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180 SRIVASTAVA. A Course on Borel Sets. 181 KRESS. Numerical Analysis.

S.M. Srivastava

A Course on Borel Sets

With 11 Illustrations

Springer

S.M. Srivastava Stat-Math Unit Indian Statistical Institute 203 B.T. Road Calcutta, 700 035 India

Editorial Board

S. Axler F.W. Gehring K.A. Ribet Department of Department of Department of

Mathematics Mathematics Mathematics San Francisco State University of Michigan University of California

University Ann Arbor, MI 48109 at Berkeley San Francisco, CA 94132 USA Berkeley, CA 94720 USA USA

Mathematics Subject Classification (1991): 04-01, 04A15, 28A05, 54H05

Library of Congress Cataloging-in-Publication Data Srivastava, S.M. (Sashi Mohan)

A course on Borel sets / S.M. Srivastava. p. cm. — (Graduate texts in mathematics ; 180)

Includes index. ISBN 0-387-98412-7 (hard : alk. paper) 1. Borel sets. I. Title. 11. Series.

QA248.S74 1998 511,3'2—dc21 97-43726

© 1998 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electtonic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-387-98412-7 Springer-Verlag New York Berlin Heidelberg SPIN 10660569

This book is dedicated to the memory of my beloved wife, Kiran

who passed away soon after this book was completed.

Acknowledgments

I am grateful to many people who have suggested improvements in theoriginal manuscript for this book. In particular I would like to thank S.C. Bagchi, R. Barua, S. Gangopadhyay (nee Bhattacharya), J. K. Ghosh,M. G. Nadkarni, and B. V. Rao. My deepest feelings of gratitude and ap-preciation are reserved for H. Sarbadhikari who very patiently read severalversions of this book and helped in all possible ways to bring the book toits present form. It is a pleasure to record my appreciation for A. Maitrawho showed the beauty and power of Borel sets to a generation of Indianmathematicians including me. I also thank him for his suggestions duringthe planning stage of the book.

I thank P. Bandyopadhyay who helped me immensely to sort out all theLATEX problems. Thanks are also due to R. Kar for preparing the LATEXfiles for the illustrations in the book.

I am indebted to S. B. Rao, Director of the Indian Statistical Institute forextending excellent moral and material support. All my colleagues in theStat – Math Unit also lent a much needed and invaluable moral supportduring the long and difficult period that the book was written. I thankthem all.

I take this opportunity to express my sincere feelings of gratitude to mychildren, Rosy and Ravi, for their great understanding of the task I tookonto myself. What they missed during the period the book was written willbe known to only the three of us. Finally, I pay homage to my late wife,Kiran who really understood what mathematics meant to me.

S. M. Srivastava

Contents

Acknowledgments vii

Introduction xi

About This Book xv

1 Cardinal and Ordinal Numbers 11.1 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Order of Infinity . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . 71.4 More on Equinumerosity . . . . . . . . . . . . . . . . . . . . 111.5 Arithmetic of Cardinal Numbers . . . . . . . . . . . . . . . 131.6 Well-Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . 151.7 Transfinite Induction . . . . . . . . . . . . . . . . . . . . . . 181.8 Ordinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . 211.9 Alephs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.10 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.11 Induction on Trees . . . . . . . . . . . . . . . . . . . . . . . 291.12 The Souslin Operation . . . . . . . . . . . . . . . . . . . . . 311.13 Idempotence of the Souslin Operation . . . . . . . . . . . . 34

2 Topological Preliminaries 392.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 Polish Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3 Compact Metric Spaces . . . . . . . . . . . . . . . . . . . . 572.4 More Examples . . . . . . . . . . . . . . . . . . . . . . . . . 63

x Contents

2.5 The Baire Category Theorem . . . . . . . . . . . . . . . . . 692.6 Transfer Theorems . . . . . . . . . . . . . . . . . . . . . . . 74

3 Standard Borel Spaces 813.1 Measurable Sets and Functions . . . . . . . . . . . . . . . . 813.2 Borel-Generated Topologies . . . . . . . . . . . . . . . . . . 913.3 The Borel Isomorphism Theorem . . . . . . . . . . . . . . . 943.4 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5 Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.6 Borel Pointclasses . . . . . . . . . . . . . . . . . . . . . . . . 115

