A Course on Borel Sets

271 Pages·2001·18.785 MB·English

by  S.M. Srivastava

180
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
S.M. Srivastava
A Course on
Borel Sets
With 11 D1ustrations
~ 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 Berkelq, 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 1 S.M. Srivastava.
p. cm. - (Graduate texts in mathematics; 180)
Includes index.
ISBN 978-3-642-85475-0 ISBN 978-3-642-85473-6 (eBook)
DOI 10.1007/978-3-642-85473-6
1. Borel sets. J. Title. 11. Series.
QA248.S741998
511,3'2-dc21 97-43726
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ISBN 978-3-642-85475-0
Srivastava: A Course on Borel Sets
e 1998 Springer-Verlag New York, Inc.
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This book is dedicated to the memory 0/
my beloved wile, Kiran
who passed away soon after this book was completed.
Acknowledgments
I am grateful to many people who have suggested improvements in the
original 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 several
versions of this book and helped in all possible ways to bring the book to
its present form. It is a pleasure to record my appreciation for A. Maitra
who showed the beauty and power of Borel sets to a generation of Indian
mathematicians including me. I also thank him for his suggestions during
the planning stage of the book.
I thank P. Bandyopadhyay who helped me immensely to sort out all the
OOEX problems. Thanks are also due to R. Kar for preparing the OOEX
files for the illustrations in the book.
I am indebted to S. B. Rao, Director of the Indian Statistical Institute for
extending excellent moral and material support. All my colleagues in the
Stat - Math Unit also lent a much needed and invaluable moral support
during the long and difficult period that the book was written. I thank
them all.
I take this opportunity to express my sincere feelings of gratitude to my
children, Rosy and Ravi, for their great understanding of the task I took
onto myself. What they missed during the period the book was written will
be 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. SriVb8tava
Contents
Acknowledgments vii
Introduction xi
About This Book xv
1 Cardinal and Ordinal Numbers 1
1.1 Countable Sets .... 1
1.2 Order of Infinity ..... 4
1.3 The Axiom of Choice. . . 7
1.4 More on Equinumerosity . 11
1.5 Arithmetic of Cardinal Numbers 13
1.6 Well-Ordered Sets .. 15
1.7 Transfinite Induction . 18
1.8 Ordinal Numbers . 21
1.9 Alephs ....... 24
1.10 Trees ........ 26
1.11 Induction on Trees 29
1.12 The Souslin Operation 31
1.13 Idempotence of the Souslin Operation 34
2 Topological Preliminaries 39
2.1 Metric Spaces . . . . . . 39
2.2 Polish Spaces ...... 52
2.3 Compact Metric Spaces 57
2.4 More Examples . . . . . 63
x Contents
2.5 The Haire Category Theorem 69
2.6 Transfer Theorems . . 74
3 Standard Borel Spaces 81
3.1 Measurable Sets and Functions 81
3.2 Borel-Generated Topologies . . . 91
3.3 The Borel Isomorphism Theorem 94
3.4 Measures ..... . 100
3.5 Category............ 107
3.6 Borel Pointclasses. . . . . . . 115
4 Analytic and Coanalytic Sets 127
4.1 Projective Sets ............ . 127
nf
4.2 E~ and Complete Sets . . 135
4.3 Regularity Properties. . . . . 141
4.4 The First Separation Theorem .... 147
4.5 One-to-One Borel Functions . . . . . . 150
4.6 The Generalized First Separation Theorem 155
4.7 Borel Sets with Compact Sections . 157
4.8 Polish Groups . . . . . . . . . . . 160
4.9 Reduction Theorems . . . . . . . 164
4.10 Choquet Capacitability Theorem 172
4.11 The Second Separation Theorem 175
4.12 Countable-to-One Borel Functions . 178
5 Selection and U niformization Theorems 183
5.1 Preliminaries .. . . . . . . . . . . . . . . 184
5.2 Kuratowski and Ryll-Nardzewski's Theorem. 189
5.3 Dubins - Savage Selection Theorems 194
5.4 Partitions into Closed Sets. . . . . . . . . . 195
5.5 Von Neumann's Theorem ........ . 198
5.6 A Selection Theorem for Group Actions . 200
5.7 Borel Sets with Small Sections . 204-
5.8 Borel Sets with Large Sections .... . 206
5.9 Partitions into G6 Sets ........ . 212
5.10 Reflection Phenomenon ....... . 216
5.11 Complementation in Borel Structures. 218
5.12 Borel Sets with u-Compact Sections 219
5.13 Topological Vaught Conjecture 227
