A Course on Borel Sets

274 Pages·1998·1.406 MB·English

by  S. M. Srivastava (auth.)

A Course on
Borel Sets
S.M. Srivastava
Springer

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 (n´ee 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
LATEX problems. Thanks are also due to R. Kar for preparing the LATEX
files for the illustrations in the book.
IamindebtedtoS.B.Rao,DirectoroftheIndianStatisticalInstitutefor
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
ontomyself.Whattheymissedduringtheperiodthebookwaswrittenwill
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. Srivastava
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 Baire 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
4.2 Σ1 and Π1 Complete Sets . . . . . . . . . . . . . . . . . . . 135
1 1
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 Uniformization 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 G Sets . . . . . . . . . . . . . . . . . . . . . 212
δ
5.10 Reflection Phenomenon . . . . . . . . . . . . . . . . . . . . 216
5.11 Complementation in Borel Structures. . . . . . . . . . . . . 218
5.12 Borel Sets with σ-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
functionsthatonestudiedweredeterminedbysimpleanalyticexpressions,
Bairedelineatedthosefunctionsthatcan 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 were
defined for the first time. In his paper, Lebesgue made a systematic study
ofBairefunctionsandintroducedmanytoolsandtechniquesthatareused
even today. Among other results, he showed that Borel functions coincide
withBairefunctions.ThestudyofBorelsetsgotanimpetusfromanerror
inLebesgue’spaper,whichwasspottedbySouslin.Lebesguewastryingto
prove the following:
Suppose f : R2 −→ R is a Baire function such that for every x, the
equation
f(x,y)=0
has a unique solution. Then y as a function of 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
operationsofunionandintersectiononintervals.Heshowedthatthereare