ≥ Nε and consequently y ∈ Ma .

.

(4.6)

.(c) Finally, it follows from (4.5) that, for each .ε > 0, there exists an .Nε ∈ N such that .xn − ya ≤ ε for any integer .n ≥ Nε . In view of (4.6), this fact implies that each Cauchy sequence in .Ma converges in .Ma , i.e., the inner product space .Ma is complete. .2

What follows is a basic definition we are familiar with, but for the sake of completeness of the book, we now define precisely the .da -isometry between subsets of .Ma . Definition 4.4. Let .E1 and .E2 be nonempty subsets of .Ma . (i) A function .f : E1 → E2 is called a .da -isometry provided .da (f (x), f (y)) = da (x, y) for all .x, y ∈ E1 .

.

(ii) .E1 is said to be .da -isometric to .E2 provided there exists a surjective .da isometry .f : E1 → E2 .

.

Let .(Ma , Ta ) be the topological space generated by the metric .da . In view of Remark 4.1 .(ii), (iii) and using Theorem 1.43, it is easy to prove the following remarks. In view of Theorem 1.35, the topological space .(Rω , T ) is a Hausdorff space. Since .[−c, c]ω ⊂ Ma for any fixed .c > 0, we may consider the families of open sets which are included in .Ma only to prove Remark 4.5 .(iii). This idea, together with Remark 4.1 .(iii), implies the validity of Remark 4.5 .(iii). Remark 4.5. We note that (i) .(Ma , ·, ·a ) is a Hilbert space over .R.

.

(ii) .(Ma , Ta ) is a Hausdorff space as a subspace of the Hausdorff space .(Rω , T ).

.

4.2. FIRST-ORDER GENERALIZED SPAN

95

(iii) .(I ω , Tω ) is a compact subspace of .(Ma , Ta ).

.

(iv) .(I ω , Tω ) is a closed subset of .(Ma , Ta ).

.

Definition 4.6. Given .c ∈ Ma the translation by c is the mapping .Tc : Ma → Ma defined by .Tc (x) = x + c for all .x ∈ Ma .

4.2 First-Order Generalized Span In [10, Theorem 2.5], we were able to extend the domain of a .da -isometry f to the whole space when the domain of f is a non-degenerate basic cylinder (see Definition 4.20 for the exact definition of non-degenerate basic cylinders). However, we shall see in Definition 4.33 and Theorem 4.34 that the domain of a .da -isometry f can be extended to the whole space whenever f is defined on a bounded set which contains more than one element. From now on, it is assumed that E, .E1 , and .E2 are subsets of .Ma , each of them contains more than one element, unless specifically stated for their cardinalities, and that they are bounded because the Hilbert cube .I ω is a bounded subset of .Ma and the involved sets are indeed (Borel) subsets of .I ω in the main theorems. If the set has only one element or no element, this case will not be covered here because the results derived from this case are trivial and uninteresting. Definition 4.7. Assume that E is a nonempty bounded subset of .Ma and p is a fixed element of E. We define the first-order generalized span of E with respect to p as  ∞ m   GS(E, p) = p + αij (xij − p) ∈ Ma : m ∈ N ; i=1 j=1 .



xij ∈ E and αij ∈ R for all i and j .

We remark that if a bounded subset E of .Ma contains more than one element, then E is a proper subset of its first-order generalized span .GS(E, p), because .x = p + (x − p) ∈ GS(E, p) for any .x ∈ E and .p + α(x − p) ∈ GS(E, p) for any .α ∈ R, which implies that .GS(E, p) is unbounded. Moreover, we note that .αx + βy ∈ Ma for all .x, y ∈ Ma and .α, β ∈ R, because .αx + βya ≤ |α|xa + |β|ya < ∞. Therefore, .GS(E, p) − p is a real vector space, because the double sum in the definition of .GS(E, p) guarantees .αx + βy ∈ GS(E, p) − p for all .x, y ∈ GS(E, p) − p and .α, β ∈ R and because .GS(E, p) − p is a subset of a real vector space .Ma (cf. Lemma 4.17 .(i) below).

CHAPTER 4. EXTENSION OF ISOMETRIES

96

We remark that the smallest flat containing E was introduced in [4] for any subset E of an n-dimensional Euclidean space somewhat similarly to the firstorder generalized span as follows:  n n   H(E) = αi xi : n ∈ N; xi ∈ E and αi ∈ R for all i with αi = 1 

.

=

i=1

p+

i=1 n 



αi (xi − p) : n ∈ N; p, xi ∈ E and αi ∈ R for all i .

i=1

Given a .p = (p1 , p2 , . . .) ∈ Ma and an .n ∈ N, if we set   n .Rp = (x1 , x2 , . . .) ∈ Ma : xi ∈ R for 1 ≤ i ≤ n and xi = pi for i > n , then it is easy to see that .H(E) ⊂ GS(E, p) for any set .E ⊂ Rnp . However, it is obvious that .GS(E, p) ⊂ H(E) for some .E ⊂ Rnp . For each .i ∈N, we set .ei = (0, . . . , 0, 1, 0, . . .), where 1 is in the ith position. Then . a1i ei i∈N is a complete orthonormal sequence in .Ma . The following definition introduces  theconcepts of index and .β-index based on the “standard” coordinate system . a1i ei i∈N and another one .{βi }i∈N , respectively. Definition 4.8. Let E be a nonempty subset of .Ma . (i) We define the index set of E by  Λ(E) = i ∈ N : there are an x ∈ E and an α ∈ R \ {0}  . satisfying x + αei ∈ E .

.

Each .i ∈ Λ(E) is called an index of E. If .Λ(E) = N, then the set E is called degenerate. Otherwise, E is called non-degenerate. (ii) Let .β = {βi }i∈N be another complete orthonormal sequence in .Ma . We define the .β-index set of E by  Λβ (E) = i ∈ N : there are an x ∈ E and an α ∈ R \ {0}  . satisfying x + αβi ∈ E .

.

Each .i ∈ Λβ (E) is called a .β-index of E. Here we should not be confused with the terminologies “index” and “index set” used in Chap. 1. Because the meanings used in Chap. 1 are very different, there will be no confusion even if we use the terms in Definition 4.8. Therefore, we will continue to use the terminologies given in the above definition.

4.2. FIRST-ORDER GENERALIZED SPAN

97

We will find that the concept of index set in Hilbert space sometimes takes over the role that the concept of dimension plays in vector space. According to the definition above, if i is an index of E, i.e., .i ∈ Λ(E), then there are .x ∈ E and .x + αei ∈ E for some .α = 0. Since .x =  x + αei , we remark that if .Λ(E) = ∅, then the set E contains at least two elements. In the following lemma, we prove that if i is an index of E and .p ∈ E, then the first-order generalized span .GS(E, p) contains the line through p in the direction .ei . Lemma 4.9. Let .β = {βi }i∈N be a complete orthonormal sequence in .Ma . Assume that E is a bounded subset of .Ma and .GS(E, p) is the first-order generalized span of E with respect to a fixed element .p ∈ E. If .i ∈ Λβ (E), then .p+αβi ∈ GS(E, p) for all .α ∈ R. Proof. By Definition 4.8 .(ii), if .i ∈ Λβ (E) then there exists an .x ∈ E and an α0 = 0, which satisfy .x + α0 βi ∈ E. Since .x ∈ E and .x + α0 βi ∈ E, by Definition 4.7, we get

.

p + α0 γβi = p + γ(x + α0 βi − p) − γ(x − p) ∈ GS(E, p)

.

for all .γ ∈ R. Setting .α = α0 γ in the above relation, we obtain .p+αβi ∈ GS(E, p) for any .α ∈ R. .2 We now introduce a lemma, which is a generalized version of [10, Lemma 2.3] and whose proof runs in the same way. We prove that the function .T−q ◦ f ◦ Tp : E1 −p → E2 −q preserves the inner product. This property is important for proving several theorems in this chapter as a necessary condition for f to be a .da -isometry. Lemma 4.10. Assume that .E1 and .E2 are bounded subsets of .Ma that are .da isometric to each other via a surjective .da -isometry .f : E1 → E2 . Assume that p is an element of .E1 and q is an element of .E2 with .q = f (p). Then the function .T−q ◦ f ◦ Tp : E1 − p → E2 − q preserves the inner product, i.e.,

. (T−q ◦ f ◦ Tp )(x − p), (T−q ◦ f ◦ Tp )(y − p) = x − p, y − pa a for all .x, y ∈ E1 . Proof. Since .T−q ◦ f ◦ Tp : E1 − p → E2 − q is a .da -isometry, we have (T−q ◦ f ◦ Tp )(x − p) − (T−q ◦ f ◦ Tp )(y − p)2a = (x − p) − (y − p)2a

.

for any .x, y ∈ E1 . If we put .y = p in the last equality, then we get (T−q ◦ f ◦ Tp )(x − p)2a = x − p2a

.

CHAPTER 4. EXTENSION OF ISOMETRIES

98

for each .x ∈ E1 . Moreover, it follows from the previous equality that (T−q ◦ f ◦ Tp )(x − p) − (T−q ◦ f ◦ Tp )(y − p)2a = (T−q ◦ f ◦ Tp )(x − p) − (T−q ◦ f ◦ Tp )(y − p),

. (T−q ◦ f ◦ Tp )(x − p) − (T−q ◦ f ◦ Tp )(y − p) a

= x − p2a − 2 (T−q ◦ f ◦ Tp )(x − p), (T−q ◦ f ◦ Tp )(y − p) a + y − p2a and .



(x − p) − (y − p)2a = (x − p) − (y − p), (x − p) − (y − p) a = x − p2a − 2x − p, y − pa + y − p2a .

Finally, comparing the last two equalities yields the validity of our assertion.

2

.

4.3 First-Order Extension of Isometries In the last section, we made all the necessary preparations to extend the domain .E1 of the surjective .da -isometry .f : E1 → E2 to its first-order generalized span .GS(E1 , p). Although .E1 is a bounded subset of .Ma , the translation of its first-order generalized span by .−p, .GS(E1 , p) − p, is a real vector space. Now we will extend the .da -isometry .T−q ◦ f ◦ Tp defined on the bounded set .E1 − p to the .da -isometry .T−q ◦ F ◦ Tp defined on the vector space .GS(E1 , p) − p. Comparing their “sizes” of .E1 − p and .GS(E1 , p) − p, or considering that .GS(E1 , p) − p is an algebraically closed space, it is a great achievement to extend the .da -isometry .T−q ◦ f ◦ Tp defined on the bounded set .E1 − p to the .da -isometry defined on the vector space .GS(E1 , p) − p. Definition 4.11. Assume that .E1 and .E2 are nonempty bounded subsets of .Ma that are .da -isometric to each other via a surjective .da -isometry .f : E1 → E2 . Let p be a fixed element of .E1 and let q be an element of .E2 that satisfies .q = f (p). We define a function .F : GS(E1 , p) → Ma as  m ∞   (T−q ◦ F ◦ Tp ) αij (xij − p) i=1 j=1 .

=

∞ m  

αij (T−q ◦ f ◦ Tp )(xij − p)

i=1 j=1

for any .m ∈ N, .xij ∈ E1 , and for all .αij ∈ R satisfying .

m  ∞ 

i=1 j=1

αij (xij −p) ∈ Ma .

4.3. FIRST-ORDER EXTENSION OF ISOMETRIES

99

We note that in the definition above, it is important for the argument of .T−q ◦ F ◦ Tp to belong to .Ma . Now we show that the function .F : GS(E1 , p) → Ma is well defined. Lemma 4.12. Assume that .E1 and .E2 are bounded subsets of .Ma that are .da isometric to each other via a surjective .da -isometry .f : E1 → E2 . Let p be an element of .E1 and let q be an element of .E2 that satisfy .q = f (p). The function .F : GS(E1 , p) → Ma given in Definition 4.11 is well defined. Proof. First, we will check that the range of F is a subset of .Ma . For any m, n1 , n2 ∈ N with .n2 > n1 , .xij ∈ E1 , and for all .αij ∈ R, it follows from Lemma 4.10 that 2  m n n1 m  2      αij (T−q ◦ f ◦ Tp )(xij − p) − αij (T−q ◦ f ◦ Tp )(xij − p)    i=1 j=1 i=1 j=1 a  m n2   = αij (T−q ◦ f ◦ Tp )(xij − p),

.

i=1 j=n1 +1 n2 m  

.

=

 αk (T−q ◦ f ◦ Tp )(xk − p)

k=1 =n1 +1 n2 m m  

a

αij

i=1 k=1 j=n1 +1 n2  

αk (T−q ◦ f ◦ Tp )(xij − p), (T−q ◦ f ◦ Tp )(xk − p)

=

=n1 +1 n2 m m   i=1 k=1 j=n1 +1 n2 m  

 =

αij

n2 

a

αk xij − p, xk − p a

=n1 +1

αij (xij − p),

i=1 j=n1 +1



n2 m  

 αk (xk − p)

k=1 =n1 +1

. a

Hence, we have  m n 2 n1 m  2      αij (T−q ◦ f ◦ Tp )(xij − p) − αij (T−q ◦ f ◦ Tp )(xij − p)    i=1 j=1 i=1 j=1 a  2 n m 2      = αij (xij − p) . (4.7)   i=1 j=n1 +1 a  m n 2 n1 m  2      = αij (xij − p) − αij (xij − p) .   i=1 j=1

i=1 j=1

a

CHAPTER 4. EXTENSION OF ISOMETRIES

100

Indeed, the equality (4.7) holds for all .m, n1 , n2 ∈ N. m  ∞  We now assume that . αij (xij −p) ∈ Ma for some .xij ∈ E1 and .αij ∈ R, i=1 j=1

where m is a fixed positive integer. Since .(Ma , Ta ) is a Hausdorff space on account of Remark 4.5 .(ii) and the topology .Ta is consistent with the metric .da and with the   m  n αij (xij − p) converges norm . · a (cf. Remark 4.1 .(i)), the sequence . n i=1 j=1   m  m  ∞ n  to . αij (xij − p) (in .Ma ) and hence, the sequence . αij (xij − p) i=1 j=1

n

i=1 j=1

is a Cauchy sequence in .Ma . We know by (4.7) and the definition of Cauchy sequences that for each .ε > 0 there exists an integer .Nε > 0 such that  m n  n1 m  2      α (T ◦ f ◦ T )(x − p) − α (T ◦ f ◦ T )(x − p)   ij −q p ij ij −q p ij   i=1 j=1 i=1 j=1 a   . n2 n m  m 1      = αij (xij − p) − αij (xij − p) < ε   i=1 j=1

i=1 j=1

a

 m  n

for all .n1 , n2 > Nε , which implies that .

i=1 j=1

αij (T−q ◦ f ◦ Tp )(xij − p)

 n

is also a Cauchy sequence in .Ma . As we proved in [10, Theorem 2.1] or by Remark 4.5 .(i), we observe that .(Ma , ·, ·a ) is a real Hilbert space when the sequence .a = {ai }i∈N satisfies the condition (4.1). Thus, .Ma is complete, so the Cauchy   m  n sequence . αij (T−q ◦ f ◦ Tp )(xij − p) converges in .Ma , i.e., by Definin

i=1 j=1

tion 4.11, we have



(T−q ◦ F ◦ Tp )

∞ m  

 αij (xij − p)

i=1 j=1

= .

∞ m  

αij (T−q i=1 j=1 n m  

αij (T−q ◦ f ◦ Tp )(xij − p)

= lim

n→∞

◦ f ◦ Tp )(xij − p)

i=1 j=1

∈ Ma , which implies

 F p+

m  ∞ 

.

i=1 j=1

 αij (xij − p)

∈ Ma + q = Ma

4.3. FIRST-ORDER EXTENSION OF ISOMETRIES for all .xij ∈ E1 and .αij ∈ R with .

m  ∞ 

101

αij (xij − p) ∈ Ma , i.e., the image of

i=1 j=1

each element of .GS(E1 , p) under F belongs to .Ma . m m ∞ ∞ 1  2  We now assume that . αij (xij − p) = βij (yij − p) ∈ Ma for i=1 j=1

i=1 j=1

some .m1 , m2 ∈ N, .xij , yij ∈ E1 , and for some .αij , βij ∈ R. It then follows from Definition 4.11 and Lemma 4.10 that   m1   ∞     αij (xij − p)  T−q ◦ F ◦ Tp  i=1 j=1

 m2   ∞ 2     − T−q ◦ F ◦ Tp βij (yij − p)   i=1 j=1 a m 2 m ∞ ∞ 1  2          = αij T−q ◦ f ◦ Tp (xij − p) − βij T−q ◦ f ◦ Tp (yij − p)   i=1 j=1 i=1 j=1 a m ∞ m ∞ 1 2       = αij T−q ◦ f ◦ Tp (xij − p) − βij T−q ◦ f ◦ Tp (yij − p), i=1 j=1 m ∞ 1  

i=1 j=1 m ∞ 2  

k=1 =1 m ∞ 1  

k=1 =1

  αk T−q ◦ f ◦ Tp (xk − p) −

.

=

αij (xij − p) −

i=1 j=1 m1  ∞ 

m2  ∞ 

αk (xk − p) −

a

 βk (yk − p)

m ∞ 2 m2  ∞ 1       = αij (xij − p) − βij (yij − p)   i=1 j=1

a

a

= 0, which implies that  .

T−q ◦ F ◦ Tp 

m ∞ 1   



αij (xij i=1 j=1 m ∞ 2  

= T−q ◦ F ◦ Tp



βk T−q ◦ f ◦ Tp (yk − p)

k=1 =1

i=1 j=1



βij (yij − p),

i=1 j=1 m2  ∞ 

k=1 =1





− p)

βij (yij − p)

i=1 j=1



CHAPTER 4. EXTENSION OF ISOMETRIES

102

for all .m1 , m2 ∈ N, .xij , yij ∈ E1 , and for all .αij , βij ∈ R which satisfy the m m ∞ ∞ 1  2  αij (xij − p) = βij (yij − p) ∈ Ma . .2 condition . i=1 j=1

i=1 j=1

In [6, Theorem 2.2], we were able to extend the domain of a .da -isometry f : J → K to the whole space .Ma when J is a non-degenerate basic cylinder, while we prove in the following theorem that the domain of a .da -isometry .f : E1 → E2 can be extended to the first-order generalized span .GS(E1 , p) whenever .E1 is a nonempty bounded subset of .Ma , whether degenerate or nondegenerate. Therefore, Theorem 4.13 is a generalization of [6, Theorem 2.2]. In the proof, we use the fact that .GS(E1 , p) − p is a real vector space. This fact is self-evident, as briefly mentioned earlier. .

Theorem 4.13. Assume that .E1 and .E2 are bounded subsets of .Ma that are .da isometric to each other via a surjective .da -isometry .f : E1 → E2 . Assume that p is an element of .E1 and q is an element of .E2 with .q = f (p). The function .F : GS(E1 , p) → Ma defined in Definition 4.11 is a .da -isometry and the function .T−q ◦ F ◦ Tp : GS(E1 , p) − p → Ma is a linear .da -isometry. In particular, F is an extension of f . Proof. .(a) Let u and v be arbitrary elements of the first-order generalized span GS(E1 , p) of .E1 with respect to p. Then

.

u−p= .

v−p=

m  ∞  i=1 j=1 ∞ n  

αij (xij − p) ∈ Ma , (4.8) βij (yij − p) ∈ Ma

i=1 j=1

for some .m, n ∈ N, some .xij , yij ∈ E1 , and for some .αij , βij ∈ R. Then, according to Definition 4.11, we have (T−q ◦ F ◦ Tp )(u − p) = .

(T−q ◦ F ◦ Tp )(v − p) =

∞ m   i=1 j=1 ∞ n   i=1 j=1

αij (T−q ◦ f ◦ Tp )(xij − p), (4.9) βij (T−q ◦ f ◦ Tp )(yij − p).

4.3. FIRST-ORDER EXTENSION OF ISOMETRIES

103

(b) By Lemma 4.10, (4.8), and (4.9), we get

(T−q ◦ F ◦ Tp )(u − p), (T−q ◦ F ◦ Tp )(v − p) a  m ∞  = αij (T−q ◦ f ◦ Tp )(xij − p),

.

i=1 j=1 ∞ n  

 βk (T−q ◦ f ◦ Tp )(yk − p)

k=1 =1

=

∞ n  m  

αij

i=1 k=1 j=1

.

∞ 

a



βk (T−q ◦ f ◦ Tp )(xij − p),

=1

(T−q ◦ f ◦ Tp )(yk − p) =

=

∞ n  m  

αij

i=1 k=1 j=1  m ∞ 

∞ 

a

βk xij − p, yk − pa

=1

αij (xij − p),

i=1 j=1

(4.10)



∞ n  

 βk (yk − p)

k=1 =1

a

= u − p, v − pa for all .u, v ∈ GS(E1 , p). That is, .T−q ◦ F ◦ Tp preserves the inner product. Indeed, equality (4.10) is an extended version of Lemma 4.10. .(c) By using equality (4.10), we further obtain  2 da F (u), F (v)

.

= F (u) − F (v)2a  2 = (T−q ◦ F ◦ Tp )(u − p) − (T−q ◦ F ◦ Tp )(v − p)a = (T−q ◦ F ◦ Tp )(u − p) − (T−q ◦ F ◦ Tp )(v − p),

(T−q ◦ F ◦ Tp )(u − p) − (T−q ◦ F ◦ Tp )(v − p) a = u − p, u − pa − u − p, v − pa − v − p, u − pa + v − p, v − pa

= (u − p) − (v − p), (u − p) − (v − p) a = (u − p) − (v − p)2a = u − v2a = da (u, v)2

for all .u, v ∈ GS(E1 , p), i.e., F is a .da -isometry. .(d) Now, let u and v be arbitrary elements of .GS(E1 , p). Then, it holds that .u − p ∈ GS(E1 , p) − p, .v − p ∈ GS(E1 , p) − p, and .α(u − p) + β(v − p) ∈ GS(E1 , p) − p for any .α, β ∈ R, because .GS(E1 , p) − p is a real vector space.

CHAPTER 4. EXTENSION OF ISOMETRIES

104

We get    (T−q ◦ F ◦ Tp ) α(u − p) + β(v − p)

2 − α(T−q ◦ F ◦ Tp )(u − p) − β(T−q ◦ F ◦ Tp )(v − p)a    = (T−q ◦ F ◦ Tp ) α(u − p) + β(v − p)

.

− α(T−q ◦ F ◦ Tp )(u − p) − β(T−q ◦ F ◦ Tp )(v − p),   (T−q ◦ F ◦ Tp ) α(u − p) + β(v − p)  − α(T−q ◦ F ◦ Tp )(u − p) − β(T−q ◦ F ◦ Tp )(v − p) . a

Since .α(u − p) + β(v − p) = w − p for some .w ∈ GS(E1 , p), we further use (4.10) to obtain    (T−q ◦ F ◦ Tp ) α(u − p) + β(v − p) 2 − α(T−q ◦ F ◦ Tp )(u − p) − β(T−q ◦ F ◦ Tp )(v − p)a .

= w − p, w − pa − αw − p, u − pa − βw − p, v − pa − αu − p, w − pa + α2 u − p, u − pa + αβu − p, v − pa − βv − p, w − pa + αβv − p, u − pa + β 2 v − p, v − pa = 0,

which implies that the function .T−q ◦ F ◦ Tp : GS(E1 , p) − p → Ma is linear. .(e) Finally, we set .α11 = 1, .αij = 0 for any .(i, j) = (1, 1), and .x11 = x in (4.8) and (4.9) to see (T−q ◦ F ◦ Tp )(x − p) = (T−q ◦ f ◦ Tp )(x − p)

.

for every .x ∈ E1 , which implies that .F (x) = f (x) for every .x ∈ E1 , i.e., F is an .2 extension of f .

4.4 Second-Order Generalized Span For any element x of .Ma and .r > 0, we denote by .Br (x) the open ball defined by Br (x) = {y ∈ Ma : y − xa < r}. Definitions 4.7 and 4.11 will be generalized to the cases of .n ≥ 2 in the following definition. We introduce the concept of nth-order generalized span .GSn (E1 , p), which generalizes the concept of first-order generalized span .GS(E1 , p). Moreover, we define the .da -isometry .Fn which extends the domain .E1 of a .da -isometry f to .GSn (E1 , p).

.

4.4. SECOND-ORDER GENERALIZED SPAN

105

It is surprising, however, that this process of generalization does not go far. Indeed, we will find in Proposition 4.18 and Theorem 4.28 that .GS2 (E1 , p) and .F2 are their limits. Definition 4.14. Let .E1 be a nonempty bounded subset of .Ma that is .da -isometric to a subset .E2 of .Ma via a surjective .da -isometry .f : E1 → E2 . Let p be an element of .E1 and q an element of .E2 with .q = f (p). Assume that r is a positive real number satisfying .E1 ⊂ Br (p). (i) We define .GS0 (E1 , p) = E1 and .GS1 (E1 , p) = GS(E1 , p). In general, we define the nth-order generalized span of .E1 with respect to p as .GSn (E1 , p) = GS(GSn−1 (E1 , p) ∩ Br (p), p) for all .n ∈ N.