5.14 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 to
come to grips with the abstract notion of a function introduced by Dirich
let and Riemann. According to them, a function was to be an arbitrary
correspondence between objects without giving any method or procedure
by which the correspondence could be established. Since all the specific
functions 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 0/ pointwise limit on a se
quence 0/ functions. These functions are now known as Baire functions.
Lebesgue [65] and Borel [19] continued this work. In [19], Borel sets were
defined for the first time. In his paper, Lebesgue made a systematic study
of Baire functions and introduced many tools and techniques that are used
even today. Among other results, he showed that Borel functions coincide
with Baire functions. The study of Borel sets got an impetus from an error
in Lebesgue's paper, which was spotted by Souslin. Lebesgue was trying to
prove the following:
Suppose / : )R2 -- R is a Baire function such that for every x, the
equation
/(x,y) = 0
has a. unique solution. Then y as a function 0/ x defined by the above
equation is Baire.
The wrong step in the proof was hidden in a lemma stating that a set
of 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 a
Borel set analytic because such a set can be constructed using analytical
operations of union and intersection on intervals. He showed that there are
xii Introduction
analytic sets that are not Borel. Immediately after this, Souslin [111 J and
Lusin [67J made a deep study of analytic sets and established most of the
basic results about them. Their results showed that analytic sets are of
fundamental importance to the theory of Borel sets and give it its power.
For instance, Souslin proved that Borel sets are precisely those analytic sets
whose complements are also analytic. Lusin showed that the image of a
Borel set under a one-to-one Borel map is BoreL It follows that Lebesgue's
thoerem-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-one
correspondence with the real line. Alexandrov showed that every uncount
able Borel set of reals is in one-to-one correspondence with the real line [2J.
In other words, a Borel set cannot be a counterexample to the continuum
hypothesis.
Unfortunately, Souslin died in 1919. The work on this new-found topic
was continued by Lusin and his students in Moscow and by Sierpinski and
his collaborators in Warsaw.
The next important step was the introduction of projective sets by
Lusin [68J, [69], [70J and Sierpinski [105J in 1925: A set is called projective
if it can be constructed starting with Borel sets and iterating the operations
of projection and complementation. Since Borel sets as well as projective
sets are sets that can be described using simple sets like intervals and
simple set operations, their theory came to be known as descriptive set
theory. It was clear from the beginning that the theory of projective sets
was riddled with problems that did not seem to admit simple solutions. As
it turned out, logicians did show later that most of the regularity properties
of projective sets, e.g., whether they satisfy the continuum hypothesis or
not or whether they are Lebesgue measurable and have the property of
Baire or not, are independent of the axioms of classical set theory.
Just as Alexandrov was trying to determine the status of the continuum
hypothesis within Borel sets, Lusin [71J considered the status of the axiom
of choice within "Borel families." He raised a very fundamental and difficult
question on Borel sets that enriched its theory significantly. Let B be a
subset of the plane. A subset C of B uniformizes B if it is the graph of a
function such that its projection on the line is the same as that of B. (See
Figure 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 this
problem and gave sufficient conditions under which Lusin's question has
a positive answer. For instance, a Borel set admits a Borel uniformization
if the sections of B are countable (Lusln [71j) or compact (Novikov [90J)
or CT-compact (Arsenin [$J ancl Kunugui [60j) or nonmeager (Kechris [SfJ
and Sarbadhikari [100]). Even today these results are ranked among the