4 Analytic and Coanalytic Sets 1274.1 Projective Sets . . . . . . . . . . . . . . . . . . . . . . . . . 1274.2 Σ1

1 and Π11 Complete Sets . . . . . . . . . . . . . . . . . . . 135

4.3 Regularity Properties . . . . . . . . . . . . . . . . . . . . . . 1414.4 The First Separation Theorem . . . . . . . . . . . . . . . . 1474.5 One-to-One Borel Functions . . . . . . . . . . . . . . . . . . 1504.6 The Generalized First Separation Theorem . . . . . . . . . 1554.7 Borel Sets with Compact Sections . . . . . . . . . . . . . . 1574.8 Polish Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.9 Reduction Theorems . . . . . . . . . . . . . . . . . . . . . . 1644.10 Choquet Capacitability Theorem . . . . . . . . . . . . . . . 1724.11 The Second Separation Theorem . . . . . . . . . . . . . . . 1754.12 Countable-to-One Borel Functions . . . . . . . . . . . . . . 178

5 Selection and Uniformization Theorems 1835.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.2 Kuratowski and Ryll-Nardzewski’s Theorem . . . . . . . . . 1895.3 Dubins – Savage Selection Theorems . . . . . . . . . . . . . 1945.4 Partitions into Closed Sets . . . . . . . . . . . . . . . . . . . 1955.5 Von Neumann’s Theorem . . . . . . . . . . . . . . . . . . . 1985.6 A Selection Theorem for Group Actions . . . . . . . . . . . 2005.7 Borel Sets with Small Sections . . . . . . . . . . . . . . . . 2045.8 Borel Sets with Large Sections . . . . . . . . . . . . . . . . 2065.9 Partitions into Gδ Sets . . . . . . . . . . . . . . . . . . . . . 2125.10 Reflection Phenomenon . . . . . . . . . . . . . . . . . . . . 2165.11 Complementation in Borel Structures . . . . . . . . . . . . . 2185.12 Borel Sets with σ-Compact Sections . . . . . . . . . . . . . 2195.13 Topological Vaught Conjecture . . . . . . . . . . . . . . . . 2275.14 Uniformizing Coanalytic Sets . . . . . . . . . . . . . . . . . 236

References 241

Glossary 251

Index 253

Introduction

The roots of Borel sets go back to the work of Baire [8]. He was trying tocome to grips with the abstract notion of a function introduced by Dirich-let and Riemann. According to them, a function was to be an arbitrarycorrespondence between objects without giving any method or procedureby which the correspondence could be established. Since all the specificfunctions that one studied were determined by simple analytic expressions,Baire delineated those functions that can be constructed starting from con-tinuous functions and iterating the operation of pointwise limit on a se-quence of functions. These functions are now known as Baire functions.Lebesgue [65] and Borel [19] continued this work. In [19], Borel sets weredefined for the first time. In his paper, Lebesgue made a systematic studyof Baire functions and introduced many tools and techniques that are usedeven today. Among other results, he showed that Borel functions coincidewith Baire functions. The study of Borel sets got an impetus from an errorin Lebesgue’s paper, which was spotted by Souslin. Lebesgue was trying toprove the following:

Suppose f : R2 −→ R is a Baire function such that for every x, theequation

f(x, y) = 0

has a unique solution. Then y as a function of x defined by the aboveequation is Baire.

The wrong step in the proof was hidden in a lemma stating that a setof real numbers that is the projection of a Borel set in the plane is Borel.(Lebesgue left this as a trivial fact!) Souslin called the projection of aBorel set analytic because such a set can be constructed using analyticaloperations of union and intersection on intervals. He showed that there are

xii Introduction

analytic sets that are not Borel. Immediately after this, Souslin [111] andLusin [67] made a deep study of analytic sets and established most of thebasic results about them. Their results showed that analytic sets are offundamental importance to the theory of Borel sets and give it its power.For instance, Souslin proved that Borel sets are precisely those analytic setswhose complements are also analytic. Lusin showed that the image of aBorel set under a one-to-one Borel map is Borel. It follows that Lebesgue’sthoerem—though not the proof—was indeed true.

Around the same time Alexandrov was working on the continuum hy-pothesis of Cantor: Every uncountable set of real numbers is in one-to-onecorrespondence with the real line. Alexandrov showed that every uncount-able Borel set of reals is in one-to-one correspondence with the real line [2].In other words, a Borel set cannot be a counterexample to the continuumhypothesis.