.

(ii) We define .F0 = f and .F1 = F , where F is defined in Definition 4.11. Moreover, for any .n ∈ N, we define the function .Fn : GSn (E1 , p) → Ma by  m ∞   (T−q ◦ Fn ◦ Tp ) αij (xij − p)

.

i=1 j=1 .

=

m  ∞ 

αij (T−q ◦ Fn−1 ◦ Tp )(xij − p)

i=1 j=1

for all .m ∈ N, .xij ∈ GSn−1 (E1 , p) ∩ Br (p), and .αij ∈ R which satisfy the m  ∞  αij (xij − p) ∈ Ma . condition . i=1 j=1

Let E be a nonempty bounded subset of .Ma . It is not difficult to show that GSn (E, p) − p is a real vector space for each .n ∈ N: Obviously, .GS(E, p) − p is a real vector space. We assume that .GSn (E, p) − p is a real vector space for some .n ∈ N and .u, v are arbitrary elements of .GSn+1 (E, p) − p. Then there exist n .m1 , m2 ∈ N, .αij , βij ∈ R and .xij , yij ∈ GS (E, p) ∩ Br (p), where r is some positive real constant with .E ⊂ Br (p), such that .

u=

m1  ∞ 

.

αij (xij − p) and v =

i=1 j=1

m2  ∞ 

βij (yij − p).

i=1 j=1

Further, for any .α, β ∈ R, we see that αu + βv = .

m1  ∞  i=1 j=1 n+1

∈ GS

ααij (xij − p) +

(E, p) − p,

m2  ∞  i=1 j=1

ββij (yij − p)

CHAPTER 4. EXTENSION OF ISOMETRIES

106

which implies that .GSn+1 (E, p) − p is a real vector space as a subspace of the real vector space .Ma . Thus we see by induction conclusion that .GSn (E, p) − p is a real vector space for every .n ∈ N. Proposition 4.15. Assume that E is a nonempty bounded subset of .Ma and .p ∈ E. If s and t are positive real numbers that satisfy .E ⊂ Bs (p) ∩ Bt (p), then     GS GSn (E, p) ∩ Bs (p), p = GS GSn (E, p) ∩ Bt (p), p

.

for all .n ∈ N. Proof. Assume that .0 < s < t. Then, there exists a real number .c > 1 with s > ct and it is obvious that .Bt/c (p) ⊂ Bs (p). Assume that x is an arbitrary element of .GS(GSn (E, p) ∩ Bt (p), p). Then there exist some .m ∈ N, some .uij ∈ m  ∞  αij (uij −p) ∈ Ma . GSn (E, p)∩Bt (p) and some .αij ∈ R such that .x = p+

.

i=1 j=1

We note that       n GS (E, p) − p ∩ Bt (p) − p = u − p ∈ Ma : u ∈ GSn (E, p) ∩ Bt (p) .

.

Since .GSn (E, p) − p is a real vector space, . ct < s, and since .uij − p ∈ (GSn (E, p) − p) ∩ (Bt (p) − p) for any i and j, we have .

1 (uij − p) ∈ (GSn (E, p) − p) ∩ (Bs (p) − p). c

Hence, we can choose a .vij ∈ GSn (E, p) ∩ Bs (p) such that . 1c (uij − p) = vij − p. Thus, we get x=p+ .

m  ∞  i=1 j=1 n

αij (uij − p) = p +

m  ∞ 

cαij (vij − p)

i=1 j=1

∈ GS(GS (E, p) ∩ Bs (p), p), which implies that .GS(GSn (E, p) ∩ Bt (p), p) ⊂ GS(GSn (E, p) ∩ Bs (p), p). The reverse inclusion is obvious, since .Bs (p) ⊂ Bt (p). .2 Proposition 4.15 guarantees the logical soundness of Definition 4.14 .(i). We now generalize Lemma 4.10 and formula (4.10) in the following lemma. Indeed, we prove that the function .T−q ◦Fn ◦Tp : GSn (E1 , p)−p → Ma preserves the inner product. This property is important in proving the following theorems as a necessary condition for .Fn to be a .da -isometry.

4.4. SECOND-ORDER GENERALIZED SPAN

107

Lemma 4.16. Let .E1 be a bounded subset of .Ma that is .da -isometric to a subset E2 of .Ma via a surjective .da -isometry .f : E1 → E2 . Assume that p and q are elements of .E1 and .E2 , which satisfy .q = f (p). If .n ∈ N, then

. (T−q ◦ Fn ◦ Tp )(u − p), (T−q ◦ Fn ◦ Tp )(v − p) = u − p, v − pa a

.

for all .u, v ∈ GSn (E1 , p). Proof. Our assertion for .n = 1 was already proved in (4.10). Considering Proposition 4.15, assume that r is a positive real number satisfying .E1 ⊂ Br (p). Now we assume that the assertion is true for some .n ∈ N. Let .u, v be arbitrary elements of n+1 .GS (E1 , p). Then there exist some .m1 , m2 ∈ N, .xij , yk ∈ GSn (E1 , p)∩Br (p), and some .αij , βk ∈ R such that u−p=

m1  ∞ 

.

αij (xij − p) ∈ Ma and v − p =

i=1 j=1

m2  ∞ 

βk (yk − p) ∈ Ma .

k=1 =1

Using Definition 4.14 .(ii) and our assumption, we get

(T−q ◦ Fn+1 ◦ Tp )(u − p), (T−q ◦ Fn+1 ◦ Tp )(v − p) a  m ∞  m2  ∞ 1    = αij (T−q ◦ Fn ◦ Tp )(xij − p), βk (T−q ◦ Fn ◦ Tp )(yk − p) = .

=

=

i=1 j=1 m2  m ∞ 1 

k=1 =1

αij

i=1 k=1 j=1 m2  m1  ∞ 

∞ 

αij

i=1 k=1 j=1  m ∞ 1  

βk (T−q ◦ Fn ◦ Tp )(xij − p), (T−q ◦ Fn ◦ Tp )(yk − p) a

=1 ∞ 

βk xij − p, yk − pa

=1

αij (xij − p),

i=1 j=1

a

m2  ∞  k=1 =1

 βk (yk − p) a

= u − p, v − pa for all .u, v ∈ GSn+1 (E1 , p). By mathematical induction, we may then conclude that our assertion is true for all .n ∈ N. .2 When .n = 1 and .p = p , the first assertion in .(i) of the following lemma is self-evident, so we have used that fact several times before, omitting the proof. The assertion .(iv) in the following lemma seems to be related in some way to Proposition 4.15. Lemma 4.17. Assume that E is a bounded subset of .Ma , p and .p are elements of E, and .n ∈ N. Let r be a positive real number satisfying .E ⊂ Br (p).

CHAPTER 4. EXTENSION OF ISOMETRIES

108

(i) .GSn (E, p) − p is a real vector space.

.

(ii) .GSn (E, p) ⊂ GSn+1 (E, p).

.

(iii) .GS2 (E, p) = GS(E, p), where .GS(E, p) is the closure of .GS(E, p) in .Ma .

.

(iv) .Λβ (GSn (E, p)) = Λβ (GSn (E, p) ∩ Br (p)), where .β = {βi }i∈N is a complete orthonormal sequence in .Ma .

.

Proof. .(i) This claim is a generalization of the argument presented between Definition 4.14 and Proposition 4.15. By using Definitions 4.7 and 4.14, we prove that  .GS(E, p)−p is a real vector space. (We can prove similarly for the case of .n > 1.) Given .x, y ∈ GS(E, p)−p , we may choose some .m1 , m2 ∈ N, some .uij , vij ∈ E, m ∞ 1  αij (uij − p) ∈ Ma and and some .αij , βij ∈ R such that .x = (p − p ) + m ∞ 2 

y = (p − p ) +

.

i=1 j=1

βij (vij − p) ∈ Ma . Since .Ma is a real vector space

i=1 j=1

and .GS(E, p) − p is a subspace of .Ma , it holds that .α β

m ∞ 2 

m ∞ 1 

αij (uij − p) +

i=1 j=1

βij (vij − p) ∈ Ma for all .α, β ∈ R.

i=1 j=1

Moreover, we see that  αx + βy =



p + (1 − α − β)(p − p) +

.

+

m2  ∞ 

 ββij (vij − p)

m1  ∞ 

ααij (uij − p)

i=1 j=1

− p

i=1 j=1

∈ GS(E, p) − p for all .α, β ∈ R. Hence, .GS(E, p) − p is a real vector space as a subspace of real vector space .Ma . n .(ii) Let r be a positive real number with .E ⊂ Br (p). If .x ∈ GS (E, p) for some .n ∈ N, then .x − p ∈ GSn (E, p) − p. Since .GSn (E, p) − p is a real vector space by .(i) and .Br (p) − p = Br (0), we can choose a (sufficiently small) real number .μ = 0 such that .μ(x − p) ∈ (GSn (E, p) − p) ∩ (Br (p) − p). We notice that  n    GS (E, p) − p ∩ Br (p) − p   (4.11) . = v − p ∈ Ma : v ∈ GSn (E, p) ∩ Br (p) . Thus, we see that .μ(x − p) = v − p for some .v ∈ GSn (E, p) ∩ Br (p). Since 1 n+1 .x = p + (E, p). Therefore, we conclude that μ (v − p), it holds that .x ∈ GS n n+1 .GS (E, p) ⊂ GS (E, p) for every .n ∈ N.

4.4. SECOND-ORDER GENERALIZED SPAN

109

.(iii) Let x be an arbitrary element of .GS(E, p). Then there exists some sequence {xn } that converges to x, where .xn ∈ GS(E, p) \ {x} for all .n ∈ N. We now set .y1 = x1 and .yi = xi − xi−1 for each integer .i ≥ 2. Then we have .

xn =

n 

.

yi ,

i=1

where .yi = (xi − p) − (xi−1 − p) ∈ GS(E, p) − p for .i ≥ 2. Since .GS(E, p) − p is a real vector space and .Br (p) − p = Br (0), we can select a real number .μi = 0 such that μi yi ∈ GS(E, p) − p and μi yi ∈ Br (p) − p

.

for every integer .i ≥ 2. Thus, it follows from (4.11) that xn =

n 

.

yi = y 1 +

i=1

n n   1 1 (μi yi ) = x1 + (vi − p), μi μi i=2

i=2

where .vi ∈ GS(E, p) ∩ Br (p) for .i ≥ 2. Since the sequence .{xn } is assumed to   n  1 converge to x, the sequence . x1 + (v − p) converges to x. Hence, we i μi i=2

n

have x1 +

.

∞  1 (vi − p) = lim xn = x ∈ Ma . n→∞ μi

(4.12)

i=2

(Since .Ma is a Hausdorff space, x is the unique limit point of the sequence .{xn }.) Furthermore, there exists a real number .μ1 = 0 that satisfies .μ1 (x1 − p) ∈ GS(E, p) − p and .μ1 (x1 − p) ∈ Br (p) − p, i.e., .μ1 (x1 − p) ∈ (GS(E, p) − p) ∩ (Br (p) − p). Thus, there exists a .v1 ∈ GS(E, p) ∩ Br (p) such that .μ1 (x1 − p) = v1 − p or .x1 − p = μ11 (v1 − p). Therefore, ∞ ∞   1 1 .x = p + (x1 − p) + (vi − p) = p + (vi − p), μi μi i=2

(4.13)

i=1

where .vi ∈ GS(E, p) ∩ Br (p) for each .i ∈ N. On account of (4.12), it holds that ∞  2 1 . μi (vi − p) ∈ Ma . Thus, by (4.13), we see that .x ∈ GS (E, p), which implies i=1

that .GS(E, p) ⊂ GS2 (E, p). On the other hand, let y be an arbitrary element of .GS2 (E, p). Then there are some .m ∈ N, some .vij ∈ GS(E, p) ∩ Br (p), and some .αij ∈ R such that

CHAPTER 4. EXTENSION OF ISOMETRIES

110 y = p+

∞ m  

.

αij (vij − p) ∈ Ma . Let us define .yn = p +

i=1 j=1

n m  

αij (vij − p) for

i=1 j=1

every .n ∈ N. Since .vij − p ∈ GS(E, p) − p for all i and j and .GS(E, p) − p is a n m   real vector space, we know that .yn − p = αij (vij − p) ∈ GS(E, p) − p and i=1 j=1

hence, .yn ∈ GS(E, p) for all .n ∈ N. Since .GS(E, p) is a Hausdorff space, y is the unique element to which the sequence .{yn }n∈N is convergent. Thus, we see that y =p+

∞ m  

.

αij (vij − p) = lim yn ∈ GS(E, p), n→∞

i=1 j=1

which implies that .GS2 (E, p) ⊂ GS(E, p). n n .(iv) Let .i ∈ Λβ (GS (E, p)). By Definition 4.8 .(ii), there exist .x ∈ GS (E, p) ∞ m   αij (uij − p) for and .α = 0 with .x + αβi ∈ GSn (E, p). Further, .x = p + i=1 j=1

some .m ∈ N, some .uij ∈ GSn−1 (E, p) ∩ Br (p), and for some .αij ∈ R. Since ∞ m   . αij (uij − p) + αβi = x − p + αβi ∈ GSn (E, p) − p, where .GSn (E, p) − p i=1 j=1

is a real vector space and .Br (p) − p = Br (0), it holds that   m ∞  .μ αij (uij − p) + αβi ∈ (GSn (E, p) − p) ∩ (Br (p) − p) i=1 j=1

for any sufficiently small .μ = 0, or equivalently, it follows from (4.11) that   ∞ m   . p + μαij (uij − p) + μαβi ∈ GSn (E, p) ∩ Br (p). (4.14) i=1 j=1

On the other hand, since it holds that

∞ m   .

∞ m   .

αij (uij − p) = x − p ∈ GSn (E, p) − p,

i=1 j=1

μαij (uij − p) ∈ (GSn (E, p) − p) ∩ (Br (p) − p) for any

i=1 j=1

sufficiently small .μ = 0. Thus, it follows from (4.11) that .p +

∞ m  

μαij (uij −

i=1 j=1

p) ∈ GSn (E, p)∩Br (p) for any sufficiently small .μ = 0. Hence, by Definition 4.8 n .(ii) and (4.14), it holds that .i ∈ Λβ (GS (E, p) ∩ Br (p)), which implies that n n .Λβ (GS (E, p)) ⊂ Λβ (GS (E, p) ∩ Br (p)). Obviously, the inverse inclusion is .2 true. As we mentioned earlier, we will see that the second-order generalized span is the last step in this kind of domain extension.

4.4. SECOND-ORDER GENERALIZED SPAN

111

Proposition 4.18. If E is a bounded subset of .Ma and .p ∈ E, then E ⊂ GS(E, p) ⊂ GS(E, p) = GS2 (E, p) = GSn (E, p)

.

for any integer .n ≥ 2. Indeed, .GSn (E, p) − p is a real Hilbert space for .n ≥ 2. Proof. .(a) Since E is bounded, we can choose a real number .r > 0 that satisfies E ⊂ Br (p). Assume that .x ∈ GS3 (E, p). Then there exist some .m0 ∈ N, some m ∞ 0  2 .uij ∈ GS (E, p) ∩ Br (p), and some .αij ∈ R such that .x = p + αij (uij − .

i=1 j=1

p) ∈ Ma . We define .xm = p +

m m 0 

αij (uij − p) for each .m ∈ N. Since .uij ∈

i=1 j=1

GS2 (E, p), there exist some .mij ∈ N, some .vijk ∈ GS(E, p) ∩ Br (p), and some m ∞ ij  .βijk ∈ R such that .uij = p + βijk (vijk − p) ∈ Ma . Hence, it holds that k=1 =1

xm = p +

mij ∞ m0  m   

.

  αij βijk vijk − p ∈ Ma ,

i=1 j=1 k=1 =1

which implies that .xm ∈ GS2 (E, p) for all .m ∈ N. Thus, .{xm } is a sequence in 2 2 2 .GS (E, p) that converges to x. Therefore, .x ∈ GS (E, p) because .GS (E, p) is 3 2 closed. Thus, .GS (E, p) ⊂ GS (E, p). The inverse inclusion is of course true due to Lemma 4.17 .(ii). We have proved that .GS2 (E, p) = GS3 (E, p). .(b) According to Definition 4.14, we have  n−1  n .GS (E, p) = GS GS (E, p) ∩ Br (p), p for all .n ∈ N. If we replace n with .n + 1 in the above definition, then   n+1 .GS (E, p) = GS GSn (E, p) ∩ Br (p), p for any .n ∈ N. Hence, if .GSn−1 (E, p) = GSn (E, p) for some integer .n ≥ 3, then it follows from the last two equalities that GSn (E, p) = GSn+1 (E, p).

.

With the conclusion of mathematical induction we prove that .GSn (E, p) = GS2 .(E, p) for every integer .n ≥ 2. n .(c) Moreover, when .n ≥ 2, .GS (E, p) is complete as a closed subset of a real Hilbert space .Ma (ref. Theorem 1.41 and Remark 4.5). Therefore, .GSn (E, p) − p .2 is a real Hilbert space for .n ≥ 2.

CHAPTER 4. EXTENSION OF ISOMETRIES

112

The following lemma is an extension of Lemma 4.9 for the second-order generalized span .GS2 (E, p). Indeed, we prove that if .i ∈ Λβ (GS2 (E, p)), then the second-order generalized span of E contains all the lines through .GS(E, p) in the direction .βi . Lemma 4.19. Assume that a bounded subset E of .Ma contains at least two elements, .p ∈ E, and .β = {βi }i∈N is a complete orthonormal sequence in .Ma . If .i ∈ Λβ (GS2 (E, p)) and .p ∈ GS(E, p), then .p + αi βi ∈ GS2 (E, p) for any .αi ∈ R. Proof. Let r be a positive real number with .E ⊂ Br (p). Assume that .i ∈ Λβ (GS2 (E, p)). Considering Lemma 4.17 .(iv) and Proposition 4.18, if we substitute .GS2 (E, p) ∩ Br (p) for E in Lemma 4.9, then .p + αi βi ∈ GS3 (E, p) = GS2 (E, p) for all .αi ∈ R. Thus, there are some .m ∈ N, some .wij ∈ GS(E, p) ∩ m  ∞  Br (p), and some .γij ∈ R with . γij (wij − p) ∈ Ma such that .p + αi βi = p+

m  ∞  i=1 j=1

i=1 j=1

γij (wij − p), and hence, we have p + αi βi = p + αi βi + (p − p) m  ∞  . =p+ γij (wij − p) + (p − p).

(4.15)

i=1 j=1

Because .p −p belongs to .GS(E, p)−p, which is a real vector space by Lemma 4.17 .(i), and .Br (p) − p = Br (0), we can choose some sufficiently small real number .μ =  0 such that μ(p − p) ∈ GS(E, p) − p and μ(p − p) ∈ Br (p) − p.

.

(4.16)

Considering (4.11), (4.15) and (4.16), if we put .μ(p − p) = w − p with a .w ∈ GS(E, p) ∩ Br (p), then we have p + α i β i = p +

∞ m  

.

i=1 j=1

γij (wij − p) +

1 (w − p) ∈ GS2 (E, p) μ

for all .αi ∈ R.

2

.

4.5 Basic Cylinders and Basic Intervals We define infinite-dimensional intervals by classifying them as degenerate basic cylinders, non-degenerate basic cylinders, and basic intervals as follows.

4.5. BASIC CYLINDERS AND BASIC INTERVALS Definition 4.20. For any positive integer n, we interval by ⎧ [0, p2i ] ⎪ ⎪ ⎪ ⎪ ∞ [p ⎨ 1i , p2i ]  [p1i , 1] .J = Ji , where Ji = ⎪ ⎪ i=1 {p } ⎪ ⎪ ⎩ 1i [0, 1]

113

define the infinite-dimensional (for i ∈ Λ1 ), (for i ∈ Λ2 ), (for i ∈ Λ3 ), (for i ∈ Λ4 ), (otherwise)

for some disjoint finite subsets .Λ1 , .Λ2 , .Λ3 of .{1, 2, . . . , n} and .0 < p1i < p2i < 1 for .i ∈ Λ1 ∪ Λ2 ∪ Λ3 and .0 ≤ p1i ≤ 1 for .i ∈ Λ4 . If .Λ4 = ∅, then J is called a non-degenerate basic cylinder. When .Λ4 is a nonempty finite set, J is called a degenerate basic cylinder. If .Λ4 is an infinite set, then J will be called a basic interval. Remark 4.21. Let J be an infinite-dimensional interval. (i) In order for J to become a basic cylinder, .Λ4 must be a finite set.

.

(ii) We remark that .Λ4 = N \ Λ(J) and .Λ(J) = N \ Λ4 . That is, .N is the disjoint union of .Λ(J) and .Λ4 .(see Definition 4.8 .(i) for .Λ(J) .).

.

(iii) If .p = (p1 , p2 , . . . , pi , . . .) is an element of J, then .Ji = {pi } for each .i ∈ Λ(J).

.

We note that the basic cylinder or the basic interval J defined in Definition 4.20 can be expressed as  ∞    1 ei : αi ∈ ai Ji for all i ∈ N , .J = αi ai i=1

  where .Ji is the interval defined in Definition 4.20 and . a1i ei i∈N is a complete orthonormal sequence in .Ma . Let .β = {βi }i∈N be a complete orthonormal sequence in .Ma . We now consider ∞ ∞   αi βi for all .x = αi a1i ei ∈ a .da -isometry .U : Ma → Ma defined by .U (x) := i=1

i=1

Ma . The .da -isometry U maps . a1i ei to .βi for any .i ∈ N. Considering this example, we define the .β-basic cylinder and the .β-basic interval as follows. Definition 4.22. Let .β = {βi }i∈N be a complete orthonormal sequence in .Ma , .Ji the interval given in Definition 4.20, and let n be a positive integer. We define  ∞  .Jβ = αi βi : αi ∈ ai Ji for all i ∈ N , i=1

CHAPTER 4. EXTENSION OF ISOMETRIES

114

for some disjoint finite subsets .Λ1 , .Λ2 , .Λ3 of .{1, 2, . . . , n} and .0 < p1i < p2i < 1 for .i ∈ Λ1 ∪ Λ2 ∪ Λ3 and .0 ≤ p1i ≤ 1 for .i ∈ Λ4 . If .Λ4 = ∅, then .Jβ is called a non-degenerate .β-basic cylinder. When .Λ4 is a nonempty finite set, .Jβ is called a degenerate .β-basic cylinder. If .Λ4 is an infinite set, then .Jβ will be called a .β-basic interval. We note that .Jβ = U (J), where J is a basic cylinder or a basic interval defined in Definition 4.20 and .Jβ is a .β-basic cylinder or a .β-basic interval defined in Definition 4.22. Using Definitions 4.20 and 4.22, Remark 4.21 .(ii) is generalized to: Remark 4.23. Let .β = {βi }i∈N be a complete orthonormal sequence in .Ma and let .Jβ be a translation of a .β-basic cylinder or a translation of a .β-basic interval. (i) .Λβ (Jβ ) = N\Λ4 , where .Λ4 is given in Definitions 4.20 and 4.22 and .Λβ (Jβ ) is defined in Definition 4.8 .(ii).

.

(ii) If .p =

∞ 

.

pi βi and .x =

i=1

∞ 

xi βi are elements of .Jβ , then .pi = xi for each

i=1

i ∈ Λβ (Jβ ).

.

Proof. .(i) If .i ∈ Λ4 , then it follows from Definition 4.22 that   ∞  .x, βi a = p+ αj β j , β i = pi + αi ∈ pi + ai Ji = {pi + ai p1i } j=1

for all .x ∈ Jβ , where .p =

a ∞  i=1

pi βi is a fixed element of .Ma such that .Jβ = Tp (Jβ )

for some .β-basic cylinder or .β-basic interval .Jβ . That is, .x, βi a = pi + ai p1i for all .x ∈ Jβ . If .i ∈ Λ4 , then .x + αβi , βi a = x, βi a + α = pi + ai p1i + α = pi + ai p1i for all .x ∈ Jβ and .α = 0, which implies that .x + αβi ∈ Jβ . That is, in view of Definition 4.8 .(ii), we conclude that .i ∈ Λβ (Jβ ). We now assume that .i ∈ Λβ (Jβ ). Then by Definition 4.8 .(ii), it holds that x + αβi ∈ Jβ

(4.17)

.

for any .x ∈ Jβ and .α =  0. Using Definition 4.22, we have x + αβi = p +



.

j∈Λ4

αj β j +



aj p1j βj + αβi

(4.18)

j∈Λ4

for all .x ∈ Jβ and .α =  0 and for some .p ∈ Ma . We assume on the contrary that i ∈ Λ4 . Since .αi ∈ ai Ji and .ai Ji is an interval with nonzero Euclidean length for

.

4.5. BASIC CYLINDERS AND BASIC INTERVALS

115

i ∈ Λ4 , there exists an .α = 0 that satisfies .αi +α ∈ ai Ji . In view of Definition 4.22 and (4.18), it holds that   .x + αβi = p + αj βj + (αi + α)βi + aj p1j βj ∈ Jβ

.

j∈Λ4 ∪{i}

j∈Λ4

for some .x ∈ Jβ and .α = 0, which is contrary to (4.17). Therefore, we conclude that if .i ∈ Λβ (Jβ ), then .i ∈ Λ4 . ∞  .(ii) If .i ∈ Λβ (Jβ ) then .i ∈ Λ4 by .(i). Furthermore, if .p = pi βi ∈ Jβ and x=

∞ 

.

i=1

i=1

xi βi ∈ Jβ , then it follows from Definitions 4.20 and 4.22 that .pi = pi +

ai p1i = xi , where .p =

∞  i=1

pi βi is a fixed element of .Ma such that .Jβ = Tp (Jβ )

for some .β-basic cylinder or .β-basic interval .Jβ .

2

.

Theorem 4.24. Let .β = {βi }i∈N be a complete orthonormal sequence in .Ma and let .Jβ be either a translation of a .β-basic cylinder or a translation of a .β-basic interval and .p ∈ Jβ . Then   .GS(Jβ , p) = p+ αi βi ∈ Ma : αi ∈ R for all i ∈ Λβ (Jβ ) . i∈Λβ (Jβ )

Proof. Assume that x is an arbitrary element of .GS(Jβ , p). By Definition 4.7, we have x−p=

∞ m  

.

εij (xij − p) ∈ Ma

i=1 j=1

for some .m ∈ N, .εij ∈ R, and .xij ∈ Jβ . Furthermore, since .xij , p ∈ Jβ , by Definition 4.22, we get xij = p +

∞ 

.

γ k β k = p +

k=1



γk βk +

k∈N\Λ4



ak p1k βk

k∈Λ4

and 

p=p +

.

∞  k=1

δ k β k = p +

 k∈N\Λ4

δk β k +



ak p1k βk

k∈Λ4

for some .p ∈ Ma and .γk , δk ∈ ak Jk , where .Jβ = Tp (Jβ ) for some .β-basic cylinder or .β-basic interval .Jβ .

CHAPTER 4. EXTENSION OF ISOMETRIES

116

Since .{βi }i∈N is a complete orthonormal sequence in .Ma , it follows from Definition 4.22 and Remark 4.23 .(i) that x−p=

m  ∞ 

εij (xij − p) =

i=1 j=1



.

=



ωk β k =

m  ∞  i=1 j=1

i∈Λβ (Jβ )

k∈N\Λ4

εij

ωi β i =



(γk − δk )βk

k∈N\Λ4



ωi β i

i∈Λβ (Jβ )

for some real numbers .ωi . We note that .Λβ (Jβ ) = Λβ (Jβ ). Since .x ∈ GS(Jβ , p) ⊂ Ma , it holds that  x=p+ ωi βi (∈ Ma ) i∈Λβ (Jβ )



.

∈





p+

αi βi ∈ Ma : αi ∈ R for all i ∈ Λβ (Jβ ) ,

i∈Λβ (Jβ )

which implies that  GS(Jβ , p) ⊂

.

p+





αi βi ∈ Ma : αi ∈ R for all i ∈ Λβ (Jβ ) .

i∈Λβ (Jβ )

It remains to prove the reverse inclusion. According to the structure of .Jβ given in Definition 4.22, for each .i ∈ Λβ (Jβ ), there exists a real number .γi = 0 such that .p + γi βi ∈ Jβ . In other words, for each .i ∈ Λβ (Jβ ), there exists a .ui ∈ Jβ such that .γi βi = ui − p. Thus, if we assume that  .p + α i β i ∈ Ma i∈Λβ (Jβ )

for some .αi ∈ R, then  p+ αi β i = p + .

i∈Λβ (Jβ )

 i∈Λβ (Jβ )

αi (γi βi ) = p + γi

 i∈Λβ (Jβ )

αi (ui − p) γi

∈ GS(Jβ , p), since .ui ∈ Jβ for all .i ∈ Λβ (Jβ ), which implies that   . GS(Jβ , p) ⊃ p+ αi βi ∈ Ma : αi ∈ R for all i ∈ Λβ (Jβ ) . i∈Λβ (Jβ )

4.5. BASIC CYLINDERS AND BASIC INTERVALS

117 2

We end the proof in this way.

.

In the following theorem, we introduce an interesting inclusion property of the second-order generalized span. Theorem 4.25. Assume that H is a closed subspace of .Ma . Let .β = {βi }i∈N and .{βi }i∈Λ be complete orthonormal sequences in the Hilbert spaces .Ma and H, respectively, and let .Jβ be either a translation of a .β-basic cylinder or a translation of a .β-basic interval that satisfies .Λβ (Jβ ) ⊂ Λ. Then .Jβ −p ⊂ GS(Jβ , p)−p ⊂ H for any .p ∈ Jβ . Proof. Assume that .p ∈ Jβ and .x ∈ GS(Jβ , p) − p. Then, according to Theorem 4.24, there exist some real numbers .αi that satisfy  .x = α i β i ∈ Ma . (4.19) i∈Λβ (Jβ )

Assume that .i ∈ Λβ (Jβ ). Then .i ∈ Λ. Since H is a real vector space and .βi ∈ H, it holds that αi β i ∈ H

.

for all .i ∈ Λβ (Jβ ). Now we define xn :=



.

(4.20)

αi β i ∈ H

i∈Λn

for any .n ∈ N, where we set .Λn = {i ∈ Λβ (Jβ ) : i < n}. Since H is assumed to be closed, it follows from (4.19) that x = lim xn ∈ H,

.

n→∞

which implies that .GS(Jβ , p) − p ⊂ H.

2

.

Since in some ways index sets have some properties of dimensions in vector space, the following theorem may seem obvious. Theorem 4.26. Let .β = {βi }i∈N be a complete orthonormal sequence in .Ma . Assume that a bounded subset E of .Ma contains at least two elements and .p ∈ E. Then, .Λβ (GS2 (E, p)) = N if and only if .GS2 (E, p) = Ma . Proof. Let x be an arbitrary element of .Ma . Then there exist some real numbers αi such that

.

x=

∞ 

.

i=1

α i β i ∈ Ma .

(4.21)

CHAPTER 4. EXTENSION OF ISOMETRIES

118

If .Λβ (GS2 (E, p)) = N, then it follows from Lemma 4.19 that p + αi βi ∈ GS2 (E, p)

.

for all .i ∈ N. In other words, αi βi ∈ GS2 (E, p) − p

.

for all .i ∈ N. By Lemma 4.17 .(i), we get xn :=

n 

.

αi βi ∈ GS2 (E, p) − p

i=1

for any .n ∈ N. Due to Lemma 4.17 .(iii) and (4.21), we further obtain x=

∞ 

.

i=1

αi βi = lim xn ∈ GS2 (E, p) − p, n→∞

which implies that .Ma ⊂ GS2 (E, p) − p, or equivalently, .Ma ⊂ GS2 (E, p). The reverse inclusion is trivial.

2

.

4.6 Second-Order Extension of Isometries It was proved in Theorem 4.13 that the domain of a .da -isometry .f : E1 → E2 can be extended to the first-order generalized span .GS(E1 , p) whenever .E1 is a nonempty bounded subset of .Ma , whether degenerate or non-degenerate. Now we generalize Theorem 4.13 in the following theorem. More precisely, we prove that the domain of f can be extended to its second-order generalized span .GS2 (E1 , p). We note that .GS2 (E1 , p) = GS(E1 , p) by Lemma 4.17 .(iii). Therefore, Theorem 4.27 is a further generalization of [6, Theorem 2.2]. In the proof, we use the fact that .GSn (E1 , p) − p is a real vector space. Theorem 4.27. Let .E1 be a bounded subset of .Ma that is .da -isometric to a subset .E2 of .Ma via a surjective .da -isometry .f : E1 → E2 . Assume that p and q are elements of .E1 and .E2 , which satisfy .q = f (p). The function .F2 : GS2 (E1 , p) → Ma is a .da -isometry and the function .T−q ◦ F2 ◦ Tp : GS2 (E1 , p) − p → Ma is linear. In particular, .F2 is an extension of F . Proof. .(a) Suppose r is a positive real number satisfying .E1 ⊂ Br (p). Referring to the changes presented in the table below and following the first part of proof of Theorem 4.13, we can easily prove that .F2 is a .da -isometry.

4.6. SECOND-ORDER EXTENSION OF ISOMETRIES Theorem 4.13: Here:

.E1 GS(E1 , p) ∩ Br (p)

.GS(E1 , p) GS2 (E1 , p)

.

Theorem 4.13: Here:

.

Definition 4.11 Definition 4.14

119 f F

F F2

.

Lemma 4.10 Lemma 4.16

.(b) Referring to the changes presented in the table below and following .(d) of the proof of Theorem 4.13, we can prove the linearity of .T−q ◦ Fn ◦ Tp : GSn (E1 , p) − p → Ma in a more general setting for .n ≥ 2.

Theorem 4.13: Here:

GS(E1 , p) n .GS (E1 , p) .

F .Fn

(4.10) Lemma 4.16

(c) According to Definition 4.14 .(i), for any .m ∈ N, .xij ∈ GS(E1 , p) ∩ Br (p), m  ∞  αij (xij − p) ∈ Ma , there exists a .u ∈ GS2 (E1 , p) and any .αij ∈ R with . .

i=1 j=1

satisfying u−p=

∞ m  

.

αij (xij − p) ∈ Ma .

(4.22)

i=1 j=1

Due to Definition 4.14 .(ii), we further have (T−q ◦ F2 ◦ Tp )(u − p) =

m  ∞ 

.

αij (T−q ◦ F ◦ Tp )(xij − p).

(4.23)

i=1 j=1

If we set .α11 = 1, .αij = 0 for each .(i, j) = (1, 1), and .x11 = x in (4.22) and (4.23) to see (T−q ◦ F2 ◦ Tp )(x − p) = (T−q ◦ F ◦ Tp )(x − p)

.

(4.24)

for all .x ∈ GS(E1 , p) ∩ Br (p). Let w be an arbitrary element of .GS(E1 , p). Then, .w − p ∈ GS(E1 , p) − p. Since .GS(E1 , p) − p is a real vector space and .Br (p) − p = Br (0), there exists a  0 such that real number .μ = μ(w − p) ∈ (GS(E1 , p) − p) ∩ (Br (p) − p).

.

Thus, by (4.11), we can choose a .v ∈ GS(E1 , p)∩Br (p) such that .μ(w−p) = v−p. Since both .T−q ◦ F2 ◦ Tp and .T−q ◦ F ◦ Tp are linear and .GS(E1 , p) ⊂ GS2 (E1 , p),

CHAPTER 4. EXTENSION OF ISOMETRIES

120 it follows from (4.24) that

μ(T−q ◦ F2 ◦ Tp )(w − p) = (T−q ◦ F2 ◦ Tp )(μ(w − p)) = (T−q ◦ F2 ◦ Tp )(v − p) = (T−q ◦ F ◦ Tp )(v − p)

.

= (T−q ◦ F ◦ Tp )(μ(w − p)) = μ(T−q ◦ F ◦ Tp )(w − p). Therefore, it follows that .(T−q ◦ F2 ◦ Tp )(w − p) = (T−q ◦ F ◦ Tp )(w − p) for all w ∈ GS(E1 , p), i.e., .F2 (w) = F (w) for all .w ∈ GS(E1 , p). In other words, .F2 is an extension of F . Also, because of Theorem 4.13, we see that .F2 is obviously an extension of f . .2

.

On account of Proposition 4.18, it holds that GS2 (E1 , p) = · · · = GSn−1 (E1 , p) = GSn (E1 , p)

.

for every integer .n ≥ 3. According to this formula, the assertion of the following theorem seems obvious, but since the proof is not long, we introduce the proof here. Theorem 4.28. Let .E1 be a bounded subset of .Ma that is .da -isometric to a subset E2 of .Ma via a surjective .da -isometry .f : E1 → E2 . Assume that p and q are elements of .E1 and .E2 , which satisfy .q = f (p). Then .Fn is identically the same as .F2 for any integer .n ≥ 3, where .F2 and .Fn are defined in Definition 4.14. .

Proof. Let r be a fixed positive real number satisfying .E1 ⊂ Br (p). We assume that .F2 ≡ F3 ≡ · · · ≡ Fn−1 on .GS2 (E1 , p). Let x be an arbitrary element of n .GS (E1 , p). Then, in view of (4.11), there exists a real number .μ = 0 and an element u of .GSn (E1 , p) ∩ Br (p) such that u − p = μ(x − p) ∈ (GSn (E1 , p) − p) ∩ (Br (p) − p).

.

If we put .α11 = 1, .αij = 0 for all .(i, j) =  (1, 1), and .x11 = v in Definition 4.14 (ii), then we get

.

(T−q ◦ Fn ◦ Tp )(v − p) = (T−q ◦ Fn−1 ◦ Tp )(v − p)

.

(4.25)

for all .v ∈ GSn−1 (E1 , p) ∩ Br (p) = GSn (E1 , p) ∩ Br (p). We note by Proposition 4.18 that .GSn (E1 , p) = GSn−1 (E1 , p) = · · · = GS2 (E1 , p).

4.6. SECOND-ORDER EXTENSION OF ISOMETRIES

121

Since .T−q ◦ Fn ◦ Tp is linear by .(b) in the proof of Theorem 4.27, it follows from (4.25) that μ(T−q ◦ Fn ◦ Tp )(x − p) = (T−q ◦ Fn ◦ Tp )(u − p) = (T−q ◦ Fn−1 ◦ Tp )(u − p)

.

= (T−q ◦ F2 ◦ Tp )(u − p) = μ(T−q ◦ F2 ◦ Tp )(x − p),

i.e., .Fn (x) = F2 (x) for every .x ∈ GSn (E1 , p) = GS2 (E1 , p). By mathematical induction, we conclude that .Fn is identically the same as .F2 for every integer .n ≥ 3. .2 In the following two remarks, let .β = {βi }i∈N be a complete orthonormal sequence in .Ma and let .Jβ be either a translation of a .β-basic cylinder or a translation of a .β-basic interval and .p ∈ Jβ . Due to Definition 4.22, Remark 4.23, and Theorem 4.24, .GS(Jβ , p) is a closed subset of .Ma . Remark 4.29. .GS(Jβ , p) is a closed subset of .Ma . Proof. Assume that .p =

∞ 

pi βi is a fixed element of .Jβ , where .Jβ is a translation

i=1

of a .β-basic cylinder or a translation of a .β-basic interval. In view of Definition 4.7 ∞  and Remark 4.23 .(ii), we note that .xi = pi for each .x = xi βi ∈ GS(Jβ , p) and i=1

each .i ∈ Λβ (Jβ ). Assume that .{zn }n∈N is a sequence of elements in .GS(Jβ , p), which converges ∞ ∞   zi βi of .Ma , where we set .zn = zni βi for any .n ∈ N. to an element .z = i=1

i=1

Since .zn ∈ GS(Jβ , p) for every .n ∈ N, the previous argument implies that .zni = pi for each .i ∈ Λβ (Jβ ). Thus, we conclude that .zi = pi for each .i ∈ Λβ (Jβ ). This fact, together with Theorem 4.24, implies that .z ∈ GS(Jβ , p). Therefore, we .2 conclude that .GS(Jβ , p) is a closed subset of .Ma . On account of Theorem 4.24, we note that .Λβ (Jβ ) = Λβ (GS(Jβ , p)). Remark 4.30. .GS2 (Jβ , p) = GS(Jβ , p). Proof. We note that .GS(Jβ , p) is a closed subset of .Ma by Remark 4.29. Referring to the changes presented in the table below Proposition 4.18: Here:

GS(E, p) ∩ Br (p) .Jβ

.

GS2 (E, p) .GS(Jβ , p)

.

GS3 (E, p) 2 .GS (Jβ , p) .

x u

xm .um .

CHAPTER 4. EXTENSION OF ISOMETRIES

122

and following the part .(a) in the proof of Proposition 4.18, we can easily show that GS2 (Jβ , p) = GS(Jβ , p). .2

.

Hence, by Theorem 4.24 with .β =

1



ai ei i∈N

∞   1 u − p, ei u−p= ai

=

i=1 ∞ 

ai (ui − pi )

i=1

.

=



=

 

u − p,

i∈Λ(J)

1 ei a ai

1 ei ai

ai (ui − pi )

i∈Λ(J)

and Remark 4.30, we have

1 ei ai

1 ei ai

(4.26)

1 ei a a i

for all .u ∈ GS2 (J, p) = GSn (J, p), where .n ∈ N. Using a similar approach to the proof of [10, Theorem 2.4], we can apply Lemma 4.16 to prove the following theorem.

Theorem 4.31. Assume that J is either a translation of a basic cylinder or a translation of a basic interval, K is a subset of .Ma , and that there exists a surjective .da -isometry .f : J → K. Suppose p is an element of J and q is an element of K with .q = f (p). For any .n ∈ N, the .da -isometry .Fn : GSn (J, p) → Ma given in Definition 4.14 satisfies

(T−q ◦ Fn ◦ Tp )(u − p) =

 

.

i∈Λ(J)

for all .u ∈ GSn (J, p).

u − p,

1 ei ai

1 (T−q ◦ Fn ◦ Tp )(ei ) a ai

4.6. SECOND-ORDER EXTENSION OF ISOMETRIES

123

Proof. Since .p + ei ∈ GSn (J, p) for each .i ∈ Λ(J), it follows from Lemma 4.16 that    1 1 (T−q ◦ Fn ◦ Tp )(u − p) − u − p, ei (T−q ◦ Fn ◦ Tp )(ei ), ai ai a i∈Λ(J)    1 1 u − p, ej (T−q ◦ Fn ◦ Tp )(ej ) (T−q ◦ Fn ◦ Tp )(u − p) − aj aj a j∈Λ(J) a

= (T−q ◦ Fn ◦ Tp )(u − p), (T−q ◦ Fn ◦ Tp )(u − p) a  

1 1 u − p, ej − (T−q ◦ Fn ◦ Tp )(u − p), (T−q ◦ Fn ◦ Tp )(ej ) a aj a aj j∈Λ(J)  

1 1 (T−q ◦ Fn ◦ Tp )(ei ), (T−q ◦ Fn ◦ Tp )(u − p) a u − p, ei − ai a ai i∈Λ(J)     1 1 u − p, ej × u − p, ei + . ai aj a a i∈Λ(J) j∈Λ(J)

1 (T−q ◦ Fn ◦ Tp )(ei ), (T−q ◦ Fn ◦ Tp )(ej ) a ai aj   

1 1 u − p, ej u − p, ej = u − p, u − p a − a aj j a a j∈Λ(J)    1 1 u − p, ei − ei , u − p ai a ai a i∈Λ(J)      1 1 1 1 u − p, ei u − p, ej + ei , ej ai aj aj a a ai a i∈Λ(J) j∈Λ(J)   

1 1 u − p, ej u − p, ej = u − p, u − p a − aj aj a a ×

j∈Λ(J)

  for all .u ∈ GSn (J, p), since . a1i ei i∈N is an orthonormal sequence in .Ma . Furthermore, we note that each .u ∈ GSn (J, p) has the expression given in (4.26). Hence, if we replace .u − p in the previous equalities with the expression (4.26), then we have  2     1 1   u − p, ei .(T−q ◦ Fn ◦ Tp )(u − p) − (T−q ◦ Fn ◦ Tp )(ei ) = 0   a ai a i i∈Λ(J)

for all .u ∈ GSn (J, p), which implies the validity of our assertion.

a

2

.

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124

According to the following theorem, the image of the first-order generalized span of .E1 with respect to p under the .da -isometry F is just the first-order generalized span of .F (E1 ) with respect to .F (p). This assertion holds also for the second-order generalized span and .F2 . According to Proposition 4.18 and Theorem 4.28, the argument of the following theorem only makes sense when .n = 1 or 2. Theorem 4.32. Assume that .E1 and .E2 are bounded subsets of .Ma that are .da isometric to each other via a surjective .da -isometry .f : E1 → E2 . Suppose p is an element of .E1 and q is an element of .E2 with .q = f (p). If .Fn : GSn (E1 , p) → Ma is the extension of f defined in Definition 4.14, then .GSn (E2 , q) = Fn (GSn (E1 , p)) for every .n ∈ N. Proof. .(a) First, we prove that our assertion is true for .n = 1, i.e., we prove that .GS(E2 , q) = F (GS(E1 , p)). Let r be a fixed positive real number satisfying .E1 ⊂ Br (p). .(b) Due to Definition 4.7, for any .y ∈ F (GS(E1 , p)), there exists an element .x ∈ GS(E1 , p) with   m  ∞    .y = F (x) = F p+ αij uij − p i=1 j=1

for some .m ∈ N, .uij ∈ E1 ∩ Br (p), and some .αij ∈ R which satisfy the condition m  ∞  .x = p + αij (uij − p) ∈ Ma . i=1 j=1

On the other hand, by Definition 4.11, we have   m ∞ m  ∞     = .(T−q ◦ F ◦ Tp ) αij uij − p αij (T−q ◦ f ◦ Tp )(uij − p) i=1 j=1

i=1 j=1

which is equivalent to   ∞ ∞ m  m        .F (x) − q = F p+ αij uij − p − q = αij f (uij ) − q . i=1 j=1

i=1 j=1

Since .uij ∈ E1 for all i and j, it holds that .f (uij ) ∈ f (E1 ) = E2 for each i and j. Moreover, since .uij ∈ E1 ∩ Br (p) for all i and j, it follows from Lemma 4.10 that f (uij ) − q2a = (T−q ◦ f ◦ Tp )(uij − p)2a

= (T−q ◦ f ◦ Tp )(uij − p), (T−q ◦ f ◦ Tp )(uij − p) a .

= uij − p, uij − pa = uij − p2a < r2

4.6. SECOND-ORDER EXTENSION OF ISOMETRIES

125

for all i and j. Hence, .f (uij ) ∈ E2 ∩ Br (q) for all i and j. Furthermore, it follows from Lemma 4.10 that 2  m ∞      αij f (uij ) − q     i=1 j=1 a  2 ∞ m      = αij (T−q ◦ f ◦ Tp )(uij − p)   i=1 j=1 a  m ∞  m  ∞   = αij (T−q ◦ f ◦ Tp )(uij − p), αk (T−q ◦ f ◦ Tp )(uk − p) =

i=1 j=1 m ∞ m  

.

=

=

k=1 =1

αij

∞ 

i=1 k=1 j=1

=1

m  ∞ m  

∞ 

i=1 k=1 j=1  m ∞ 

αij

αk (T−q ◦ f ◦ Tp )(uij − p), (T−q ◦ f ◦ Tp )(uk − p) a αk uij − p, uk − pa

=1

αij (uij − p),

i=1 j=1

m  ∞ 

 αk (uk − p)

k=1 =1

 m ∞ 2     = αij (uij − p)   i=1 j=1

a

a

a

< ∞, since . that .

m  ∞ 

αij (uij − p) = x − p ∈ Ma . Thus, on account of Remark 4.2, we see

i=1 j=1 m  ∞ 

αij (f (uij ) − q) ∈ Ma . Therefore, in view of Definition 4.7, we get

i=1 j=1

y = F (x) ∞ m     αij f (uij ) − q =q+ . i=1 j=1

∈ GS(E2 , q) and we conclude that .F (GS(E1 , p)) ⊂ GS(E2 , q). .(c) Now we assume that .y ∈ GS(E2 , q). By Definition 4.7, there exist some .m ∈ N, .vij ∈ E2 ∩ Br (q), and some .αij ∈ R which satisfy the condition .y − q = m  ∞  αij (vij −q) ∈ Ma . Since .f : E1 → E2 is surjective, there exists a .uij ∈ E1 i=1 j=1

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126

satisfying .vij = f (uij ) for any i and j. Moreover, by Lemma 4.10, we have

.

uij − p2a = uij − p, uij − pa

= (T−q ◦ f ◦ Tp )(uij − p), (T−q ◦ f ◦ Tp )(uij − p) a

= f (uij ) − q, f (uij ) − q a = vij − q, vij − qa = vij − q2a < r2

for any i and j. So we conclude that .uij ∈ E1 ∩ Br (p) and .vij = f (uij ) for all i and j. On the other hand, using Lemma 4.10, we have  m ∞ 2     α (u − p)   ij ij   i=1 j=1 a  m ∞  m  ∞   = αij (uij − p), αk (uk − p) =

=

i=1 j=1 m  m ∞  i=1 k=1 j=1 m  ∞ m   i=1 k=1 j=1 ∞ m  

 .