Unfortunately, Souslin died in 1919. The work on this new-found topicwas continued by Lusin and his students in Moscow and by Sierpinski andhis collaborators in Warsaw.

The next important step was the introduction of projective sets byLusin [68], [69], [70] and Sierpinski [105] in 1925: A set is called projectiveif it can be constructed starting with Borel sets and iterating the operationsof projection and complementation. Since Borel sets as well as projectivesets are sets that can be described using simple sets like intervals andsimple set operations, their theory came to be known as descriptive settheory. It was clear from the beginning that the theory of projective setswas riddled with problems that did not seem to admit simple solutions. Asit turned out, logicians did show later that most of the regularity propertiesof projective sets, e.g., whether they satisfy the continuum hypothesis ornot or whether they are Lebesgue measurable and have the property ofBaire or not, are independent of the axioms of classical set theory.

Just as Alexandrov was trying to determine the status of the continuumhypothesis within Borel sets, Lusin [71] considered the status of the axiomof choice within “Borel families.” He raised a very fundamental and difficultquestion on Borel sets that enriched its theory significantly. Let B be asubset of the plane. A subset C of B uniformizes B if it is the graph of afunction such that its projection on the line is the same as that of B. (SeeFigure 1.)

Lusin asked, When does a Borel set B in the plane admit a Borel uni-formization? By Lusin’s theorem stated earlier, if B admits a Borel uni-formization, its projection to the line must be Borel. In [16] Blackwell [16]showed that this condition is not sufficient. Several authors considered thisproblem and gave sufficient conditions under which Lusin’s question hasa positive answer. For instance, a Borel set admits a Borel uniformizationif the sections of B are countable (Lusin [71]) or compact (Novikov [90])or σ-compact (Arsenin [3] and Kunugui [60]) or nonmeager (Kechris [52]and Sarbadhikari [100]). Even today these results are ranked among the

Introduction xiii

Y

X

C

B

Figure 1. Uniformization

finest results on Borel sets. For the uniformization of Borel sets in general,the most important result proved before the war is due to Von Neumann[124]: For every Borel subset B of the square [0, 1] × [0, 1], there is a nullset N and a Borel function f : [0, 1] \N −→ [0, 1] whose graph is containedin B. As expected, this result has found important applications in severalbranches of mathematics.

So far we have mainly been giving an account of the theory developedbefore the war; i.e., up to 1940. Then for some time there was a lull, notonly in the theory of Borel sets, but in the whole of descriptive set theory.This was mainly because most of the mathematicians working in this areaat that time were trying to extend the theory to higher projective classes,which, as we know now, is not possible within Zermelo – Fraenkel set theory.Fortunately, around the same time significant developments were takingplace in logic that brought about a great revival of descriptive set theorythat benefited the theory of Borel sets too. The fundamental work of Godelon the incompleteness of formal systems [44] ultimately gave rise to a richand powerful theory of recursive functions. Addison [1] established a strongconnection between descriptive set theory and recursive function theory.This led to the development of a more general theory called effectivedescriptive set theory. (The theory as developed by Lusin and othershas become known as classical descriptive set theory.)

From the beginning it was apparent that the effective theory is morepowerful than the classical theory. However, the first concrete evidence ofthis came in the late seventies when Louveau [66] proved a beautiful the-orem on Borel sets in product spaces. Since then several classical resultshave been proved using effective methods for which no classical proof isknown yet; see, e.g., [47]. Forcing, a powerful set-theoretic technique (in-vented by Cohen to show the independence of the continuum hypothesisand the axiom of choice from other axioms of set theory [31]), and otherset-theoretic tools such as determinacy and constructibility, have been veryeffectively used to make the theory of Borel sets a very powerful theory.(See Bartoszynski and Judah [9], Jech [49], Kechris [53], and Moschovakis[88].)