=

k=1 =1

αij αij

∞  =1 ∞ 

a



αk uij − p, uk − p

=1

αij (T−q ◦ f ◦ Tp )(uij − p),

m  ∞ 

k=1 =1 2 

 m ∞    = αij (T−q ◦ f ◦ Tp )(uij − p)   i=1 j=1 a  m ∞ 2      = αij f (uij ) − q    i=1 j=1 a  m ∞ 2     = αij (vij − q)   < ∞,

a



αk (T−q ◦ f ◦ Tp )(uij − p), (T−q ◦ f ◦ Tp )(uk − p) a

i=1 j=1

i=1 j=1

a

 αk (T−q ◦ f ◦ Tp )(uk − p) a

4.7. EXTENSION OF ISOMETRIES TO THE ENTIRE SPACE since . that .

m  ∞ 

127

αij (vij − q) = y − q ∈ Ma . Therefore, it follows from Remark 4.2

i=1 j=1 m  ∞ 

αij (uij − p) ∈ Ma .

i=1 j=1

Hence, it follows from Definition 4.11 that y=q+

=q+

∞ m   i=1 j=1 m  ∞ 

  αij f (uij ) − q αij (T−q ◦ f ◦ Tp )(uij − p)

i=1 j=1 .



= q + (T−q ◦ F ◦ Tp )

∞ m  

 αij (uij − p)

i=1 j=1

 =F p+

∞ m  



αij (uij − p)

i=1 j=1

∈ F (GS(E1 , p)). Thus, we conclude that .GS(E2 , q) ⊂ F (GS(E1 , p)). .(d) Similarly, referring to the changes presented in the tables below and following the previous parts .(b) and .(c) in this proof, we can prove that .GS2 (E2 , q) = F2 (GS2 (E1 , p)). The case .n = 1: The case .n = 2: The case .n = 1: The case .n = 2:

E1 .GS(E1 , p) .

F .F2

E2 .GS(E2 , q) .

Definition 4.7 Definition 4.14 .(i)

GS(E1 , p) 2 .GS (E1 , p) .

GS(E2 , q) 2 .GS (E2 , q) .

Definition 4.11 Definition 4.14 .(ii)

f F

Lemma 4.10 (4.10)

.(e) Finally, according to Proposition 4.18, Theorem 4.28, and .(d), we further have

GSn (E2 , q) = GS2 (E2 , q) = F2 (GS2 (E1 , p)) = Fn (GSn (E1 , p))

.

for any integer .n ≥ 3.

2

.

4.7 Extension of Isometries to the Entire Space Let .I ω =

∞  i=1

I be the Hilbert cube, where .I = [0, 1] is the unit closed interval.

From now on, we assume that .E1 and .E2 are nonempty subsets of .I ω . It is clear that they are bounded.

CHAPTER 4. EXTENSION OF ISOMETRIES

128

In Theorem 4.34, we will prove that the domain .E1 of a local .da -isometry f : E1 → E2 can be extended to any real Hilbert space including the domain .E1 of f .

.

Definition 4.33. Let .E1 be a nonempty subset of .I ω that is .da -isometric to a subset ω via a surjective .d -isometry .f : E → E . Let p be an element of .E .E2 of .I a 1 2  1  and q an element of .E2 with .q = f (p). Assume that . a1i ei i∈Λα is a complete orthonormal sequence in the Hilbert space .GS2 (E1 , p)−p, where .Λα is a nonempty proper subset of .N. Moreover, assume that .{βi }i∈N is a complete orthonormal sequence in the Hilbert space .Ma such that .βi = a1i (T−q ◦ F2 ◦ Tp )(ei ) for each 2 .i ∈ Λα , where .F2 : GS (E1 , p) → Ma is defined in Definition 4.14. Let .pi be the ∞  pi ei . For any set .Λ satisfying .Λα ⊂ Λ ⊂ N, we ith component of p, i.e., .p = i=1

define a basic cylinder or a basic interval .J˜ by ! ∞  [0, 1] (for i ∈ Λ), ˜ ˜ ˜ Ji , where Ji = .J = {pi } (for i ∈ Λ). i=1

˜ p) → Moreover, referring to Theorem 4.31, we define the function .G2 : GS2 (J, Ma by   1 u − p, ei βi .(T−q ◦ G2 ◦ Tp )(u − p) = (4.27) ai a ˜ i∈Λ(J)

˜ p). for all .u ∈ GS2 (J, The following theorem states that the domain of a local .da -isometry can be extended to any real Hilbert space including the domain of the local .da -isometry. Theorem 4.34. Let .E1 be a bounded subset of .I ω that contains at least two elements. Suppose .E1 is .da -isometric to a subset .E2 of .I ω via a surjective .da -isometry .f : E1 → E2 . Let p and q be elements of .E1 and .E2 satisfying .q = f (p).   Assume that . a1i ei i∈Λα is a complete orthonormal sequence in the Hilbert space 2 .GS (E1 , p) − p, where .Λα is a nonempty proper subset of .N. Moreover, assume that .{βi }i∈N is a complete orthonormal sequence in the Hilbert space .Ma such that 1 .βi = ai (T−q ◦ F2 ◦ Tp )(ei ) for each .i ∈ Λα . Let .Λ be a set satisfying .Λα ⊂ Λ ⊂ N ˜ p) → Ma and let .J˜ be defined as in Definition 4.33. Then the function .G2 : GS2 (J, 2 ˜ is a .da -isometry and the function .T−q ◦ G2 ◦ Tp : GS (J, p) − p → Ma is linear. In particular, .G2 is an extension of .F2 . ˜ p) − p → Proof. .(a) First, we assert that the function .T−q ◦ G2 ◦ Tp : GS2 (J, Ma preserves the inner product. Assume that u and v are arbitrary elements of

4.7. EXTENSION OF ISOMETRIES TO THE ENTIRE SPACE

129

˜ p). Since .Λ = Λ(J), ˜ it follows from (4.26), (4.27), and the orthonormality GS2 (J,  1 of . ai ei i∈N and .{βi }i∈N that

.



.

(T−q ◦ G2 ◦ Tp )(u − p), (T−q ◦ G2 ◦ Tp )(v − p) a     1 1 = u − p, ei βi , v − p, ej βj ai aj a a i∈Λ j∈Λ a     1 1 = u − p, ei v − p, ej βi , βj a ai aj a j∈Λ a i∈Λ      1 1 1 1 = ei , ej u − p, ei v − p, ej ai aj aj a j∈Λ a ai a i∈Λ     1 1 1 1 = ei , ej u − p, ei v − p, ej ai a a a j a i a j i∈Λ

j∈Λ

a

= u − p, v − pa ˜ p), i.e., .T−q ◦ G2 ◦ Tp preserves the inner product. for all .u, v ∈ GS2 (J, .(b) We assert that .G2 is a .da -isometry. Let u and v be arbitrary elements of 2 ˜ .GS (J, p). Since .T−q ◦ G2 ◦ Tp preserves the inner product by .(a), we have  2 da G2 (u), G2 (v)  2 = (T−q ◦ G2 ◦ Tp )(u − p) − (T−q ◦ G2 ◦ Tp )(v − p)a = (T−q ◦ G2 ◦ Tp )(u − p) − (T−q ◦ G2 ◦ Tp )(v − p),

(T−q ◦ G2 ◦ Tp )(u − p) − (T−q ◦ G2 ◦ Tp )(v − p) a .

= u − p, u − pa − u − p, v − pa − v − p, u − pa + v − p, v − pa

= (u − p) − (v − p), (u − p) − (v − p) a = (u − p) − (v − p)2a = u − v2a = da (u, v)2

˜ p), i.e., .G2 : GS2 (J, ˜ p) → Ma is a .da -isometry. for all .u, v ∈ GS2 (J, 2 ˜ .(c) We assert that the function .T−q ◦ G2 ◦ Tp : GS (J, p) − p → Ma is linear. 2 ˜ Assume that u and v are arbitrary elements of .GS (J, p) and .α, β are real numbers. ˜ p) − p is a real vector space, it holds that .α(u − p) + β(v − p) ∈ Since .GS2 (J, 2 ˜ ˜ p). GS (J, p) − p. Thus, .α(u − p) + β(v − p) = w − p for some .w ∈ GS2 (J, Hence, referring to the changes presented in the table below and following .(d) of the proof of Theorem 4.13, we can easily prove that .T−q ◦ G2 ◦ Tp is linear.

CHAPTER 4. EXTENSION OF ISOMETRIES

130

Theorem 4.13: Here:

GS(E1 , p) ˜ p) GS2 (J,

.

.

F G2

.

(4.10) .(a)

(d) Finally, we assert that .G2 is an extension of .F2 . Let .Jˆ be either a basic cylinder or a basic interval defined by .

ˆ= .J

∞ 

! Jˆi , where Jˆi =

i=1

[0, 1] (for i ∈ Λα ), {pi } (for i ∈ Λα ).

ˆ = Λα = Λ(GS2 (E1 , p)). We see that .p = (p1 , p2 , . . .) ∈ Jˆ ∩ E1 and .Λ(J) By Lemma 4.19, if .i ∈ Λ(GS2 (E1 , p)), then .αi ei ∈ GS2 (E1 , p) − p for all 2 .αi ∈ R. Since .GS (E1 , p) − p is a real vector space, if we set .Λn = {i ∈ 2 Λ(GS (E1 , p)) : i < n}, then we have  . αi ei ∈ GS2 (E1 , p) − p i∈Λn

for all.n ∈ N and for all .αi ∈ R. For now, with all .αi ’s fixed, we define .xn = p+ αi ei for any .n ∈ N. Then .{xn } is a sequence in .GS2 (E1 , p). When .{xn } i∈Λn

converges in .Ma , it holds that  .p +

αi ei = lim xn ∈ GS2 (E1 , p), n→∞

i∈Λ(GS2 (E1 ,p))

because .GS2 (E1 , p) is closed by Lemma 4.17 .(iii). That is,    2  p+ αi ei ∈ Ma : αi ∈ R for all i ∈ Λ GS (E1 , p) .

i∈Λ(GS2 (E1 ,p))

⊂ GS2 (E1 , p). Hence, by the previous inclusion and Theorem 4.24 with .β = ˆ we get .Jβ = J,   ˆ p) − p = ˆ GS(J, αi ei ∈ Ma : αi ∈ R for all i ∈ Λ(J)  .

=

1



ai ei i∈N

and

ˆ i∈Λ(J)





αi ei ∈ Ma : αi ∈ R for all i ∈ Λ GS2 (E1 , p)

i∈Λ(GS2 (E1 ,p))

⊂ GS2 (E1 , p) − p.





4.7. EXTENSION OF ISOMETRIES TO THE ENTIRE SPACE

131

So we have ˆ p) ∩ Br (p) ⊂ GS2 (E1 , p) ∩ Br (p) ˆ ∩ Br (p) ⊂ GS(J, .J for some real number .r > 0 and hence, we further have ˆ p) ⊂ GS2 (J, ˆ p) ⊂ GS3 (E1 , p) = GS2 (E1 , p). .GS(J, ˆ p) = GS(J, ˆ p). Hence, we have Moreover, by Remark 4.30, we know that .GS2 (J, ˆ p) = GS2 (J, ˆ p) ⊂ GS2 (E1 , p). GS(J, 1  On the other hand, since . ai ei i∈Λα is a complete orthonormal sequence in 1  2 .GS (E1 , p) − p, it follows from Theorem 4.24 with .β = ai ei i∈N that   1 1 ˆ p) − p = GS2 (J, ˆ p) − p x, ei .x = ei ∈ GS(J, ai a i a .

i∈Λα

ˆ p) = for all .x ∈ GS2 (E1 , p) − p, which implies that .GS2 (E1 , p) = GS2 (J, ˆ GS(J, p). Let u be an arbitrary element of .GS2 (E1 , p). Then by (4.26) with .Jˆ instead of J, we have   1 1 u − p, ei .u − p = ei (4.28) ai a ai ˆ i∈Λ(J)

and since .T−q ◦ F2 ◦ Tp is linear and continuous, we use (4.27), (4.28), and the facts ˆ p) = GS(J, ˆ p) and .Λ(J) ˆ = Λα = Λ(GS2 (E1 , p)) to have GS2 (E1 , p) = GS2 (J,

.

(T−q ◦ G2 ◦ Tp )(u − p)   1 = u − p, ei βi ai a ˜ i∈Λ(J)   1 = u − p, ei βi ai a ˆ i∈Λ(J)   1 1 u − p, ei (T−q ◦ F2 ◦ Tp )(ei ) = ai a ai ˆ i∈Λ(J) .   1 1 u − p, ei (T−q ◦ F2 ◦ Tp )(ei ) = lim n→∞ ai a ai ˆ i∈Λn (J)     1 1 u − p, ei ei = lim (T−q ◦ F2 ◦ Tp ) n→∞ ai ai a ˆ i∈Λn (J)     1 1 u − p, ei ei = (T−q ◦ F2 ◦ Tp ) ai a ai ˆ i∈Λ(J)

= (T−q ◦ F2 ◦ Tp )(u − p),

132

CHAPTER 4. EXTENSION OF ISOMETRIES

ˆ = {i ∈ Λ(J) ˆ : i < n} for every .n ∈ N. where we set .Λn (J) Therefore, it follows that .(T−q ◦ G2 ◦ Tp )(u − p) = (T−q ◦ F2 ◦ Tp )(u − p) for all .u ∈ GS2 (E1 , p), i.e., .G2 (u) = F2 (u) for all .u ∈ GS2 (E1 , p). In other words, .G2 is an extension of .F2 . .2 In Theorem 4.34, .GS2 (E1 , p) − p is closed in .Ma . Hence, .GS2 (E1 , p) − p is a closed subspace of the real Hilbert space .Ma , i.e., .GS2 (E1 , p) − p is itself a ˜ p) − p is a real Hilbert space. Since the .da real Hilbert space. Similarly, .GS2 (J, 2 ˜ ˜ p))−q is a homeomorphism isometry .T−q ◦G2 ◦Tp : GS (J, p)−p → G2 (GS2 (J, 2 ˜ 2 ˜ ˜ p)) − q is and .(T−q ◦ G2 ◦ Tp )(GS (J, p) − p) = G2 (GS (J, p)) − q, .G2 (GS2 (J, a closed subspace of the real Hilbert space .Ma , i.e., it is also a real Hilbert space. Similarly, .GS2 (E2 , q) − q is a real Hilbert space. Assuming the axiom of choice, it is well known that (i) Every Hilbert space has a complete orthonormal sequence (ref. [2, p. 140]).

.

(ii) The union of a complete orthonormal sequence in the Hilbert space H and a complete orthonormal sequence in the orthogonal complement of H is a complete orthonormal sequence in the Hilbert space .H ⊕ H ⊥ (see Remark 2.46 .(ii) or [3, p. 124]).   Indeed, .{βi }i∈Λα = a1i (T−q ◦ F2 ◦ Tp )(ei ) i∈Λα is a complete orthonormal sequence in the real Hilbert space .GS2 (E2 , q) − q. In view of .(i), we can choose a complete orthonormal sequence .{βi }i∈Λ\Λα in the real Hilbert space 2 2 ˜ ⊥ .(GS (E2 , q)−q) ∩(G2 (GS (J, p))−q). We can also choose a complete orthonor˜ p)) − q. mal sequence .{βi }i∈N\Λ in the orthogonal complement of .G2 (GS2 (J, We note that   ˜ p)) − q) Ma = (GS2 (E2 , q) − q) ⊕ (GS2 (E2 , q) − q)⊥ ∩ (G2 (GS2 (J, . ˜ p)) − q)⊥ , ⊕ (G2 (GS2 (J, .

˜ p)) − q)⊥ are the orthogonal complewhere .(GS2 (E2 , q) − q)⊥ and .(G2 (GS2 (J, 2 ˜ p)) − q, respectively. ments of the Hilbert spaces .GS (E2 , q) − q and .G2 (GS2 (J, 2 2 ˜ Since .GS (E1 , p) ⊂ GS (J, p), it follows from Theorem 4.32 that .GS2 (E2 , q) = ˜ p)) ⊂ Ma . Hence, F2 (GS2 (E1 , p)) ⊂ G2 (GS2 (J, ! " 1 .{βi }i∈N = (T−q ◦ F2 ◦ Tp )(ei ) ∪ {βi }i∈Λ\Λα ∪ {βi }i∈N\Λ ai i∈Λα is a complete orthonormal sequence in the real Hilbert space .Ma . The pair .(H1 , H2 ) of Hilbert spaces is said to have the isometric extension property if for every isometry f from an arbitrary subset S of .H1 into .H2 , there

4.7. EXTENSION OF ISOMETRIES TO THE ENTIRE SPACE

133

exists an isometry F of .H1 into .H2 such that the restriction of F to S coincides with f . The following theorem is a well known result due to [23, Theorem 11.4]. Theorem 4.35 (Wells and Williams). If H is a Hilbert space, then .(H, H) has the isometric extension property if and only if H is finite-dimensional. In general, if .S ⊂ H and .f : S → H is an isometry, then f can be extended as an isometry to the closed linear span of S. We note that Theorem 4.35 does not imply Theorem 4.34. For example, assume that .E1 and .E2 are subsets of the Hilbert cube .I ω and .Λ is a proper subset of .N and ˜ p)−p is a proper subspace of the real Hilbert a proper superset of .Λα . Then .GS2 (J, 2 ˜ p)−p. Nevertheless, space .Ma and .GS (E1 , p)−p is a proper subspace of .GS2 (J, it follows from Theorem 4.34 that every surjective isometry .f : E1 → E2 can be ˜ p) → Ma . On the other hand, we cannot extended to an isometry .G2 : GS2 (J, expect to obtain this result using Theorem 4.35, since the closed linear span of .E1 ˜ p) − p, which implies that Theorem 4.34 is not only is a proper subset of .GS2 (J, different from Theorem 4.35 but also has many advantages over it. Moreover, for any bounded subset S of .I ω , it is clear that .span(S) ⊂ GS2 (S, p). But it is not yet clear whether .span(S) = GS2 (S, p), where .span(S) denotes the closed linear span of S. If .span(S) =  GS2 (S, p) is correct, Theorem 4.34 has more advantages than Theorem 4.35. According to Theorem 4.34, the domain of a local .da -isometry can be extended to any real Hilbert space containing that domain.

Chapter 5

History of Ulam’s Conjecture A conjecture of Ulam states that the standard product probability measure .π on the Hilbert cube .I ω is .da -invariant when the sequence .a = {ai }i∈N of positive numbers satisfies ∞  the condition . a2i < ∞. In 1974 and 1977, J. Mycielski published the first papers on i=1

this topic. Indeed, he proved the conjecture of Ulam affirmatively under the additional assumption that the sets are open. In 1982, J. W. Fickett succeeded in partially proving Ulam’s conjecture by proving the following statement in a different way than J. Mycielski: If the .ai ’s tend to 0 very rapidly, then any two Borel subsets of the Hilbert cube which are .da -isometric have the same standard product probability measure. About 40 years later, in 2018, S.-M. Jung and E. Kim jointly studied Ulam’s conjecture and improved the result of J. W. Fickett by partially proving Ulam’s conjecture. In this chapter, we will introduce the historical process of solving Ulam’s conjecture by presenting a summary of these papers. In some cases, the reader may skip reading this chapter.

5.1 Historical Background S. M. Ulam raised the conjecture on the invariance of measures defined in the compact metric space (see [21]): Let X be a compact metric space. Does there exist a finitely additive measure .μ defined for at least all the Borel subsets of X, such that .μ(X) = 1, .μ(p) = 0 for all points p of X, and such that congruent subsets of X have equal measure? Thereafter, J. Mycielski [17] confined the question of Ulam to the Hilbert cube .I ω and reformulated it using modern mathematical terms: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4 5

135

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

136

The standard product probability measure .π on .I ω is .da -invariant. The above statement is widely known today as the conjecture of Ulam. (For the definition of the standard product probability measure, see Definition 3.38.) In 1974, by using the axiom of choice, J. Mycielski [16, 17] answered the question of Ulam affirmatively under the additional assumption that the sets are open. In addition, he asked in [16] whether one can prove the conjecture of Ulam under the assumption that the sets are closed. J. W. Fickett [4], one step further, showed in 1982 that Ulam’s conjecture is true when the sequence .a = {ai }i∈N decreases very rapidly to 0 such that 1/2i+1

ai+1 . ai

→ 0 as i → ∞.

In 2018, S.-M. Jung and E. Kim proved in their paper [10] that Ulam’s conjecture is true when the sequence .a = {ai }i∈N of positive real numbers is monotone decreasing and satisfies the condition   a √i .ai+1 = o as i → ∞ i (see also [5, 11]). It is evident that the last condition is much weaker than that of Fickett. In 2020, S.-M. Jung combined various methods to prove Ulam’s conjecture, among which he developed and applied a method to extend the domain of local isometries, and he was able to completely prove that Ulam’s conjecture is true. A detailed explanation of this is deferred to the last chapter.

5.2 Basic Definitions We denote by .Rω the infinite-dimensional real vector space defined as   ω .R = (x1 , x2 , . . .) : xi ∈ R for all i ∈ N . In addition, .(Rω , T ) denotes the product space .

∞ 

i=1

R, where .(R, TR ) is the usual

topological space. Let .I = [0, 1] be the unit closed interval and .I ω =

∞ 

i=1 on .I ω .

I the Hilbert cube, and

We denote by .(I ω , Tω ) let .π be the standard product probability measure ω the (topological) subspace of .(R , T ). Then, .Tω is the relative topology for .I ω induced by .T .

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137

As in Sect. 4.1, let .a = {ai }i∈N be a sequence of positive real numbers that satisfies the condition (4.1). Using this sequence a, we define the metric on .I ω by formula (4.2), i.e.,  da (x, y) =

∞

.

1/2 a2i (xi − yi )2

i=1

for all .x = (x1 , x2 , . . .) ∈ I ω and .y = (y1 , y2 , . . .) ∈ I ω . As in Sect. 4.1 (see Theorem 4.3), we define the real Hilbert space .Ma by  ∞ ω 2 2 .Ma = (x1 , x2 , . . .) ∈ R : ai xi < ∞ i=1

and we define an inner product .·, ·a on .Ma by x, ya =

∞

.

a2i xi yi

i=1

for all .x = (x1 , x2 , . . .) and .y = (y1 , y2 , . . .) of .Ma . Then this inner product induces the norm

.xa = x, xa for all .x ∈ Ma . In view of definition (4.2), the metric .da on .I ω can be extended to the metric on .Ma , i.e.,

.da (x, y) = x − y, x − ya for all .x, y ∈ Ma .

5.3 Mycielski’s Partial Solution Using the axiom of choice, J. Mycielski [16, 17] answered Ulam’s question in the affirmative with the additional assumption that the sets are open. Definition 5.1. Let .(X, d) be a metric space. We define an entropy by   .E(K, t) = min card U : U is a t-covering of K for all compact subsets K of X and .0 < t ≤ 1.

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138

(i) A compact subset K of X is called thin if for any .ε > 0 there exist a .δ > 0 and an open subset U of X including K such that

.

E(C, t) ≤ εE(X, t)

.

for all compact subsets C of X with .C ⊂ U and for all .0 < t < δ. (ii) A nonempty compact subset K of X is called thick in X if there exists an open subset U of X including K and a real number .α > 0 such that

.

E(C, t) ≤ αE(K, t)

.

for all compact subsets C of X with .C ⊂ U and for all .0 < t ≤ 1. Now we introduce the definitions of filter and ultrafilter. Definition 5.2. Let X be a set. A nonempty collection .F of nonempty subsets of X is called a filter on X if it has the properties: (i) If .A, B ∈ F, then .A ∩ B ∈ F.

.

(ii) If .A ∈ F and .A ⊂ B, then .B ∈ F.

.

A filter .F on X is called a proper filter if it is different from the power set .P(X) of X. A proper filter .F on X is said to be an ultrafilter or a maximal filter if no other proper filter on X contains .F as a proper subset. An ultrafilter .F is called principal if  .kerF = A = {x} ∈ F A∈F

for some .x ∈ X. Let X be a topological space. For any element .x ∈ X, the neighborhood system .Nx of x is called the neighborhood filter of x. Definition 5.3. Let .(X, T ) be a topological space, let .x ∈ X, and let .F be a filter on X. Then .F is said to converge to x if .F contains the neighborhood filter of x. For any element x of a topological space X, the neighborhood filter .Nx of x converges to x. Remark 5.4. Let .[0, ∞] be the topological space equipped with the natural compact topology and let .F be an ultrafilter of subsets of .N. For each sequence .{xi }i∈N , let . lim xi be the unique .x ∈ [0, ∞] such that .{i ∈ N : xi ∈ U } ∈ F for i→F

any neighborhood U of x. This generalized limit has the following properties:

5.3. MYCIELSKI’S PARTIAL SOLUTION

139

(i) For each given .n ∈ N, let .f : [0, ∞]n → [0, ∞] be a continuous function. If .xi,j ∈ [0, ∞] for all .i ∈ N and .j ∈ {1, 2, . . . , n}, then     lim xi,1 , lim xi,2 , . . . , lim xi,n . . lim f xi,1 , xi,2 , . . . , xi,n = f

.

i→F

i→F

i→F

i→F

(ii) For any sequences .{xi }i∈N and .{yi }i∈N in .[0, ∞], it holds that

.

.

lim (xi + yi ) = lim xi + lim yi ,

i→F

i→F

i→F

where we set .x + ∞ = ∞ for all .x ∈ [0, ∞]. We remember that the Borel measure is the measure .μ defined on the .σ-algebra of Borel sets. Theorem 5.5. Let .(X, d) be a metric space. If K is a compact subset of X that is thick in X, then there exists a Borel measure .μ on X with the properties: (i) .μ(K) = 1.

.

(ii) If U and V are open subsets of X that are isometric to each other, then .μ(U ) = μ(V ).

.

Proof. Let .C be the collection of all compact subsets of X. Then it holds that .K ∈ C. .(a) We assert that there exists a function .λ : C → [0, ∞] with the following properties: (i) .λ(K) = 1.

.

(ii) There are an open subset U of X and a constant .0 < α < ∞ such that .K ⊂ U and .λ(A) < α for all .A ∈ C with .A ⊂ U .

.

(iii) .λ(A ∪ B) ≤ λ(A) + λ(B) for all .A, B ∈ C.

.

(iv) .λ(A ∪ B) = λ(A) + λ(B) for all .A, B ∈ C with .A ∩ B = ∅.

.

(v) .λ(A) = λ(B) for all .A, B ∈ C that are isometric to each other.

.

For the proof of the assertion above, we put λ(A) = lim

.

i→F

E(A, 1/i) E(K, 1/i)

for all .A ∈ C, where .F is a non-principal ultrafilter of subsets of .N. All properties from .(i) to .(v) are obvious from this definition and the assumption that K is thick.

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

140 (b) We define

.

  μ0 (U ) = sup λ(A) : A ⊂ U and A ∈ C ,

.

for all open subsets U of X, and   ∗ .μ (V ) = inf μ0 (U ) : V ⊂ U and U is open in X for any subset V of X. We assert that .μ∗ has the following properties: (i) .μ∗ (∅) = 0.

.

(ii) .1 ≤ μ∗ (K) < ∞.

.

(iii) If .V1 and .V2 are subsets of X with .V1 ⊂ V2 , then .μ∗ (V1 ) ≤ μ∗ (V2 ).

.

(iv) If .{Vi }i∈N is a sequence of subsets of X, then  ∞

∞  ∗ .μ Vi ≤ μ∗ (Vi ).

.

i=1

i=1

(v) If .V1 and .V2 are subsets of X that are positively separated, then .μ∗ (V1 ∪ V2 ) = μ∗ (V1 ) + μ∗ (V2 ).

.

(vi) If .U1 and .U2 are open subsets of X that are isometric to each other, then ∗ ∗ .μ (U1 ) = μ (U2 ).

.

Due to .(i) in .(a), if we put .A = K and .B = ∅ in .(iv) of .(a), then we have λ(∅) = 0. Thus, .(i) is true. Furthermore, .(ii) is true by .(i) and .(ii) of .(a). .(iii) is obviously true by the definition of .μ∗ . Now we focus on the proof of .(iv). We prove that  ∞

∞  .μ0 Ui ≤ μ0 (Ui ) (5.1)

.

i=1

i=1

for any sequence .{Ui }i∈N of open subsets of X. Assume that .A ∈ C with A⊂

∞ 

.

Ui .

i=1

Then, since A is a compact subset of X, there exists an .m ∈ N such that A⊂

m 

.

i=1

Ui

5.3. MYCIELSKI’S PARTIAL SOLUTION

141

and there exist sets .A1 , A2 , . . . , Am ∈ C such that A=

m 

.

Ai and Aj ⊂ Uj

i=1

for each .j ∈ {1, 2, . . . , m}. Hence, the inequality (5.1) follows from .(iii) of .(a) and the definition of .μ0 . Now, for any .ε > 0, we can choose open subsets .Ui of X with .Vi ⊂ Ui and μ0 (Ui ) ≤ μ∗ (Vi ) +

.

1 ε. 2i

Then it follows from (5.1) that  ∞

 ∞

∞ ∞   ∗ .μ Vi ≤ μ0 Ui ≤ μ0 (Ui ) ≤ μ∗ (Vi ) + ε, i=1

i=1

i=1

i=1

which implies the validity of .(iv). To the proof of .(v), since .V1 and .V2 are positively separated, there exist open subsets .U1 and .U2 of X such that .V1 ⊂ U1 , .V2 ⊂ U2 , and .U1 ∩ U2 = ∅. For each .A ∈ C with .A ⊂ U1 ∪ U2 , we have .A ∩ U1 ∈ C and .A ∩ U2 ∈ C. Thus, by .(iv) in ∗ ∗ ∗ .(a), we get .μ (V1 ∪ V2 ) ≥ μ (V1 ) + μ (V2 ) and .(v) follows from .(iv). Finally, .(vi) follows from .(v) of .(a). ∗ .(c) Due to .(i), .(iii), .(iv), and .(v) of .(b), .μ is a metric outer measure on X. Thus, by Theorem 3.30, every Borel set in X is .μ∗ -measurable. In view of .(ii) in .(b), we can write μ(A) =

.

μ∗ (A) μ∗ (K)

for all .A ∈ B, where we use .B to denote the .σ-algebra of all Borel sets in X. Therefore, .μ is a Borel measure on .B and .μ(K) = 1. The last requirement of our theorem is satisfied by .(vi) of .(b). .2 According to Remark 4.1 .(iii), .(I ω , Tω ) is a compact subspace of .(Rω , T ). Thus, .I ω is a compact subspace of .I ω . Moreover, we note that .Tω is the topology for .I ω generated by the metric .da and .I ω is thick in itself. If we substitute .I ω for both X and K in Theorem 5.5, then we immediately obtain .(i) and .(iii) of the following theorem. Theorem 5.6. There exists a Borel measure .μ on .I ω with the following properties: (i) .μ(I ω ) = 1.

.

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

142

(ii) If K is a thin subset of .I ω , then .μ(K) = 0.

.

(iii) If U and V are open subsets of .I ω that are .da -isometric to each other, then .μ(U ) = μ(V ).

.

In particular, the measure .μ is the Borel measure given in Theorem 5.5. Proof. We only have to prove .(ii). Assume that K is a thin subset of .I ω . In view of Definition 5.1 .(i), for any .ε > 0, there exist a .δε > 0 and an open subset .Uε of ω such that .K ⊂ U and .I ε E(C, t) ≤ εE(I ω , t)

.

for all compact subsets C of .Uε and .0 < t < δε . Furthermore, we get λ(C) = lim

.

i→F

E(C, 1/i) ≤ε E(I ω , 1/i)

for all compact subsets C of .Uε , where .λ is defined in .(a) of the proof of Theorem 5.5, which implies that .λ(C) = 0 for all compact subsets C of .Uε . Therefore, it holds that   μ0 (Uε ) = sup λ(C) : C is a compact subset of Uε = 0,

.

where .μ0 is defined in .(b) of the proof of Theorem 5.5. Finally, by (c) in the proof of Theorem 5.5, we obtain μ∗ (K) μ∗ (I ω )   inf μ0 (U ) : K ⊂ U and U is open in I ω   = inf μ0 (U ) : I ω ⊂ U and U is open in I ω μ0 (Uε ) ≤ μ0 (I ω ) = 0,

μ(K) =

.

as required.

2

.

Proposition 5.7. If K is a compact subset of .I ω and for each .n ∈ N there exists an open subset U of .I ω such that .K ⊂ U and there are disjoint subsets .U1 , U2 , . . . , Un of .I ω , all .da -isometric to U , then K is thin.

5.3. MYCIELSKI’S PARTIAL SOLUTION

143

Proof. Given any .ε > 0, we select an integer n not less than . 1ε . Let U be an open subset of .I ω such that .K ⊂ U and there exist disjoint subsets .U1 , U2 , . . . , Un of ω .I , all .da -isometric to U . Then for any compact subset C of U , there are n disjoint compact subsets .Ci of .Ui , all .da -isometric to C. We set δ=

.

  1 min da (x, y) : x ∈ Ci and y ∈ Cj for i = j . 2

Then, we have E(C, t) ≤

.

1 E(I ω , t) ≤ εE(I ω , t) n

for all .0 < t < δ. Thus, it follows that K is thin.

2

.

We note that the metric .da on the Hilbert cube .I ω is translation-invariant, i.e., ω with .x + z, y + z ∈ I ω . .da (x, y) = da (x + z, y + z) for all .x, y, z ∈ I We denote by the symbol .π the standard product probability measure on .I ω . Lemma 5.8. The Borel measure .μ of Theorem 5.6 is unique and .μ = π. Proof. Since .da is translation-invariant, so is .μ over all open subsets of .I ω . We now consider a covering of .I n with .mn isometric n-cubes with non-overlapping interiors. We denote by .Cmn the collection of .mn open cylinders in .I ω over the interiors of those n-cubes. We will prove that μ(K) =

.

1 = π(K) mn

(5.2)

for all cylinders .K ∈ Cmn . Since .μ is translation-invariant over all open subsets ∈ Cmn . On of .I ω , we have .μ(K1 ) = μ(K2 ) for all cylinders .K1 , K2  the other hand, it follows from Proposition 5.7 that the boundary .∂( Cmn ) of . Cmn is a finiteunion of thin subsets of .I ω . Thus, by Theorem 5.6 .(ii), we conclude that .μ(∂( Cmn )) = 0. ∞  Cmn generates the Borel .σSince the collection of all open cylinders of . m,n=1

algebra over .I ω , by well-known facts, (5.2) implies .μ = π.

2

.

J. Mycielski proved the conjecture of Ulam affirmatively under the additional assumption that the sets are open. Theorem 5.9. Let .U1 and .U2 be open subsets of the Hilbert cube .I ω . If .U1 is .da -isometric to .U2 , then .π(U1 ) = π(U2 ).

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

144

Proof. The assertion of this theorem can easily be proved by using Theorem 5.6 and Lemma 5.8. .2 As we have seen in this section, J. Mycielski has positively proved Ulam’s conjecture under the additional assumption that the related sets are open. And he asked whether one could prove Ulam’s conjecture by assuming that the related sets are closed instead of open.

5.4 Fickett’s Partial Solution In 1982, J. W. Fickett succeeded in partially proving Ulam’s conjecture by proving the following in a different way from J. Mycielski: if the .ai ’s tend to 0 fast enough so that 1/2i+1

ai+1 . ai

→ 0 as i → ∞,

(5.3)

then any two Borel subsets of the Hilbert cube .I ω which are .da -isometric have the same standard product probability measure. For any subset E of .Rn , we denote by .H(E) the smallest flat containing E, i.e.,  H(E) = λ0 p0 + λ1 p1 + · · · + λn pn : p0 , p1 , . . . , pn ∈ E;  . λ0 , λ1 , . . . , λn ∈ R and λ0 + λ1 + · · · + λn = 1 . We will write .H(p0 , p1 , . . . , pk ) instead of .H({p0 , p1 , . . . , pk }). The points .p0 , p1 , . . . , pk of .Rn are said to be independent if .H(p0 , p1 , . . . , pk ) is k-dimensional. For example, three points are independent if and only if they are not collinear. We use .de (·, ·) to denote the usual Euclidean distance for .Rn . It is easy to check that if .p0 , p1 , . . . , pn are independent in .Rn and .d0 , d1 , . . . , dn ≥ 0, then the equations .de (x, pi ) = di , for all .i ∈ {0, 1, . . . , n}, have at most one solution n .x ∈ R . The following is also easy to prove. Proposition 5.10. If .p0 , p1 , . . . , pn−2 are independent in .Rn and .d0 , d1 , . . . , dn−2 ≥ 0, then the subset S of .Rn defined by   S = x ∈ Rn : de (x, pi ) = di for i ∈ {0, 1, . . . , n − 2}

.

is a .(possibly degenerate.) circle with center in .H(p0 , p1 , . . . , pn−2 ). If S is nondegenerate, then .H(S) is perpendicular to .H(p0 , p1 , . . . , pn−2 ).

5.4. FICKETT’S PARTIAL SOLUTION

145

The following lemma is a special case of Lemma 5.12. The latter is the main tool for the proof of Theorem 5.13. Lemma 5.11. Let S and c be a circle and a point in .R3 , respectively. Assume that the perpendicular projection of c on the plane of S lies outside of S or on S. Let d be given such that .0 < d ≤ 1 and the ball with center c and radius d intersects S. For any .0 < ε ≤ 1 and any point p on S with .|d − de (p, c)| ≤ ε, there exists a √ point q on S with .de (c, q) = d and .de (p, q) ≤ 2 ε. Proof. Consider the ball with center c and radius d and also the spheres with center c and radii .d − ε and .d + ε, and intersect them with the plane of S. Then the ball and spheres become a circle .S  and an annulus about .S  , respectively. We note that    .S has its center .c outside of S or on S. Let q be one of the two points of .S ∩ S closer to p. The worst case is when p lies on the outer boundary of the annulus and the line segment .qp is tangent to .S  . In this case, we get the final result by applying .2 the Pythagorean theorem to the triangle .Δcpq. Lemma 5.12. Assume that .p0 , p1 , . . . , pn−1 are independent points of .Rn . Define n that is no further away .H = H(p0 , p1 , . . . , pn−2 ) and let .pn be any point of .R from H as .pn−1 . Define .di = de (pn , pi ) for all .i ∈ {0, 1, . . . , n − 2}. Assume that .di ≤ 1 for .i ∈ {0, 1, . . . , n − 2} and that .dn−1 and .0 < ε ≤ 1 are given with .|dn−1 − de (pn , pn−1 )| ≤ ε. If there exists a point q with .de (q, pi ) = di for all √ ε. .i ∈ {0, 1, . . . , n − 1}, then there exists such a q with .de (pn , q) ≤ 2 Proof. Due to Proposition 5.10, the set   .S = x ∈ Rn : de (x, pi ) = di for each i ∈ {0, 1, . . . , n − 2} is a circle with radius r and center in H. If .r = 0, then .pn = q, so we are done. So let us assume that .r > 0. Then, by Proposition 5.10, H is perpendicular to S and passes through the center of circle S. Let K be a three-dimensional flat containing S and .pn−1 . Then .H ∩ K is a line through the center of S, perpendicular to the plane of S. Moreover, .pn−1 is at least r from this line, and .pn is on S. The ball of radius .dn−1 about .pn−1 intersects S. So we may apply Lemma 5.11 with S, p, c, d, and q there equal to S, .pn , .pn−1 , .dn−1 , and q here, respectively, to get the desired result. .2 Let .(X, d1 ) and .(Y, d2 ) be metric spaces, and let .δ > 0. We remember that a function .g : X → Y is called a .δ-isometry if     .d2 g(x), g(y) − d1 (x, y) ≤ δ for all .x, y ∈ X. That is, a .δ-isometry is a function that preserves distances within δ.

.

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

146

First we describe a method for converting a .δ-isometry, which maps a bounded subset of .Rn into .Rn , into an isometry: let S be a bounded subset of .Rn and let n .g : S → R be a .δ-isometry for some .δ > 0. Step 1. We extend the .δ-isometry g to .S, where .S denotes the closure of S. We define a function .g1 : S → Rn by ⎧ ∞ ⎨  g y ∈ S : d (x, y) < 1  (for x ∈ S \ S), e i .g1 (x) ∈ ⎩ i=1 {g(x)} (for x ∈ S). Then we can easily show that .g1 is also a .δ-isometry. Step 2. Assume that .H(S) is k-dimensional. We select .s0 , s1 , . . . , sk ∈ S such that   de (s0 , s1 ) = d S ,    (5.4)  . de si , H(s0 , s1 , . . . , si−1 ) = sup de (s, H(s0 , s1 , . . . , si−1 )) : s ∈ S for all .i ∈ {2, 3, . . . , k}. We note that the conditions in (5.4) force .s0 , s1 , . . . , sk to be independent. Step 3. We define inductively f (s0 ) = g1 (s0 ), .

f (si ) = a point as close as possible to g1 (si ) satisfying

(5.5)

de (f (si ), f (sj )) = de (si , sj ) for 0 ≤ j ≤ i ≤ k. And finally, for any .s ∈ S, we define f (s) = the unique point in H(f (s0 ), f (s1 ), . . . , f (sk )) satisfying .

de (f (s), f (si )) = de (s, si ) for 0 ≤ i ≤ k.

(5.6)

We remember that we use the symbol .d(S) to denote the diameter of any subset S of .Rn , i.e., .d(S) = sup{de (x, y) : x, y ∈ S}. For any .δ ≥ 0, we define 1−i

.

K0 (δ) = K1 (δ) = δ, K2 (δ) = 3(3δ)1/2 , and Ki (δ) = 27δ 2

(5.7)

for each integer .i ≥ 3. Theorem 5.13. Let S be a bounded subset of .Rn and .g : S → Rn a .δ-isometry, δ where .0 ≤ δ ≤ 1 and .0 ≤ 3Kn ( d(S) ) ≤ 1. Then the isometry .f : S → Rn gotten by applying the above construction to g satisfies     δ d(S). . sup de (f (s), g(s)) : s ∈ S ≤ Kn+1 d(S)

5.4. FICKETT’S PARTIAL SOLUTION

147

Proof. The general case can be reduced to the case .d(S) = 1 by a homothety argument. Apply the above construction to g, and let .g1 , .s0 , s1 , . . . , sk and f be as given there. We apply induction on m to prove that if .t0 , t1 , . . . , tm ∈ S, with .t0 , t1 , . . . , .tm−1 independent, satisfy     de ti , H(t0 , t1 , . . . , ti−1 ) ≥ de tj , H(t0 , t1 , . . . , ti−1 ) ,

.

(5.8)

for all integers .1 ≤ i ≤ j ≤ m, and if .h : {t0 , t1 , . . . , tm } → Rn is defined inductively by h(ti ) is a point as close as possible to g1 (ti ) satisfying .

de (h(ti ), h(tj )) = de (ti , tj ) for all integers 0 ≤ j ≤ i ≤ m,

then   de h(ti ), g1 (ti ) ≤ Km (δ)

.

for all .i ∈ {0, 1, . . . , m}. This assertion is obviously true for .m ∈ {0, 1}. For some integer .m ≥ 2, we assume the truth of the assertion for m-point sets, and let .t0 , t1 , . . . , tm and h be as described. Then each of .{t0 , . . . , tm−2 , tm−1 } and .{t0 , t1 , . . . , tm−2 , tm } satisfies an m-point version of (5.8). Let us define ⎧ (for 0 ≤ i ≤ m − 1), ⎪ ⎪ h(ti ) ⎨ a point as close as possible to g1 (tm ) .h1 (ti ) = satisfying de (h1 (tm ), h1 (ti )) = de (tm , ti ) (for i = m). ⎪ ⎪ ⎩ for 0 ≤ i ≤ m − 2 It can easily be proved by induction that .h1 (tm ) ∈ H(h(t0 ), h(t1 ), . . . , h(tm )). By the induction hypothesis, we have   de h1 (ti ), g1 (ti ) ≤ Km−1 (δ)

.

for any .i ∈ {0, 1, . . . , m}. Hence, we get     de h1 (tm−1 ), h1 (tm ) − de (tm−1 , tm ) ≤ δ + 2Km−1 (δ) ≤ 3Km−1 (δ).

.

Thus, we can apply Lemma 5.12 with .p0 , p1 , . . . , pm , .dm−1 , .ε, and .Rm there equal to .h1 (t0 ), h1 (t1 ), . . . , h1 (tm ), .de (tm , tm−1 ), .3Km−1 (δ), and .H(h(t0 ), . . . , h(tm ))

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

148

to get a .q ∈ H(h(t0 ), h(t1 ), . . . , h(tm )) with .de (q, h(ti )) = de (tm , ti ) for all i ∈ {0, 1, . . . , m − 2} and .de (q, h(tm )) ≤ 2(3Km−1 (δ))1/2 . Thus, we have     de g1 (tm ), h(tm ) ≤ de g1 (tm ), q     ≤ de g1 (tm ), h1 (tm ) + de h1 (tm ), q .  1/2 ≤ Km−1 (δ) + 2 3Km−1 (δ)

.

≤ Km (δ), which completes the proof of our first assertion. Finally, let .s ∈ S be arbitrary. By (5.4), (5.5), and (5.6), we can apply the proceeding with .m = k + 1, .t0 = s0 , t1 = s1 , . . . , tk = sk , .tm = s, and .h = f to conclude that   .de f (s), g1 (s) ≤ Kk+1 (δ) ≤ Kn+1 (δ) for all .s ∈ S.

2

.

In the following theorem, a measure .μ defined on the Borel .σ-algebra of subsets of the Hilbert cube .I ω is said to be .da -invariant if .da -isometric Borel sets have the same .μ-measure. Theorem 5.14. Let .{ai }i∈N0 be a sequence of positive real numbers that satisfies the condition (5.3). Then the standard product probability measure .π on .I ω is .da -invariant. Proof. The proof uses Theorem 5.13 and a modification of the reduction of Ulam’s conjecture proposed by J. Mycielski in [17, Theorem 5]. Assume that .a = {ai }i∈N0 is a sequence of positive real numbers that satisfies the condition (5.3). We set  rn =

∞

.

1/2 a2i

n

1−1/2n

and εn = 27rn1/2 r0

i=n

for all .n ∈ N0 . We note that .εn = r0 Kn+1 ( rrn0 ), where .Kn+1 is defined in (5.7). Let us define the parallelepiped .Pn in .Rn by Pn = [0, a0 ] × [0, a1 ] × · · · × [0, an−1 ].

.

Let   Pn = x ∈ Rn : de (x, Pn ) ≤ εn ,

.

5.4. FICKETT’S PARTIAL SOLUTION

149

where .de is the usual Euclidean distance for .Rn , and δn =

.

μn (Pn ) − 1, μn (Pn )

where .μn denotes the n-dimensional Lebesgue measure. We assert that δn → 0 as n → ∞.

(5.9)

.

Indeed, it easily follows from (5.3) that an ≤

.

1 an−1 2n

(5.10)

and hence rn2

=

∞

a2i

i=n a2n +

a2n+1 + a2n+2 + · · · 1 1 ≤ a2n + n+1 a2n + n+2 a2n+1 + · · · 4 4 1 1 1 ≤ a2n + n+1 a2n + n+2 n+1 a2n + · · · 4 4 4 ≤ 4a2n =

.

for all sufficiently large integers n. Hence, it follows from (5.3) that 

εn an−1

2

2/2n 2−2/2n

272 rn r0 = a2n−1

n

2−2/2n

272 (4a2n )1/2 r0 ≤ a2n−1

.

n

(5.11) 1/2n−1

2−2/2n an

= 272 41/2 r0

a2n−1

→0 as .n → ∞. Thus, there exists an .n0 ∈ N such that .

for all integers .n > n0 .

εn an−1 εn 1 or 2 ≤ ≤ 2 ai ai an−1

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

150 Therefore, we have

(a0 + 2εn )(a1 + 2εn ) · · · (an−1 + 2εn ) μn (Pn ) ≤ μn (Pn ) a0 a1 · · · an−1   n−1  εn 1+2 = . ai i=0   n−2   n0    εn an−1 εn 1+2 1+ ≤ 1+2 an−1 ai ai i=0

i=n0 +1

for all integers .n > n0 . It follows from (5.10) that an−1 an−1 = an−2 ai 1 ≤ n−1 2 . =

an−2 ai+1 ··· an−3 ai 1 1 · n−2 · · · i+1 2 2 1 ·

2((n−1)n−i(i+1))/2 1 ≤ n−1 2

for all .i ∈ {n0 + 1, n0 + 2, . . . , n − 2}. Hence, we have 1+

.

an−1 ≤ 1 + 21−n , ai

for all .i ∈ {n0 + 1, n0 + 2, . . . , n − 2}, and μn (Pn ) ≤ . μn (Pn )

   

n0  εn εn 1+2 1+2 (1 + 21−n )n−n0 −2 an−1 ai i=0

for all integers .n > n0 , which tends to 1 as n tends to .∞, and thus, (5.9) holds. We define a function .ωn : I ω → Pn by ωn (x) = (a0 x0 , a1 x1 , . . . , an−1 xn−1 )

.

and νn (E) =

.

for any Borel set .E ⊂ Pn .

1 μn (E) a0 a1 · · · an−1

5.4. FICKETT’S PARTIAL SOLUTION

151

Furthermore, we define    ω : d (x, E) ≤ t ω x ∈ I a (t)  (for E ⊂ I ), .E =  x ∈ Pn : de (x, E) ≤ t (for E ⊂ Pn ) for any .t ≥ 0. Thus it holds that  .

ωn (E)

(t)

  = ωn E (t)

(5.12)

for .E ⊂ I ω . We assert that    νn ωn S (εn ) → π(S) as n → ∞

.

(5.13)

for all compact subsets S of .I ω . Let .η > 0 be given. Since .π is regular, we can select .t > 0 such that   1 π S (t) < π(S) + η. 2

.