xiv Introduction

Much of the interest in Borel sets also stems from the applications thatits theory has found in areas such as probability theory, mathematicalstatistics, functional analysis, dynamic programming, harmonic analysis,representation theory of groups, and C∗-algebras. For instance, Blackwellshowed the importance of these sets in avoiding certain inherent pathologiesin Kolmogorov’s foundations of probability theory [13]; in Blackwell’s modelof dynamic programming [14] the existence of optimal strategies has beenshown to be related to the existence of measurable selections (Maitra [74]);Mackey made use of these sets in problems regarding group representations,and in particular in defining topologies on measurable groups [72]; Choquet[30], [34] used these sets in potential theory; and so on. The theory of Borelsets has found uses in diverse applied areas such as optimization, controltheory, mathematical economics, and mathematical statistics [5], [10], [32],[42], [91], [55]. These applications, in turn, have enriched the theory ofBorel sets itself considerably. For example, most of the measurable selectiontheorems arose in various applications, and now there is a rich supply ofthem. Some of these, such as the cross-section theorems for Borel partitionsof Polish spaces due to Mackey, Effros, and Srivastava are basic results onBorel sets.

Thus, today the theory of Borel sets stands on its own as a powerful,deep, and beautiful theory. This book is an introduction to this theory.

About This Book

This book can be used in various ways. It can be used as a stepping stoneto descriptive set theory. From this point of view, our audience can beundergraduate or beginning graduate students who are still exploring areasof mathematics for their research. In this book they will get a reasonablythorough introduction to Borel sets and measurable selections. They willalso find the kind of questions that a descriptive set theorist asks. Thoughwe stick to Borel sets only, we present quite a few important techniques,such as universal sets, prewellordering, and scales, used in descriptive settheory. We hope that students will find the mathematics presented in thisbook solid and exciting.

Secondly, this book is addressed to mathematicians requiring Borel sets,measurable selections, etc., in their work. Therefore, we have tried our bestto make it a convenient reference book. Some applications are also givenjust to show the way that the results presented here are used.

Finally, we desire that the book be accessible to all mathematicians.Hence the book has been made self-contained and has been written inan easygoing style. We have refrained from displaying various advancedtechniques such as games, recursive functions, and forcing. We use onlynaive set theory, general topology, some analysis, and some algebra, whichare commonly known.

The book is divided into five chapters. In the first chapter we give the set-theoretic preliminaries. In the first part of this chapter we present cardinalarithmetic, methods of transfinite induction, and ordinal numbers. Thenwe introduce trees and the Souslin operation. Topological preliminaries arepresented in Chapter 2. We later develop the theory of Borel sets in the

xvi About This Book

general context of Polish spaces. Hence we give a fairly complete account ofPolish spaces in this chapter. In the last section of this chapter we prove sev-eral theorems that help in transferring many problems from general Polishspaces to the space of sequences NN or the Cantor space 2N. We introduceBorel sets in Chapter 3. Here we develop the theory of Borel sets as muchas possible without using analytic sets. In the last section of this chapterwe introduce the usual hierarchy of Borel sets. For the first time, readerswill see some of the standard methods of descriptive set theory, such asuniversal sets, reduction, and separation principles. Chapter 4 is central tothis book, and the results proved here bring out the inherent power of Borelsets. In this chapter we introduce analytic and coanalytic sets and provemost of their basic properties. That these concepts are of fundamental im-portance to Borel sets is amply demonstrated in this chapter. In Chapter5 we present most of the major measurable selection and uniformizationtheorems. These results are particularly important for applications. Weclose this chapter with a discussion on Vaught’s conjecture—an outstand-ing open problem in descriptive set theory, and with a proof of Kondo’suniformization of coanalytic sets.

The exercises given in this book are an integral part of the theory, andreaders are advised not to skip them. Many exercises are later treated asproved theorems.

Since this book is intended to be introductory only, many results onBorel sets that we would have much liked to include have been omitted.For instance, Martin’s determinacy of Borel games [80], Silver’s theorem oncounting the number of equivalence classes of a Borel equivalence relation[106], and Louveau’s theorem on Borel sets in the product [66] have not beenincluded. Similarly, other results requiring such set-theoretic techniquesas constructibility, large cardinals, and forcing are not given here. In ourinsistence on sticking to Borel sets, we have made only a passing mention ofhigher projective classes. We are sure that this will leave many descriptiveset theorists dissatisfied.

We have not been able to give many applications, to do justice to whichwe would have had to enter many areas of mathematics, sometimes evendelving deep into the theories. Clearly, this would have increased the sizeof the book enormously and made it unwieldy. We hope that users will findthe passing remarks and references given helpful enough to see how resultsproved here are used in their respective disciplines.