Since .π is a product measure, there exists an .N ∈ N such that      1 νn ωn S (t) < π S (t) + η < π(S) + η 2

.

for all .n > N . The condition (5.3) or (5.11) implies that .εn → 0 as .n → ∞. Thus, we choose an .M > N such that .εn < t for .n > M . Then we have       (εn ) < νn ωn S (t) < π(S) + η .π(S) < νn ωn S for all .n > M , which proves (5.13). Now we fix a compact subset S of .I ω and a .da -isometry .f : S → I ω . Let −1 .qn : ωn (S) → S be any function satisfying .qn (x) ∈ ωn (x) for all .x ∈ ωn (S). Moreover, we define a function .Fn : ωn (S) → Pn by Fn = ωn ◦ f ◦ qn .

.

Then     Fn ωn (S) ⊂ ωn f (S) .

.

Since   de (x, y) ≤ da qn (x), qn (y) ≤ de (x, y) + rn

.

(5.14)

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

152 and

    da qn (x), qn (y) − rn = da f (qn (x)), f (qn (y)) − rn   ≤ de Fn (x), Fn (y)   . ≤ da f (qn (x)), f (qn (y))   = da qn (x), qn (y)

for all .x, y ∈ ωn (S), it holds that .Fn is an .rn -isometry. Hence, due to Theorem 5.13, there exists an isometry .fn : ωn (S) → Rn within .εn of .Fn . Therefore, it holds that .fn (ωn (S)) ⊂ Pn . Since       μn fn (ωn (S)) ∩ Pn ≥ μn fn (ωn (S)) − μn Pn \ Pn   . ≥ μn ωn (S) − δn μn (Pn ), we have μn

.

  (ε )   Fn (ωn (S)) n ≥ μn ωn (S) − δn μn (Pn )

or νn

.



Fn (ωn (S))

(εn ) 

  ≥ νn ωn (S) − δn .

Thus, given .ε > 0, we select an .N ∈ N such that      π f (S) + ε ≥ νn ωn f (S)(εn )  (ε )  = νn ωn (f (S)) n  (ε )  . ≥ νn Fn (ωn (S)) n   ≥ νn ωn (S) − δn

(5.15)

by (5.13) by (5.12) by (5.14) by (5.15)

≥ π(S) − δn for all .n ≥ N . It follows from (5.9) that .δn → 0 as .n → ∞. Hence, we have π(f (S)) ≥ π(S). By its symmetric property, it holds that .π(f (S)) = π(S). Thus, ω have the same .π-measure, and so, by regularity .da -isometric compact subsets of .I of .π, .π is .da -invariant. .2 .

5.5 Jung and Kim’s Partial Solution S.-M. Jung and E. Kim proved that Ulam’s conjecture is true if the sequence .a = {ai }i∈N of positive real numbers is monotone decreasing and satisfies the condition 

ai+1

.

ai =o √ i

 as i → ∞.

5.5. JUNG AND KIM’S PARTIAL SOLUTION

153

In other words, they proved that Ulam’s conjecture is true if the sequence .a = {ai }i∈N satisfies the equivalent condition √ i ai+1 . → 0 as i → ∞. (5.16) ai Obviously the last condition is much weaker than the condition (5.3) of Fickett. In this section, we assume that .a = {ai }i∈N is a monotone decreasing sequence of positive real numbers satisfying the condition (5.16) and we define a sequence .{δi }i∈N by δi2 =

∞

.

a2j + a2i+1

(5.17)

j=i+1

for any .i ∈ N. We set .I = [0, 1] and   .G = G ⊂ Ma : there exist an F ∈ F and a da -isometry of F onto G , (5.18) where .F is the set of all non-degenerate basic cylinders in .I ω , i.e.,  ∞  F= Ii : Ii is a non-degenerate closed interval in I; .

i=1



Ii = I for at most finitely many i . We recall that we use the symbol .da (E) to denote the diameter of any subset E of .Ma , i.e., .d(E) = sup{da (x, y) : x, y ∈ E}. Moreover, we set U=

∞  

.

   G ∈ G : da (G) = δi ∪ ∅

(5.19)

i=1

and define

  Fδ = F ∈ F : da (F ) < δ ,   G ∈ G : da (G) < δ , . Gδ =   Uδ = U ∈ U : da (U ) < δ

for all .δ > 0. For a fixed compact subset K of a metric space .(X, d), let .E(K, δ) be the entropy that denotes the least number of sets of diameter .< δ necessary to cover K. With this notation, J. Mycielski has introduced the following pre-measure: h(G) =

.

1 E(K, d(G))

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

154

to construct some special Hausdorff measures (see Sect. 5.3). We apply this definition in the modified form τ (G) =

.

1 E  (I ω , d

a (G))

and τ (∅) = 0,

(5.20)

where .E  (I ω , δ) denotes the least number of sets in .Uδ necessary to cover .I ω and ∞  Uδk . where .τ is defined on the collection of all cylinders in . k=1

Let J be a (closed) basic cylinder from .F and let K be an arbitrary cylinder in .G such that there exists a .da -isometry f of J onto K. Since J is a compact subset of .Ma as a closed subset of a compact set .I ω , Theorem 1.23 implies that K is also a compact subset of .Ma as the continuous image of a compact set J. Moreover, by Theorem 1.43, we conclude that K is a closed subset of .Ma as a compact subset of the Hausdorff space .Ma . Hence, we come to an important consequence. Remark 5.15. (i) Every cylinder in .G is closed in .I ω .

.

(ii) .G and .U are closed under the actions of .da -isometries.

.

It is worth noting that .τ does not depend on the shape of sets, but it depends on the diameter of the sets only. According to Theorem 3.24, an outer measure .μ∗ may be constructed by applying the “Method II” from the pre-measure .τ as we see in the following:  ∗  ∗ .μ (C) = sup μk (C) : k ∈ N for all subsets C of .Ma , where  ∞ ∗ .μk (C) = inf τ (Ui ) : {Ui }i∈N ⊂ Uδk is a covering of C . i=1

Remark 5.16. According to Theorems 3.26 and 3.30, we remark that (i) The outer measure .μ∗ is metric.

.

(ii) Each Borel set in .I ω is .μ∗ -measurable.

.

(iii) The translation invariance of .da implies that .μ∗ on .I ω is also invariant under translation.

.

For any cylinder .G ∈ G, we denote by .|G|i , .i ∈ N, the (Euclidean) length of ∞  |G|i the volume of G. the ith edge of G and by .vol(G) = i=1

5.5. JUNG AND KIM’S PARTIAL SOLUTION

155

Which of the cylinders .G ∈ G of a given diameter D has the largest volume? The following two lemmas provide a partial answer to the above question. Lemma 5.17. Let D be a sufficiently small positive number, .m ∈ N, and .G ∈ G any cylinder. Then ⎧ D ⎨ √ (for i ∈ {1, 2, . . . , m}), (5.21) .|G|i = a m ⎩ 1i (for i > m) if and only if  vol(G) = max vol(G ) : G ∈ G is a cylinder satisfying .



m

a2i |G |2i

(5.22)



= D and |G |i = 1 for i > m . 2

i=1

Proof. There exist an .F ∈ F and a .da -isometry of F onto G and, according to Remark 6.5, it holds that .vol(F ) = vol(G). Hence, it suffices to prove our lemma for the basic cylinders in .F. m  First, under the assumption (5.21), we prove that if .G ∈ F and . a2i |G |2i = i=1

D2 (we temporarily neglect the condition, .|G |i = 1 for .i > m), then m  .

i=1

|G |i ≤

m 

|G|i

(5.23)

i=1

for all .m ∈ N and .D > 0. This assertion is true for .m = 1 and for any sufficiently small .D > 0. Now, assume that our assertion is valid for .m = q − 1 (.q ≥ 2), i.e., q−1  2 2 ai |G |i = D2 , then we assume that if .D > 0, .G ∈ F, and . i=1

q−1  .

i=1



|G |i ≤

q−1  i=1

1 |G|i = a1 a2 · · · aq−1



D √ q−1

q−1 .

(5.24)

Let x be a sufficiently small positive number and let .G ∈ F be a basic cylinder q q−1   2 2 satisfying . a2i |G |2i = D2 and .|G |q = x. Then, . ai |G |i = D2 − a2q x2 > 0 i=1

i=1

(we except the degenerate case, .|G |i = 0 for some i, from our consideration).

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

156

Since we assumed  that inequality (5.24) holds for all sufficiently small .D > 0, if we replace D with . D2 − a2q x2 in (5.24), then q−1  .

i=1

1 |G |i ≤ a1 a2 · · · aq−1



D2 − a2q x2 q−1

(q−1)/2

and q  .

|G |i = x

i=1

q−1  i=1

x |G |i ≤ a1 a2 · · · aq−1



D2 − a2q x2 q−1

(q−1)/2 =: v(x). (5.25)

If we differentiate .v(x), then 1  .v (x) = a1 a2 · · · aq−1



(q−3)/2

D2 − a2q x2 q−1

·

Since .D2 − a2q x2 > 0, .v(x) has at .x = x0 := 1 .v(x0 ) = a1 a2 · · · aq



D √ q

q =

D √ aq q

D2 − qa2q x2 . q−1

its maximum

q q   D |G|i . √ = ai q i=1

(5.26)

i=1

In view of (5.25) and (5.26), we proved that q  .

|G |i ≤ v(x) ≤ v(x0 ) =

i=1

q 

|G|i

i=1

for all basic cylinders .G ∈ F with .

q 

i=1

a2q |G |2i = D2 and for .m = q. Indeed, we

proved the validity of inequality (5.23) for all .m ∈ N. Considering vol(G ) =

∞ 

.

i=1

|G |i ≤

m  i=1

|G |i ≤

m  i=1

|G|i =

∞  i=1

|G|i = vol(G)

5.5. JUNG AND KIM’S PARTIAL SOLUTION

157

and m .

a2i |G|2i = D2 ,

i=1

we conclude that (5.21) implies (5.22). Finally, we will prove that (5.22) implies (5.21). Obviously, (5.22) implies (5.21) for .m = 1. We now assume that .m > 1 and .G ∈ G is a cylinder with .|G|i = 1 for all .i > m. Then we have vol(G) =

∞ 

.

|G|i =

i=1

m  i=1

|G|i =

m 

xi =: f (x1 , x2 , . . . , xm ),

i=1

where we temporarily set .xi := |G|i > 0 for every .i ∈ {1, 2, . . . , m}. √ holds for each .i ∈ {1, 2, . . . , m} under the It is to prove that .xi = a D i m condition g(x1 , x2 , . . . , xm ) :=

m

.

a2i x2i = D2 ,

(5.27)

i=1

which is given in (5.22). In other words, we will prove that if G is a cylinder satisfying the condition (5.22), then G has the form of (5.21). We apply the method of Lagrange multipliers to maximize the value of .f (x1 , x2 , . . . , xm ) subject to the condition (5.27) (ref. [22]). We introduce a new variable .λ and define the Lagrange function by   2 .L(x1 , x2 , . . . , xm , λ) := f (x1 , x2 , . . . , xm ) − λ g(x1 , x2 , . . . , xm ) − D and solve .

∂ ∂ L(x1 , x2 , . . . , xm , λ) = 0 L(x1 , x2 , . . . , xm , λ) = 0 and ∂λ ∂xi

for all .i ∈ {1, 2, . . . , m}. Thus, we have .

m x1 x2 · · · xm − 2λa2i xi = 0 and a2j x2j = D2 xi j=1

for any .i ∈ {1, 2, . . . , m}. From the first equalities of (5.28), we get λ=

.

x1 x2 · · · xm or a21 x21 = a22 x22 = · · · = a2m x2m 2a2i x2i

(5.28)

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

158

for any .i ∈ {1, 2, . . . , m}. By the second equality of (5.28), we obtain |G|i = xi =

.

D √ ai m

for all .i ∈ {1, 2, . . . , m}, which implies that G has the form of (5.21).

2

.

Lemma 5.18. Let .a = {ai }i∈N be a monotone decreasing sequence of positive real numbers satisfying the condition (5.16). Let .j ∈ N be sufficiently large. Assume ∞  a2i + a2j+1 and that U is a cylinder in .U with .da (U )2 = i=j+1

⎧ √ 2 aj+1 ⎨ √ .|U |i = ⎩ ai j + 1 1

(for i ∈ {1, 2, . . . , j + 1}), (for i > j + 1).

Then   1 sup vol(U  ) : U  ∈ Gda (U ) . 2

vol(U ) ≥

.

Proof. In view of the definition of .U and Remark 6.5, it suffices to prove our lemma for the intervals in .F. Let .U  ∈ Fda (U ) be given with ∞  .

|U  |i = c ≤ 1.

i=j+2

First, assume .c = 1. Since .da (U  )2 < da (U )2 and .|U  |i = 1 for all integers .i ≥ j + 2, it follows that j+1 .

a2i |U  |2i +

i=1

∞

a2i = da (U  )2 < da (U )2 = 2a2j+1 +

i=j+2

∞

a2i

i=j+2

or equivalently that j+1 .

i=1

a2i |U  |2i < 2a2j+1 .

√ If we set .G = U , .m = j + 1, and .D = 2 aj+1 in Lemma 5.17, we conclude that our assertion is true for .c = 1. Now, let .c < 1. By Lemma 5.17, we immediately have √ j+1 j+1  2 aj+1  da (U  ) √ √ and vol(U  ) < c , .vol(U ) = a j+1 a j+1 i=1 i i=1 i

5.5. JUNG AND KIM’S PARTIAL SOLUTION

159

since j+1 .

a2i |U  |2i
0 be sufficiently small and let .{Ui }i∈N ⊂ Uδk be a covering of .I ω satisfying ∞ .

  τ (Ui ) ≤ μ∗k I ω + ε.

(5.30)

i=1

By (5.17) and (5.19), there exists, for every .i ∈ N, a sufficiently large .j(i) ∈ N such that ∞

da (Ui )2 =

2 a2n + a2j(i)+1 = δj(i) .

.

n=j(i)+1

Now, let .U ⊂ I ω be a basic cylinder satisfying ⎧ √ ⎪ 2 aj(i)+1 ⎨

(for n ∈ {1, 2, . . . , j(i) + 1}), .|U |n = a j(i) + 1 n ⎪ ⎩ 1 (for n > j(i) + 1). Then .U ∈ U and da (U )2 =

∞

a2n |U |2n

n=1



j(i)+1

= .

a2n

n=1

2a2j(i)+1 a2n (j(i) + 1)

= 2a2j(i)+1 +

∞ n=j(i)+2

= da (Ui )2 .

a2n

+

∞ n=j(i)+2

a2n

(5.31)

5.5. JUNG AND KIM’S PARTIAL SOLUTION

161

ω can In view of (5.31), for each √ .n ∈ {1, 2, . . . , j(i) + 1}, the nth edge of .I a j(i)+1 intervals of length .|U |n . Hence, by (5.20) be covered with at most .1 + √n2 a j(i)+1

and (5.31), we obtain  j(i)+1  

−1

an j(i) + 1 1+ √ τ (Ui ) ≥ 2 aj(i)+1 n=1 . √ j(i)+1  2 aj(i)+1

√ . = 2 aj(i)+1 + an j(i) + 1 n=1

(5.32)

It now follows from (5.16), (5.30), and (5.32) that μ∗k



.

I

ω



∞

∞ j(i)+1 

√

2 aj(i)+1

−ε 2 aj(i)+1 + an j(i) + 1 i=1 i=1 n=1  j(i)+1 √ √ 

 ∞  2 aj(i)+1 2 aj(i)+1

−ε ≥ · 1− an j(i) + 1 n=1 an j(i) + 1 i=1  j(i)+1 √

j(i)

 √ ∞  2 aj(i)+1 2 aj(i)+1 (5.33)

≥ 1− · a a j(i) + 1 j(i) + 1 n j(i) n=1 i=1

 √ 2 −ε · 1− j(i) + 1 ∞ j(i)+1 √ 2 aj(i)+1 1 

− ε, ≥ 2 an j(i) + 1 ≥

τ (Ui ) − ε ≥

√

i=1 n=1

since it follows from (5.16) that  .

√ ⎞

j ⎛  √ √ − √aj j+1 − 2 a 2 aj+1 2 aj+1 j+1 ⎠ 1− √ =⎝ 1− √ aj j + 1 aj j + 1

√ 2 jaj+1 √ aj j+1

≥

1 2

for any sufficiently large j. Since .{Ui }i∈N is a covering of .I ω , by (5.33) and Lemma 5.18, we get ∞    1  1 1 sup vol(U  ) : U  ∈ Gda (Ui ) − ε ≥ − ε μ∗k I ω ≥ 2 4 2

.

i=1

and       1 μ∗ I ω = sup μ∗k I ω : k ∈ N ≥ − ε. 4

.

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

162

Since .ε > 0 can be chosen arbitrarily small, this implies   1 μ∗ I ω ≥ . 4

.

On the other hand, for each sufficiently large .k ∈ N, there exists a covering .{Fi }i∈{1,2,...,n} .⊂ Fδ of .I ω such that .da (Fi ) = δk for all .i ∈ {1, 2, . . . , n} and k−1  ω .n = N (I , δk−1 ). Since a is a monotone decreasing sequence of positive real ∞ ∞   2 a2j + a2k+1 ≤ δk−1 = a2j + a2k , .Uδk ⊂ Uδk−1 , and since numbers, .δk2 =  ω .N (I , δ n 

k)

≥

1 τ (Fi ) n, . i=1

j=k+1 N  (I ω , δ

≤

n  i=1

1 n

k−1 ),

we

j=k  ω get .N (I , da (Fi ))

≥ n, .τ (Fi ) =

= 1, and thus, .μ∗ (I ω ) ≤ 1, as desired.

≤

1 N  (I ω ,da (Fi ))

2

.

In the following lemma, we show that the restriction of .μ∗ to the .σ-algebra of all .μ∗ -measurable subsets of .I ω , denoted by .μ , is .da -invariant. We remark that each Borel set in .I ω is .μ∗ -measurable (see Remark 5.16 .(ii)). We note that the paper [11] has been improved by replacing [11, Lemma 2.10] and [11, Lemma 3.1] with Theorem 4.26 and the following lemma, respectively. Lemma 5.20. If the monotone decreasing sequence .a = {ai }i∈N of positive real numbers satisfies the condition (5.16), then .μ is .da -invariant. Proof. Let A and B be Borel subsets of .I ω that are .da -isometric to each other via a (fixed) surjective .da -isometry .f : A → B. Without loss of generality, we assume that A is non-degenerate. Then, by Theorem 4.26, .GS2 (A, p) includes every cylinder from .G, where p is a (fixed) element of A. (When A is degenerate, we can revise Theorem 4.26 by replacing .N with .Λ(A). Then, .GS2 (A, p) − p is a proper subspace of .Ma .) If .F2 : GS2 (A, p) → Ma is given as in Definition 4.14, then .F2 is a .da -isometry and it extends the surjective .da -isometry .f : A → B by Theorem 4.27. If .{Ui }i∈N is a covering of A with each .Ui from .G, then it follows from (5.18) that there exist a .Bi from .F and a surjective .da -isometry .fi : Bi → Ui for each i. In particular, the cylinder .Ui = fi (Bi ) from .G is included in .GS2 (A, p), the domain of .F2 , by the argument in the first part (or by Theorem 4.26). Furthermore, we know that .{F2 (Ui )}i∈N = {(F2 ◦fi )(Bi )}i∈N is a covering of B with .da (Ui ) = da (F2 (Ui )).

5.5. JUNG AND KIM’S PARTIAL SOLUTION

163

Due to the definitions of .μ and .Uδ and by the previous argument, we have μ (A) = sup μ∗k (A) k∈N  ∞ τ (Ui ) : {Ui }i∈N ⊂ Uδk is a covering of A = sup inf k∈N



= sup inf k∈N .



= sup inf k∈N

≥ sup inf k∈N



i=1 ∞   τ F2 (Ui ) : {Ui }i∈N ⊂ Uδk is a covering of A



i=1 ∞   τ F2 (Ui ) : {F2 (Ui )}i∈N ⊂ Uδk is a covering of B i=1 ∞



τ (Vi ) : {Vi }i∈N ⊂ Uδk is a covering of B

i=1

= μ (B), since some coverings of B may not be represented as images of coverings of A under .F2 . Analogously, we can easily prove the opposite inequality. .2 For all Borel sets .C ⊂ I ω , let ν(C) =

.

μ (C) . μ (I ω )

Clearly, in view of Lemma 5.20, the measure .ν is .da -invariant with .ν(I ω ) = 1. The proof of the following lemma is comparable to that of [17, Lemma 1]. Lemma 5.21. Assume that the monotone decreasing sequence .a = {ai }i∈N of positive real numbers satisfies the condition (5.16). Then, the measure .ν coincides with the standard product probability measure .π on the Borel subsets of .I ω . Proof. Given any positive integers m and n, let .Zmn be the collection of the nondegenerate basic cylinders C (in .I ω ) defined by C=

∞ 

.

Ci ,

i=1

where ⎧" # ⎨ j j+1 , for some j ∈ {0, 1, . . . , m − 1} (for i ≤ n), .Ci = m m ⎩ [0, 1] (for i > n).

CHAPTER 5. HISTORY OF ULAM’S CONJECTURE

164

The translation invariance of .da implies that of .ν. Thus we have ν(C1 ) = ν(C2 )

.

(5.34)

for all .C1 , C2 ∈ Zmn . Let C be an arbitrary basic cylinder in .Zmn . In general, we let   .C = (x1 , x2 , . . .) ∈ I ω : bi ≤ xi ≤ ei for all i ∈ N , where .0 ≤ bi < ei ≤ 1 for .i ∈ {1, 2, . . . , n} and .bi = 0 and .ei = 1 for .i > n. Let .∂C denote the boundary of C. Then, there are at most countably many sets .H1 , H2 , . . . of the forms   .H2i−1 = (x1 , x2 , . . .) ∈ I ω : xi = bi and   H2i = (x1 , x2 , . . .) ∈ I ω : xi = ei

.

satisfying ∂C ⊂

∞ 

.

Hk .

(5.35)

k=1

It is obvious that there are infinitely many disjoint translates of .Hk in .I ω for every .k ∈ N. Since . 14 ≤ μ∗ (I ω ) ≤ 1 by Lemma 5.19, we obtain μ∗ (Hk ) = 0

.

for each .k ∈ N. Due to Theorem 3.6, .Hk is .μ∗ -measurable. Hence, .μ (Hk ) = 0 and ν(Hk ) =

.

μ (Hk ) = 0. μ (I ω )

(5.36)

It follows from (5.35) and (5.36) that ν(∂C) = 0

.

(5.37)

for all .C ∈ Zmn . In addition, on account of Remark 5.16 .(ii), we conclude that all Borel subsets of .I ω are .μ∗ -measurable. Hence, by (5.37), we have  ◦ .ν(C) = ν C

5.5. JUNG AND KIM’S PARTIAL SOLUTION

165

for each .C ∈ Zmn . Therefore, we get





       .ν C◦ ≤ ν Iω ≤ ν C ≤ν C◦ , C∈Zmn

C∈Zmn

C∈Zmn

and by (5.34), we have   1 ν(C) = ν C ◦ = n = π(C) m

.

(5.38)

for all .C ∈ Zmn . Finally, given an .n ∈ N, let .Jn be an arbitrary non-degenerate basic cylinder defined by ∞ 

Jn =

.

[p1i , p2i ],

i=1

where .0 ≤ p1i < p2i ≤ 1 for any .i ∈ {1, 2, . . . , n} and .p1i = 0, .p2i = 1 for any .i > n. For all positive integers m, we define the finite collection .Cmn of basic cylinders by   .Cmn = C ∈ Zmn : C ∩ Jn = ∅ . Then, every non-degenerate basic cylinder .Jn can be expressed as ∞ 

Jn =



.

C,

m=1 C∈Cmn

where the finite union .



C of basic cylinders is closed.

C∈Cmn

Since the collection of all non-degenerate basic cylinders .Jn for all .n ∈ N, together with the empty set, generates the Borel .σ-algebra over .I ω , we conclude ∞  Zmn , together with the empty set, generate the that the basic cylinders from . m,n=1

Borel .σ-algebra over .I ω . Therefore, (5.38) implies that .ν coincides with .π on the .2 Borel subsets of .I ω . As we already mentioned in the paragraph following Lemma 5.20, the measure .ν is .da -invariant. Using Lemma 5.21, we obtain the following result: Theorem 5.22. For any monotone decreasing sequence .a = {ai }i∈N of positive real numbers satisfying (5.16), the standard product probability measure .π on .I ω is .da -invariant.

Chapter 6

Ulam’s Conjecture The conjecture of Ulam states that the standard product probability measure .π on the Hilbert cube .I ω is invariant under the induced metric .da when the sequence .a = {ai }i∈N of positive numbers satisfies condition (4.1). This conjecture was proved in [6] when .E1 is a non-degenerate subset of .Ma . In this chapter, we will completely prove Ulam’s conjecture to be true by considering both non-degenerate as well as degenerate cases. More precisely, under the assumption that the axiom of choice is accepted and the sequence .a = {ai }i∈N ∞  of positive real numbers satisfies the condition . a2i < ∞, we prove that .π(E1 ) = π(E2 ) i=1

for all Borel subsets .E1 and .E2 of .I ω which are .da -isometric to each other, where .π is the standard product probability measure on .I ω .

6.1 Basic Definitions As we did in Sects. 4.1 and 5.2, we denote by .Rω the infinite-dimensional real vector space defined as   Rω = (x1 , x2 , . . .) : xi ∈ R for all i ∈ N .

.

In addition, .(Rω , T ) denotes the product space .

∞ 

i=1

R, where .(R, TR ) is the usual

topological space. Let .I = [0, 1] be the unit closed interval, .I ω =

∞ 

I the Hilbert cube, and

i=1 on .I ω .

let .π be the standard product probability measure We denote by .(I ω , Tω ) ω the (topological) subspace of .(R , T ). Then, .Tω is the relative topology for .I ω induced by .T . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4 6

167

CHAPTER 6. ULAM’S CONJECTURE

168

As in Sect. 4.1, let .a = {ai }i∈N be a sequence of positive real numbers that satisfies the condition ∞  .

a2i < ∞.

i=1

Using this sequence a, we define the metric on .I ω by  da (x, y) =

∞ 

.

1/2 a2i (xi − yi )2

i=1

for all .x = (x1 , x2 , . . .) ∈ I ω and .y = (y1 , y2 , . . .) ∈ I ω . As in Sect. 4.1, we define the real Hilbert space .Ma by

∞  .Ma = (x1 , x2 , . . .) ∈ Rω : a2i x2i < ∞ i=1

and we define an inner product .·, ·a on .Ma by x, ya =

∞ 

.

a2i xi yi

i=1

for all .x = (x1 , x2 , . . .) and .y = (y1 , y2 , . . .) of .Ma . Then this inner product induces the norm  .xa = x, xa for all .x ∈ Ma . In view of definition (4.2), the metric .da on .I ω can be extended to the metric on .Ma , i.e.,  .da (x, y) = x − y, x − ya for all .x, y ∈ Ma .

6.2 Cylinders For each positive integer n, let .Bn be the set of all basic cylinders .J =

∞ 

Ji i=1 .Λ1 , .Λ2 , .Λ3 , .Λ4 of

defined by Definition 4.20 for some disjoint finite subsets {1, 2, . . . , n} and .0 < p1i < p2i < 1 for .i ∈ Λ1 ∪ Λ2 ∪ Λ3 and .0 ≤ p1i ≤ 1 for .i ∈ Λ4 . This definition of basic cylinders is a slight modification of Definition 4.20, .

6.2. CYLINDERS

169

but the two definitions are essentially the same. We note that .Λ1 ∪ Λ2 ∪ Λ3 ∪ Λ4 ⊂ {1, 2, . . . , n} and at most n edges of each basic cylinder of .Bn have a Euclidean length of less than 1, and all remaining edges have a Euclidean length of 1. On the other hand, in some cases the set .Λ1 ∪ Λ2 ∪ Λ3 ∪ Λ4 can be infinite. In this case, the corresponding interval will be called the extraordinary basic cylinder. ∞ 

Example 6.1. Let .J = {0} × K =

∞ 

.

i=1

[0, 1] be a degenerate basic cylinder and let

i=2

be an infinite-dimensional interval. If we define a function 0, ai+1 ai

f : J → K by

.

 f

.



1 ai+1

ei+1

=

1 ei , ai

for all .i ∈ N, and by f (x) =

∞ 

.

ai+1 xi+1

i=1

for all .x =

∞ 

x i ei =

i=1



f (x) − f (y)2a = .

=

for all .x =

∞ 

∞  i=1

1 ei ai

ai xi a1i ei ∈ J, then

∞ 

∞  1 1 aj+1 (xj+1 − yj+1 ) ej ai+1 (xi+1 − yi+1 ) ei , aj ai

i=1 ∞  a2i (xi i=2

j=1

from J onto K. Thus, .K =

a

− yi )2 = x − y2a

xi ei ∈ J and .y =

i=1



∞  i=1

∞ 

yi ei ∈ J. That is, f is a .da -isometry

i=1

is a cylinder as an isometric image of a 0, ai+1 ai

degenerate basic cylinder J. We note that K is an extraordinary basic cylinder. For .i ∈ {1, 2, . . . , n}, each .Ji is a compact subinterval of .[0, 1] with respect to the relative topology for .[0, 1]. Thus, by Theorems 1.20, 1.29, and 1.36, the infinite-dimensional interval   . [0, 1] × Ji × [0, 1] 1≤ji

is a compact subset of the Hilbert cube .I ω . Since .I ω is a Hausdorff space, Theorem 1.43 implies that the infinite-dimensional interval above is a closed subset of

CHAPTER 6. ULAM’S CONJECTURE

170

the Hilbert cube .I ω for each .i ∈ {1, 2, . . . , n}. In particular, when .i = 1, we read the last expression as    . [0, 1] × J1 × [0, 1] = J1 × [0, 1]. j>1

1≤j1

Since every basic cylinder .J ∈ Bn is expressed as   ∞ n     .J = Ji = [0, 1] × Ji × [0, 1] , i=1

i=1

1≤ji

we see that each .J ∈ Bn is a closed subset of .I ω as the intersection of closed sets. Since .I ω is a closed subset of .Ma by Remark 4.5 .(iv), we use Theorem 1.46 to conclude that J is a closed subset of .Ma . Now, let us define B = {∅} ∪

∞ 

.

  Bn and Bδ = J ∈ B : da (J) < δ

n=1

for any .0 < δ < 1, where the diameter of J is defined as .da (J) = sup{da (x, y) : x, y ∈ J}. Here we need to distinguish the difference between .Bn for the positive integer n and .Bδ for the real number .0 < δ < 1. We remark that every basic cylinder in .B is a closed subset of .Ma . (Also, each extraordinary basic cylinder is likewise a closed subset of .Ma .) We denote by .C the set of every subset K of .Ma , for which there exists a basic cylinder .J ∈ B and a surjective .da -isometry .f : J → K, and we define .Cδ = {K ∈ C : da (K) < δ} for any .δ > 0. We notice that .Bδ ⊂ Cδ for every .0 < δ < 1. We note that the family .B includes not only non-degenerate basic cylinders but also degenerate ones. Assume that a basic cylinder J and a cylinder K are given such that J is .da isometric to K through a surjective .da -isometry .f : J → K. Since J is a compact subset of .Ma as a closed subset of a compact set .I ω , K is also a compact subset of .Ma as the continuous image of a compact set J (see Theorem 1.23). Moreover, K is a closed subset of .Ma as a compact subset of the Hausdorff space .Ma (see Theorem 1.43). As we did in Remark 5.15, we come to an important consequence. Remark 6.2. Every cylinder .K ∈ C is closed in .Ma . Let J be a basic cylinder that is .da -isometric to a cylinder K via a surjective da -isometry .f : J → K. Assume that .p = (p1 , p2 , . . .) is the lower left corner of J, q is an element of K with .q = f (p), .GS(J, p) is the first-order generalized span of J with respect to p, and that .F : GS(J, p) → Ma is the extension of f given in Definition 4.11. If .x = (x1 , x2 , . . .) ∈ GS(J, p), then it follows from

.

6.2. CYLINDERS

171

Theorem 4.24 that .x − p =

∞ 

(xj − pj )ej =

j=1



(xj − pj )ej ∈ Ma . By

j∈Λ(J)

Theorem 4.31, we get (T−q ◦ F ◦ Tp )(x − p)    1 1 x − p, ei (T−q ◦ F ◦ Tp )(ei ) = ai a ai . i∈Λ(J)

=



ai (xi − pi )

i∈Λ(J)

(6.1)

1 (T−q ◦ F ◦ Tp )(ei ). ai

For any .y = (y1 , y2 , . . .) ∈ GS(J, p), it follows from Theorem 4.24 that y = p + (y − p) ∞  (yi − pi )ei =p+ i=1

.

=p+



i∈Λ(J)

ai (yi − pi )

1 ei ai

∈ GS(J, p) and it further follows from (6.1) that F (y) = q + (T−q ◦ F ◦ Tp )(y − p)  1 ai (yi − pi ) (T−q ◦ F ◦ Tp )(ei ) = q + . ai i∈Λ(J)

∈ GS(K, q), since GS(K, q) =F (GS(J, p)) by Theorem 4.32. Moreover, since the sequences   1 . 1 (T ◦F ◦T )(e ) are both orthonormal, the following e and . −q p i i ai ai i∈Λ(J) i∈Λ(J) definition may be useful.

.

Definition 6.3. Every interval in .B will be called a basic cylinder and each element of .C a cylinder. Assume that J is a basic cylinder. If .Λ(J) = N, then J will be called non-degenerate. Otherwise, it will be called degenerate. Remark 6.4. The term “non-degenerate” or “degenerate” defined in relation to basic cylinders is similar to, but not identical to, the term “non-degenerate” or “degenerate” for the general sets E defined in Definition 4.8. We notice that the terms “non-degenerate” and “degenerate” are used for convenience only and are not exact mathematical terms.

CHAPTER 6. ULAM’S CONJECTURE

172

6.3 Elementary Volumes Assume that both .J1 and .J2 are two distinct basic cylinders which are .da -isometric to the same cylinder K via the surjective .da -isometries .f1 : J1 → K and .f2 : J2 → K, respectively. Moreover, assume that u is the lower left corner and x is the vertex of .J1 diagonally opposite to u, i.e., x is the upper right corner of .J1 . Analogously, let v be the lower left corner and y the vertex of .J2 diagonally opposite to v and .f1 (u) = f2 (v) =: w ∈ K. Furthermore, assume that .F(1) : GS(J1 , u) → Ma and .F(2) : GS(J2 , v) → Ma are .da -isometries given in Definition 4.11 and that they are extensions of .f1 and .f2 , respectively. Then, by (6.1), we have    (T−w ◦ F(1) ◦ Tu )(x − u) = (T−w ◦ F(1) ◦ Tu ) (xi − ui )ei =



i∈Λ(J1 )

(xi − ui )(T−w ◦ F(1) ◦ Tu )(ei ),

i∈Λ(J1 )



.

(T−w ◦ F(2) ◦ Tv )(y − v) = (T−w ◦ F(2) ◦ Tv ) =







(6.2)

(yi − vi )ei

i∈Λ(J2 )

(yi − vi )(T−w ◦ F(2) ◦ Tv )(ei ).

i∈Λ(J2 )

Further, the right hand side of the first equality expresses the vector .f1 (x) − w, since f1 (x) − w = F(1) (x) − w = (T−w ◦ F(1) ◦ Tu )(x − u).

.

Similarly, the right hand side of the second equality in (6.2) expresses the vector f2 (y) − w.   According to (4.26), Theorem 4.31 and (6.2), the coordinates . x − u, a1i ei a remain unchanged under the action of the .da -isometry .T−w ◦ F(1) ◦ Tu . Moreover, the points w and .f1 (x) are the diagonally opposite vertices of the cylinder K. The same is true for w and .f2 (y). Thus, we can conclude that .f1 (x) − w = f2 (y) − w. Since .(T−w ◦ F(1) ◦ Tu )((xi − ui )ei ) = (xi − ui )(T−w ◦ F(1) ◦ Tu )(ei ) for each .i ∈ Λ(J1 ), .F(1) maps each edge of basic cylinder .J1 onto the edge of K. Conversely, every edge of the cylinder K is an image of the edge of .J1 under the .da -isometry .F(1) . The same case is also for .F(2) and .J2 . Therefore, we conclude that there exists a permutation .σ : Λ(J1 ) → Λ(J2 ) that satisfies .

(xi − ui )(T−w ◦ F(1) ◦ Tu )(ei ) = (yσ(i) − vσ(i) )(T−w ◦ F(2) ◦ Tv )(eσ(i) )

.

6.3. ELEMENTARY VOLUMES

173

for any .i ∈ Λ(J1 ).   According to (4.10), it is obvious that both . a1i (T−w ◦ F(1) ◦ Tu )(ei ) and 1  . ai (T−w ◦ F(2) ◦ Tv )(ei ) are orthonormal sequences. Hence, we get ai |xi − ui | = aσ(i) |yσ(i) − vσ(i) |

.

(6.3)

for all .i ∈ Λ(J1 ). Since .J1 and .J2 are basic cylinders, due to the structural property of basic cylinders, we see that there is an .0 ∈ N such that .|xi − ui | = |yi − vi | = 1 for all .i > 0 . Thus, it follows from (6.3) that there exists an .m0 ∈ N (.m0 ≥ 0 ) that satisfies .ai = aσ(i) for each .i > m0 . Consequently, when .J1 is non-degenerate, we use (6.3) to show that the basic cylinders .J1 and .J2 have the same elementary volume: vol(J1 ) =

∞ 

|xi − ui | =

i=1

∞   aσ(i)  yσ(i) − vσ(i)  ai i=1

m0   aσ(i)  yσ(i) − vσ(i)  × = ai i=1

= .

= =

m0  aσ(i)

ai

i=1 m0 

i=m0 +1

∞    yσ(i) − vσ(i)  ×

aσ(i) × ai

i=1 ∞ 

∞    yσ(i) − vσ(i) 

i=1 ∞ 

|yi − vi |

i=1

|yi − vi |

i=1

= vol(J2 ). Hence, it is reasonable to define the volume of the cylinder K as the elementary volume of one of the basic cylinders which are .da -isometric to K, i.e., vol(K) = vol(J1 ) = vol(J2 ).

.

When .J1 is degenerate, we define .vol(K) = vol(J1 ) = vol(J2 ) = 0. Remark 6.5. Let .J1 be a basic cylinder and .K1 a cylinder. Assume that .J1 and K1 are .da -isometric to each other via a surjective .da -isometry .f : J1 → K1 . Assume that p is an element of .J1 and q is an element of .K1 with .q = f (p). Comparing (4.26) and Theorem 4.31 and considering the fact that .GS(J1 , p) = GSn (J1 , p) for any .n ∈ N .(see the proof of Theorem 4.34.), under the action of the .da -isometry .T−q ◦ F ◦ Tp : GS(J1 , p) − p → Ma , the following statements are true. .

CHAPTER 6. ULAM’S CONJECTURE

174

  (i) The orthonormal sequence . a1i ei i∈Λ(J1 ) is changed to another orthonormal   sequence . a1i (T−q ◦ F ◦ Tp )(ei ) i∈Λ(J1 ) .

.

  (ii) The coordinates .(or Fourier coefficients.) . u − p, a1i ei a , .i ∈ Λ(J1 ), of each element .u − p ∈ GS(J1 , p) − p remain unchanged.

.

According to Theorem 4.31, the .da -isometry .T−q ◦ F ◦ Tp transforms the ith coordinate . a1i ei into . a1i (T−q ◦ F ◦ Tp )(ei ) for every .i ∈ Λ(J1 ). Moreover, by (4.26) and Theorem 4.31, the coordinate expression  1of the image of .u − p under the action of .T−q ◦ F ◦ Tp .(in the coordinate system . ai (T−q ◦ F ◦ Tp )(ei ) i∈Λ(J1 ) ) is the same   as that of .u − p in the coordinate system . a1i ei i∈Λ(J1 ) . Therefore, .T−q ◦ F ◦ Tp preserves each m-face of each basic cylinder J contained in .GS(J1 , p) − p, where .m ∈ {0, 1, 2, . . .}. More precisely, (iii) F maps each m-face of basic cylinder J contained in .GS(J1 , p) onto an m-face of cylinder .F (J), where .m ∈ {0, 1, 2, . . .}. In particular, F maps each 1-face of J onto a 1-face of .F (J). Consequently, the “volume” of cylinder .F (J) is defined as the elementary volume of the basic cylinder J, ∞  si , where .si is the Euclidean length of the i.e., .vol(F (J)) = vol(J) =

.

i=1

ith edge of J. (iv) The adjacent edges of the cylinder .F (J) meet orthogonally, and the volume .vol(F (J)) of .F (J) is the infinite product of the Euclidean lengths of all edges of .F (J).

.

Based on Remark 6.5, we can define the elementary volume of basic cylinders and the volume of cylinders accurately. Definition 6.6. .(i) The elementary volume of a basic cylinder J is denoted by .vol(J) and defined by ⎧ ∞ ⎨  s (for Λ(J) = N), i .vol(J) = i=1 ⎩ 0 (for Λ(J) = N), where .si is the Euclidean length of the ith edge of J. .(ii) Considering Remark 6.5 .(iii), we define the volume of cylinder K by vol(K) = vol(J)

.

for any .K ∈ C for which there exists a basic cylinder .J ∈ B and a surjective .da -isometry .f : J → K.

6.4. CONSTRUCTION OF INVARIANT MEASURE

175

6.4 Construction of Invariant Measure If we set .vol(∅) = 0, then the volume “.vol” defined in Definition 6.6 is a premeasure (see Definition 3.10). According to Theorem 3.24 and (3.14), we can apply Munroe’s Method II to construct an outer measure .μ∗ from the pre-measure “.vol” by the formula μ∗ (E) = lim μ∗δ (E)

.

δ→0+

(6.4)

for all subsets E of .Ma , where ∞

∞   ∗ .μδ (E) = inf vol(Ci ) : E ⊂ Ci where Ci ∈ Cδ for all i ∈ N . i=1

i=1

We remark that since .Ma is a separable metric space, Theorem 1.12 (Lindel¨of’s theorem) allows us to consider only the countable coverings. Remark 6.7. For any extraordinary basic cylinder .Jex with .da (Jex ) < δ, there exists a basic cylinder J with .Jex ⊂ J and .da (J) < δ. For example, if .Jex is an ∞  [αi , βi ], where .0 ≤ αi ≤ βi ≤ 1 extraordinary basic cylinder given by .Jex = i=1

for all .i ∈ N, then we can choose a basic cylinder J as J=

n 

.

i=1

[αi , βi ] ×

∞ 

[0, 1].

i=n+1

It is obvious that .Jex is included in J. Moreover, if .da (Jex ) < δ then we can choose a sufficiently large integer n such that .da (J) < δ. We remind us that a collection .{Ci : i ∈ N} of sets is called a covering of ∞  Ci . If, in addition, the diameter of each .Ci is less than .δ, then the E if .E ⊂ i=1

collection .{Ci : i ∈ N} is called a .δ-covering of E. In this section, we assume that each of the sets .E1 and .E2 has uncountably many elements. We note that if .E1 and .E2 have only countably many elements, then obviously .μ∗ (E1 ) = 0 = μ∗ (E2 ). One of the most important theorems in this book is the following theorem stating that the outer measure .μ∗ is .da -invariant. However, we note that this theorem has already been proved for the non-degenerate case in [6, Theorem 3.1]. Now we will completely prove this theorem by providing the proof for the degenerate case also. Theorem 6.8. If .E1 and .E2 are subsets of .I ω that are .da -isometric to each other, then .μ∗ (E1 ) = μ∗ (E2 ).

CHAPTER 6. ULAM’S CONJECTURE

176

Proof. .(a) We assume that .E1 and .E2 are arbitrary subsets of .I ω , each of which has uncountably many elements, and that they are .da -isometric to each other via the surjective .da -isometry .f : E1 → E2 . Using the definition of F (Definition 4.11) and assuming that p is an element of .E1 and q is an element of .E2 with .q = f (p), Theorem 4.13 states that .F : GS(E1 , p) → Ma is a .da -isometry which extends the surjective .da -isometry .f : E1 → E2 . Let r be a positive real number satisfying .E1 ⊂ Br (p), where .Br (p) denotes the open ball defined as .Br (p) = {y ∈ Ma : y − pa < r}. According to Theorem 4.27, the function .F2 : GS2 (E1 , p) → Ma (defined in Definition 4.14) is a .da -isometry and it is an extension of F and so .F2 is obviously an extension of f . 2 .(b) We consider the case where .Λ(GS (E1 , p)) = N. In other words, we assume that the second-order generalized span .GS2 (E1 , p) of .E1 with respect to p is degenerate, i.e., .GS2 (E1 , p) = Ma (by Theorem 4.26). The translation .T−p : Ma → Ma is a homeomorphism and hence, it is a closed mapping. According to Lemma 4.17 .(iii), .GS2 (E1 , p) is a closed proper subset of .Ma , so is .T−p (GS2 (E1 , p)). That is, .GS2 (E1 , p) − p is a closed (proper) subspace of the real Hilbert space .Ma and hence, it follows from Theorem 1.41 that .GS2 (E1 , p) − p is itself a real Hilbert space. 1 .(b.1) In this part we assume the axiom of choice. Let .{ ai ei }i∈Λ be a complete orthonormal sequence in the Hilbert space .GS2 (E1 , p) − p, where .Λ is a nonempty proper subset of .N. Moreover, we note that .Ma can be orthogonally decomposed into    ⊥ Ma = GS2 (E1 , p) − p ⊕ GS2 (E1 , p) − p ,

.

where .(GS2 (E1 , p) − p)⊥ is also a real Hilbert space as a closed subspace of the Hilbert space .Ma . We assume that .β = {βi }i∈N is a complete orthonormal sequence in the Hilbert space .Ma such that .βi = a1i ei for each .i ∈ Λ. We note that   Λ = Λβ GS2 (E1 , p)  . = i ∈ N : there are z ∈ GS2 (E1 , p) and α ∈ R \ {0}  satisfying z + αβi ∈ GS2 (E1 , p) . Let K be either a .β-basic cylinder or a .β-basic interval defined by ∞

 K= ki βi : ki ∈ [0, ai ] for i ∈ Λ; ki = pi for i ∈ Λ

.

=

i=1

 1   ki ei + pi βi : ki ∈ [0, ai ] for i ∈ Λ , ai i∈Λ

i∈Λ

6.4. CONSTRUCTION OF INVARIANT MEASURE where we set .p =

∞ 

177

pi βi ∈ E1 . We note that if

i=1

x=

∞ 

.

x i ei =



i=1

i∈Λ

 ai xi

1 ei ai

 +



pi βi ∈ E1 ⊂ I ω ,

i∈Λ

then .ai xi ∈ [0, ai ] for all .i ∈ Λ. Hence, comparing K with Definition 4.22, we get .E1 ⊂ K. Thus, we obtain .p ∈ E1 ⊂ K and .Λβ (K) = Λ = Λβ (GS2 (E1 , p)). According to Theorem 4.25 with .H = GS2 (E1 , p) − p and .Jβ = K, it holds that 2 .K ⊂ GS(K, p) ⊂ GS (E1 , p). Hence, we have E1 ⊂ K ⊂ GS2 (E1 , p).

(6.5)

.

(b.2) If we divide K into finitely many .da -isometric translations .{K1 , . . . , Km } of a degenerate .β-basic cylinder or a .β-basic interval whose diameter is less than .δ, then .{K1 , K2 , . . . , Km } is a .δ-covering of .E1 . Hence, considering (6.5), we obtain .

μ∗δ (E1 ) ≤

m 

.

μ∗δ (Ki ) ≤

i=1

m 

vol(Ki ) = vol(K) = 0

i=1

for any .δ > 0, because K is degenerate. Therefore, .μ∗ (E1 ) = lim μ∗δ (E1 ) = 0. δ→0+

On the other hand, we see that .E2 = f (E1 ) = F2 (E1 ) ⊂ F2 (K). Since .F2 is a surjective .da -isometry and K is either a degenerate .β-basic cylinder or a .β-basic interval, it follows that .vol(F2 (K)) = vol(K) = 0. Hence, we obtain .μ∗δ (E2 ) ≤ μ∗δ (F2 (K)) = 0 for all .δ > 0. Therefore, we conclude that .μ∗ (E1 ) = 0 = μ∗ (E2 ). 2 .(c) Now, we consider the case where .Λ(GS (E1 , p)) = N, or equivalently, the case where .GS2 (E1 , p) = Ma (see Theorem 4.26). Let .δ > 0 be given. By the definition of .μ∗δ , for any .ε > 0, there exists a .δ-covering .{Ki : i ∈ N} of .E1 with cylinders from .Cδ such that ∞  .

vol(Ki ) ≤ μ∗δ (E1 ) + ε.

(6.6)

i=1

By the definitions of .Bδ and .Cδ , there exists a basic cylinder .Ji ∈ Bδ and a surjective .da -isometry .fi : Ji → Ki for each .i ∈ N. Since .F2 ◦ fi : Ji → F2 (Ki ) is a surjective .da -isometry, .vol(F2 (Ki )) = vol(Ji ) for all .i ∈ N and .{F2 (Ki ) : i ∈ N} = {(F2 ◦ fi )(Ji ) : i ∈ N} is a .δ-covering of .E2 with cylinders from .Cδ . Moreover, by applying Definition 6.6 to the surjective .da -isometry .fi : Ji → Ki , we obtain .vol(Ji ) = vol(Ki ). Thus, we have   .vol F2 (Ki ) = vol(Ki ) (6.7)

CHAPTER 6. ULAM’S CONJECTURE

178

for all .i ∈ N. Therefore, it follows from (6.6) and (6.7) that μ∗δ (E2 ) ≤

∞ 

.

∞    vol F2 (Ki ) = vol(Ki ) ≤ μ∗δ (E1 ) + ε.

i=1

i=1

Since we can choose a sufficiently small .ε > 0, we conclude that .μ∗δ (E2 ) ≤ μ∗δ (E1 ). Conversely, if we exchange the roles of .E1 and .E2 in the previous part, then we get .μ∗δ (E1 ) ≤ μ∗δ (E2 ). Hence, we conclude that .μ∗δ (E1 ) = μ∗δ (E2 ) for any ∗ ∗ .δ > 0. Therefore, it follows from (6.4) that .μ (E1 ) = μ (E2 ). .2 The following lemmas are the same as the lemmas [6, Lemma 3.2, Lemma 3.3, Lemma 3.4]. It is easy to prove that .μ∗ (I ω ) ≤ 1. Lemma 6.9. .μ∗ (I ω ) ≤ 1. Proof. We apply an idea from the proof of [17, Lemma 1]. For any .δ > 0, there exist positive integers m and n such that .I n = [0, 1]n is covered by .mn isometric n-cubes .Ci which are closed in .I n with non-overlapping interiors and .da (Ji ) < δ for all .i ∈ {1, 2, . . . , mn }, where .Ji is the cylinder in .I ω over .Ci , i.e., .Ji ∈ Bδ for each i. Then, .{J1 , J2 , . . . , Jmn } is a .δ-covering of .I ω with non-degenerate basic cylinders from .Bδ and m  n

∗ ω .μδ (I )

≤

vol(Ji ) = 1.

i=1

Hence, it follows from (6.4)

that .μ∗ (I ω )

= lim μ∗δ (I ω ) ≤ 1. δ→0+

Lemma 6.10. Given real numbers .δ and b with .0 < δ < α=

.

0 (for 0 ≤ b < δ), b − δ (for δ ≤ b ≤ 1)

and β =

1 2

2

.

and .0 ≤ b ≤ 1, define

b + δ (for 0 ≤ b ≤ 1 − δ), 1 (for 1 − δ < b ≤ 1).

Then .0 < β − α ≤ 2δ and .α ≤ b ≤ β. Proof. .(a) If .0 ≤ b < δ, then .0 ≤ b < δ < 1 − δ and we get .α = 0 and β = b + δ < 2δ. Hence, it follows that .0 < β − α < 2δ and .α ≤ b < β. .(b) If .δ ≤ b ≤ 1 − δ, then .α = b − δ and .β = b + δ. Thus, we have .0 < β − α = 2δ and .α < b < β. .(c) Finally, if .1 − δ < b ≤ 1, then we see that .α = b − δ and .β = 1. So, we .2 have .0 < β − α = 1 − b + δ < 2δ and .α < b ≤ β. .

For every basic cylinder .J ∈ B, let .∂J denote the boundary of J. In the following lemma, we will prove that .μ∗ (∂J) = 0.

6.4. CONSTRUCTION OF INVARIANT MEASURE

179

Lemma 6.11. If .J ∈ B, then .μ∗ (∂J) = 0. Proof. Similarly as in the proof of Lemma 6.9, there exist positive integers m and n such that .I n = [0, 1]n is covered by .mn isometric n-cubes .Cin .(i ∈ {1, 2, . . . , mn }) with non-overlapping interiors, where each .Cin is a closed subset of .I n . Let .δ and b be any real numbers with .0 < δ < 12 and .0 ≤ b ≤ 1, .α and .β be defined as in Lemma 6.10, .Pi = Cin × [α, β] be an .(n + 1)-dimensional rectangular parallelepiped, and let .Ji denote the cylinder in .I ω over .Pi with .da (Ji ) < δ for all n n .i ∈ {1, 2, . . . , m }, i.e., .Ji ∈ Bδ for each .i ∈ {1, 2, . . . , m }. In view of Lemma 6.10, .{J1 , J2 , . . . , Jmn } is a .δ-covering of a hyper-plane H given by   .H = (x1 , x2 , . . .) ∈ I ω : xn+1 = b . (6.8) Hence, by the definition of .μ∗δ and using Lemma 6.10 again, it holds that m  n

∗ .μδ (H)

≤

i=1

m  1 vol(Ji ) = (β − α) ≤ 2δ mn n

i=1

and further we get μ∗ (H) = lim μ∗δ (H) = 0.

.

δ→0+

(6.9)

Let J be a basic cylinder in .B. In view of Definition 4.20, without loss of generality, we will deal with the basic cylinder of the form   .J = (x1 , x2 , . . .) ∈ I ω : p1i ≤ xi ≤ p2i for all i ∈ N only, where there exists a positive integer n such that .0 ≤ p1i < p2i ≤ 1 for each i ∈ {1, 2, . . . , n} and .p1i = 0, .p2i = 1 for all .i > n. Then, there are at most countably many hyper-planes .H1 , H2 , . . . of the forms   .H2i−1 = (x1 , x2 , . . .) ∈ I ω : xi = p1i

.

and

  H2i = (x1 , x2 , . . .) ∈ I ω : xi = p2i

.

satisfying ∂J ⊂

∞ 

.

Hk .

(6.10)

k=1

Finally, it follows from (6.9) and (6.10) that  ∞  ∞   ∗ ∗ .μ (∂J) ≤ μ Hk ≤ μ∗ (Hk ) = 0, k=1

which completes the proof.

k=1

2

.

CHAPTER 6. ULAM’S CONJECTURE

180

6.5 Efficient Coverings Let J be a basic cylinder given by Definition 4.20 that is .da -isometric to a cylinder K via a surjective .da -isometry .f : J → K. We now define ⎧ [0, b∗ ] ⎪ ⎪ ∞ ⎨  [p1i , p1i + b∗ ] ∗ .J = Ji∗ , where Ji∗ = {p } ⎪ ⎪ i=1 ⎩ 1i [0, 1]

(for i ∈ Λ1 ), (for i ∈ Λ2 ∪ Λ3 ), (for i ∈ Λ4 ), (otherwise)

(6.11)

and .b∗ is a sufficiently small positive real number in comparison with each of .p2i , .p 2i − p1i , and .1 − p1i for all .i ∈ Λ1 , .i ∈ Λ2 , and .i ∈ Λ3 , respectively. We then note that .J ∗ ⊂ J. Taking Remark 6.5 .(ii) and .(iii) into account, we can cover the cylinder .K = f (J) with a finite number of translations of .f (J ∗ ) as efficiently as we wish by choosing the .b∗ sufficiently small (see the illustration and Lemma 6.12 below).

J J∗

f

 BBB    B  B  B  BB    B B K B B  BB     B B  BK2 B 3 B  B BB B BBK1 B B B B  B B B B B BB K B  BB - B B B  B B KmBB B B B B B   B  B B B B  B B  B ∗B  Bf (J )BB   B

A finite number of translations of .f (J ∗ ) cover .K = f (J) J ∗ is a basic cylinder and the restriction .f |J ∗ : J ∗ → f (J ∗ ) is a surjective .da isometry. Thus, .f (J ∗ ) is a cylinder, i.e., .f (J ∗ ) ∈ C (see Remark 6.2). Applying this argument, Remark 6.5 .(ii), .(iii) and Lemma 6.11, we obtain the following lemma that is an improved version of [6, Lemma 4.1]. This new version includes the degenerate case. .

Lemma 6.12. Let .δ > 0 and .ε > 0 be given. If K is a cylinder from .Cδ , then there exists a finite number of translations .K1 , K2 , . . . , Km of some cylinder in .Cδ .(for example, .f (J ∗ ) above and see the corresponding illustration above.) such that (i) .Ki ∩ Kj .(i =  j) is included in the union of at most countably many .(da isometric images of .) hyper-planes and these hyper-planes are of the form b ω : x = b}, for some . ∈ N and .0 ≤ b ≤ 1, with .H = {(x1 , x2 , . . .) ∈ I   ∗ b .μ (H ) = 0. 

.

6.5. EFFICIENT COVERINGS

181

(ii) .{K1 , K2 , . . . , Km } is a covering of K, i.e., .K ⊂

m 

.

Ki .

i=1

(iii)

.

m  .

vol(Ki ) ≤ vol(K) + ε.

i=1

Proof. We can choose a .J ∈ Bδ and a surjective .da -isometry .f : J → K, where J is a basic cylinder of the form given in Definition 4.20. We now define a basic cylinder .J ∗ by (6.11) such that .J ∗ ⊂ J. Then J can be covered with at most m translations of the basic cylinder .J ∗ , where we set   " #   " #   " p2j # p2j − p1j 1 − p1j +1 × +1 × +1 .m = b∗ b∗ b∗ j∈Λ1

j∈Λ2

j∈Λ3

and where .[x] denotes the largest integer not exceeding the real number x. This fact, together with Remark 6.5 .(ii) and .(iii), implies that the cylinder .K = f (J) can be covered with at most m translations of the cylinder .f (J ∗ ) which are denoted by .K1 , K2 , . . . , Km (see the previous illustration). Moreover, in view of Remark 6.5 .(iii), we have vol(Ki ) = vol(f (J ∗ )) = vol(J ∗ )  b∗ (for non-degenerate K), . j∈Λ1 ∪Λ2 ∪Λ3 = 0 (for degenerate K)

for each .i ∈ {1, 2, . . . , m} and vol(K) = vol(J) ⎧   ⎪ p2j × (p2j − p1j )× ⎪ ⎪ ⎨ j∈Λ j∈Λ2 1 . (1 − p1j ) × = ⎪ j∈Λ3 ⎪ ⎪ ⎩ 0

(for non-degenerate K), (for degenerate K).

CHAPTER 6. ULAM’S CONJECTURE

182

Thus, when K is a non-degenerate cylinder, we have #    "  " p2j # p2j − p1j +1 × +1 × vol(Ki ) = b∗ b∗ i=1 j∈Λ1 j∈Λ2   " 1 − p1j # × + 1 × vol(K1 ) b∗ j∈Λ3       p2j p2j − p1j +1 × +1 × ≤ b∗ b∗ j∈Λ1 j∈Λ2  .    1 − p1j +1 × b∗ × ∗ b j∈Λ3 j∈Λ1 ∪Λ2 ∪Λ3    = (p2j + b∗ ) × (p2j − p1j + b∗ ) × (1 − p1j + b∗ ) m 

j∈Λ1

=



p2j ×

j∈Λ1

 j∈Λ2 ∗

j∈Λ2

(p2j − p1j ) ×



j∈Λ3

(1 − p1j ) + O(b∗ )

j∈Λ3

= vol(K) + O(b ) and we choose a sufficiently small .b∗ such that the term .O(b∗ ) becomes less than .ε. When K is degenerate, we have .vol(K) = 0 and .vol(Ki ) = 0 for any .i ∈ {1, 2, . . . , m}. Hence, our assertion .(iii) holds true. .2 Using Lemmas 6.9 and 6.12, we will prove that .μ∗ (I ω ) = 1. The following theorem is equivalent to [6, Theorem 4.2], but the proof of this theorem is much more concise than that of [6, Theorem 4.2]. Hence, we will introduce the proof. Theorem 6.13. .μ∗ (I ω ) = 1. Proof. Given a .δ > 0 and an .ε > 0, let .{Ki : i ∈ N} be a .δ-covering of .I ω with cylinders from .Cδ such that ∞  .

i=1

ε vol(Ki ) ≤ μ∗δ (I ω ) + . 2

(6.12)

In view of Lemma 6.12, for each .Ki , there exist translations .Ki1 , Ki2 , . . . , Kimi of some cylinder in .Cδ such that (i) .Kij ∩ Ki .(j =  ) is included in the union of at most countably many (.da isometric images of) hyper-planes of the form .{(x1 , x2 , . . .) ∈ I ω : x = b}, for some . ∈ N and .0 ≤ b ≤ 1, whose .μ∗ -measures are 0.

.

6.6. ULAM’S CONJECTURE ON INVARIANCE OF MEASURE (ii) .Ki ⊂

m i

.

183

Kij .

j=1

(iii)

.

mi  .

vol(Kij ) ≤ vol(Ki ) +

j=1

ε 2i+1

.

We notice that each .Kij is a cylinder from .Cδ . If we replace the covering {Ki : i ∈ N} with a new .δ-covering .{Kij : i ∈ N; j ∈ {1, 2, . . . , mi }}, then it follows from .(iii) that

.

mi ∞   .

vol(Kij ) ≤

i=1 j=1

∞ 

vol(Ki ) +

i=1

ε 2

and it follows from this inequality and (6.12) that 1 = vol(I ω ) ≤

mi ∞  

.

vol(Kij ) ≤

i=1 j=1

∞ 

vol(Ki ) +

i=1

ε ≤ μ∗δ (I ω ) + ε. 2

If we take a sufficiently small value of .ε > 0, then we have .μ∗δ (I ω ) ≥ 1 and = lim μ∗δ (I ω ) ≥ 1. On the other hand, in view of Lemma 6.9, we have

∗ ω .μ (I )

δ→0+

μ∗ (I ω ) ≤ 1. Hence, we conclude that .μ∗ (I ω ) = 1.

.

2

.

6.6 Ulam’s Conjecture on Invariance of Measure According to Theorems 3.26 and 3.30, all Borel sets in .Ma are .μ∗ -measurable. Moreover, each Borel subset of .I ω is also a Borel subset of .Ma , i.e., each Borel subset of .I ω is .μ∗ -measurable. On account of Theorems 6.8 and 6.13, the outer measure .μ∗ is .da -invariant with .μ∗ (I ω ) = 1. The proof of the following lemma is the same as that of [6, Lemma 5.1]. Lemma 6.14. The outer measure .μ∗ coincides with the standard product probability measure .π on the Borel subsets of .I ω . Proof. Similarly as in the proof of Lemma 6.9, consider a covering of .I n = [0, 1]n by .mn isometric n-cubes with non-overlapping interiors, where each n-cube is closed in .I n . Let .Zmn be the collection of cylinders in .I ω over those n-cubes, i.e., ∗ .Zmn ⊂ B. The translation-invariance of .da implies that of .μ . Thus, we have μ∗ (J1 ) = μ∗ (J2 )

.

for all .J1 , J2 ∈ Zmn .

(6.13)

CHAPTER 6. ULAM’S CONJECTURE

184

Since all Borel subsets of .I ω are .μ∗ -measurable, by Lemma 6.11, we have μ∗ (J) = μ∗ (J ◦ )

(6.14)

.

for any .J ∈ Zmn and  μ

.

∗



 J

◦

 ∗

≤ μ (I ) ≤ μ ω

∗

J∈Zmn



 J .

J∈Zmn

We notice that .J1◦ ∩ J2◦ = ∅ for any distinct .J1 , J2 ∈ Zmn . Hence, it follows from Theorem 6.13, (6.13), and the last relations that mn μ∗ (J ◦ ) ≤ 1 ≤ mn μ∗ (J)

.

and it moreover follows from (6.14) that μ∗ (J) =

.

1 = π(J) mn

(6.15)

for all .J ∈ Zmn . Referring to the proof of Lemma 5.21, the basic cylinders from .

∞ ∞  

Zmn ,

m=1 n=1

together with the empty set, generate the Borel .σ-algebra over .I ω . Hence, by well known facts, (6.15) implies that .μ∗ coincides with .π on the Borel subsets of .I ω . .2 According to Theorem 6.8, assuming the axiom of choice, the outer measure is .da -invariant. Hence, assuming the axiom of choice and using Lemma 6.14, we get the following main result:

∗ .μ

Theorem 6.15. For any sequence .a = {ai }i∈N of positive real numbers satisfying ∞  the condition . a2i < ∞, the standard product probability measure .π on .I ω is i=1

da -invariant. More precisely, if .E1 and .E2 are arbitrary Borel subsets of .I ω that are .da -isometric to each other, then .π(E1 ) = π(E2 ).

.

Bibliography ´ ´ 1. E. Borel, Sur quelques points de la theorie des fonctions. Ann. Ecole. Norm. sup (3), 12, 9–55 (1895) 2. K.R. Davidson, A.P. Donsig, Real Analysis and Applications (Springer, New York, 2010) ´ 3. L. Debnath, P. Mikusinski, Introduction to Hilbert Spaces with Applications, 2nd edn. (Academic Press, New York, 2005) 4. J.W. Fickett, Approximate isometries on bounded sets with an application to measure theory. Studia Math. 72(1), 37–46 (1982) ¨ deckung von kompakten Mengen – Hausdorff¨ 5. S.-M. Jung, Gunstig e Uber maß und Dimension, Doctoral Thesis at University of Stuttgart, 1994 6. S.-M. Jung, The conjecture of Ulam on the invariance of measure on Hilbert cube. J. Math. Anal. Appl. 481(2), Article 123500 (2020) 7. S.-M. Jung, Corrigendum to “The conjecture of Ulam on the invariance of measure on Hilbert cube” [J. Math. Anal. Appl. 481 (2) (2020) 123500]. J. Math. Anal. Appl. 490(2), Article 124080 (2020) 8. S.-M. Jung, The conjecture of Ulam on the invariance of measure on Hilbert cube. http://export.arXiv.org/pdf/1807.05624 9. S.-M. Jung, Extension of isometries in real Hilbert spaces. Open Math. 20(1), 1353–1379 (2022) 10. S.-M. Jung, E. Kim, On the conjecture of Ulam on the invariance of measure in the Hilbert cube. Colloq. Math. 152(1), 79–95 (2018) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4

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11. S.-M. Jung, E. Kim, Corrigendum to “On the conjecture of Ulam on the invariance of measure in the Hilbert cube” (Colloq. Math. 152 (2018), 79–95). Colloq. Math. 164(1), 161–170 (2021) 12. S.-M. Jung, D. Nam, Some properties of interior and closure in general topology. Mathematics 7(7), 624 (2019) 13. R.H. Kasriel, Undergraduate Topology (W. B. Saunders, Philadelphia, 1971) 14. P. Mankiewicz, On extension of isometries in normed linear spaces. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 20, 367–371 (1972) 15. M.E. Munroe, Introduction to Measure and Integration (Addison-Wesley, Reading, MA, 1953) 16. J. Mycielski, Remarks on invariant measures in metric spaces. Colloq. Math. 32, 105–112 (1974) 17. J. Mycielski, A conjecture of Ulam on the invariance of measure in Hilbert’s cube. Studia Math. 60, 1–10 (1977) 18. C.A. Rogers, Hausdorff Measures (Cambridge University Press, Cambridge, 1998) 19. G.F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, London, 1963) 20. S.M. Ulam, A Collection of Mathematical Problems (Interscience, New York, 1960) 21. S.M. Ulam, Problems in Modern Mathematics (Wiley, New York, 1964) 22. E.W. Weisstein, CRC Encyclopedia of Mathematics, 3rd edn (CRC Press, Boca Raton, 2009) 23. J.H. Wells, L.R. Williams, Embeddings and Extensions in Analysis. Erg. der Math., vol. 84 (Springer, Berlin/Heidelberg, 1975)

Index Symbols n-dimensional Hausdorff measure, 86 nth-order generalized span, 105 .Rδ , 70 .Rσ , 70 .B r (x), 2, 32 .da (x, y), 92, 93, 137, 168 .d(B), 8 .C, 27 .K, 27 .N0 , 4 .N, 4 n .R , 2 ω .R , 91, 167, 136 .R, 27 .x, ya , 92, 137, 168 .{ai }i∈N , 92, 137, 168 ∗ .μ , 55 h .μ , 86 (n) , 86 .μ .dimX, 29 .spanA, 29 n .GS (E1 , p), 105 .GS(E, p), 95 .vol, 173 .π, 136 .A, 56 .Bδ , 170 .Bn , 168 .B, 170

Cδ , 170 C, 170 .TR , 91 .Tω , 91, 136, 167 .T (d), 3 .T , 3, 91, 136, 167 .Br (x), 2, 32 .Fn , 105 ω .I , 91, 127, 136, 167 .α, 50 .Ma , 92, 137, 168 .α, 50 .Sr (x), 32 .x ⊥ y, 39 .β-basic interval, 114 .β-index, 96 .β-index set, 96 .da -isometric, 94 .da -isometry, 94 .δ-covering, 175 .ε-net, 7 .λ-coordinate, 17 .λ-coordinate space, 17 .(n)-measure, 86 ∗ .μ -measurable, 56 .R-regular outer measure, 70 .σ-algebra, 56 .T1 -space, 21 .T2 -space, 18, 21 .T3 -space, 22 .T4 -space, 22 . .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-M. Jung, Ulam’s Conjecture on Invariance of Measure in the Hilbert Cube, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-30886-4

187

INDEX

188 A absolutely convergent, 34 addition, 27 axiom of choice, 132, 184 B ball, 32 Banach space, 34 basic cylinder, 168, 171 basic interval, 113 basic open covering, 13 basis, 29 Bessel inequality, 42 big, 8 Bolzano-Weierstrass property, 6 Borel measure, 139 Borel set, 71 Borel .σ-algebra, 71 Borel, E., 83 bounded, 8 C Cartesian product, 16, 17 Cauchy sequence, 19, 33 Cauchy-Schwarz inequality, 36 closed, 2, 3 closed ball, 2, 32 closed base, 12 closed in X, 2 closed rectangle, 81 closed subbase, 13 closure, 4 compact, 6 complete, 20, 33, 45 complete orthonormal sequence, 96 complex vector space, 28 continuous, 10 continuous function, 10 converge, 4, 6 convergence, 30 convex, 48

countably additive measure, 61 covering, 5, 175 cylinder, 171 D degenerate, 96, 171 degenerate basic cylinder, 113 degenerate .β-basic cylinder, 114 dense, 4 dense in X, 4 diameter, 8, 170 dimension, 29 direct sum, 51 discrete metric, 2 discrete topology, 3 E elementary volume, 81, 173, 174 entropy, 137, 153 equivalent, 30 Euclidean distance, 2 Euclidean norm, 29 extended Pythagorean theorem, 42 extraordinary basic cylinder, 169, 175 F Fickett, J. W., 136, 144 filter, 138 finite intersection property, 12 first-order generalized span, 95 flat, 96 G generalized Fourier coefficient, 43 generalized Fourier series, 43 H half-open rectangle, 81 Hausdorff function, 86 Hausdorff measure, 86 Hausdorff space, 18, 21 Hausdorff, F., 86

INDEX Heine–Borel theorem, 6, 15, 16 Hilbert cube, 91, 127, 136, 167 Hilbert space, 40 Hilbert space isomorphism, 52 I imaginary part, 50 index, 96 index set, 96 indiscrete topology, 3 inner product, 35, 92, 137, 168 inner product space, 36 interior, 4 interior point, 3 isometric extension property, 133 isomorphic, 52 J Jung, S.-M., 136, 152 K Kim, E., 136, 152 L Landau’s symbol, 83 Lebesgue measure, 82 Lebesgue number, 8 Lebesgue’s covering lemma, 8 limit point, 4 Lindel¨of’s theorem, 5 linear combination, 28 linear mapping, 52 linear subspace, 39 linearly dependent, 29 linearly independent, 28 local isometry, 91 M maximal filter, 138 measure, 61 Method I, 61 Method II, 72

189 metric, 1, 92, 93, 137, 168 metric outer measure, 76 metric space, 1 metrizable, 3 Munroe, M. E., 61 Mycielski, J., 135–137 N neighborhood filter, 138 neighborhood system, 138 non-degenerate, 96, 171 non-degenerate basic cylinder, 113 non-degenerate .β-basic cylinder, 114 norm, 29, 92, 137, 168 normal space, 22 normed space, 30 null set, 61 O open, 2, 3 open ball, 2, 32 open base, 12 open covering, 5 open in X, 2 open rectangle, 81 open subbase, 13 orthogonal, 39, 47 orthogonal complement, 47 orthogonal decomposition, 51 orthogonal system, 40 orthonormal basis, 45 orthonormal sequence, 41 orthonormal system, 40 outer measure, 55, 175 P parallelogram law, 37 Parseval’s formula, 47 polarization identity, 37 positively separated, 73 power set, 55

INDEX

190 pre-Hilbert space, 36 pre-measure, 61, 175 principal ultrafilter, 138 product measure, 85 product space, 17, 91 product topology, 17 projection function, 17 proper filter, 138 proper subspace, 28 Ptolemy inequality, 38 Pythagorean theorem, 39 R real part, 50 real vector space, 28 regular outer measure, 66 regular space, 21 relative topology, 3 S scalar, 28 scalar multiplication, 27 separable, 4, 51 separated, 56 sequentially compact, 6 space spanned by A, 29 span, 45 sphere, 32 standard product probability measure, 85, 136, 143, 167

sub-covering, 5 subbasic open covering, 13 subspace, 1, 3, 28, 39 sup-norm, 30 T thick, 138 thin, 138 topological space, 3 topology, 3 topology generated by d, 3 totally bounded, 7 translation, 95 translation-invariant, 85, 92 triangle inequality, 29 trivial topology, 3 Tychonoff’s theorem, 19 U Ulam’s conjecture, 136 Ulam, S. M., 135 ultrafilter, 138 usual topological space, 91 V vector, 28 vector space, 27 vector subspace, 28 volume, 173, 174