Graduate Texts in Mathematics S Axler Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo 180 Editorial Board F.W Gehring K.A Ribet Graduate Texts in Mathematics TAKBUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OxTOBY Measure and Category 2nd ed ScHAEFER Topological Vector Spaces HILTON/STAMMBACH A Course in 33 HIRSCH Differential Topology 34 SprrzER Principles of Random Walk 2nd ed 35 ALEXANDER/WERMER Several Complex Variables and Banach Algebras 3rd ed Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras HUMPHREYS Introduction to Lie 40 20 21 Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BBALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GoLUBiTSKY/GuiLLEMiN Stable Mappuigs and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nded HusEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups 22 BARNES/MACK An Algebraic Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LofevE Probability Theory I 4th ed 46 LofevE Probability Theory 11 4th ed 47 MoiSE Geometric Topology in Dimensions and 48 SACHSAVU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nded 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HA'RTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVER/WATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/FOX Introduction to Knot Theory 58 KoBLiTZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 10 11 12 13 14 15 16 17 18 19 Introduction to Mathematical Logic 23 GREUB Linear Algebra 4th ed 24 HOLMES Geometric Functional Analysis and Its Applications 25 HEWITT/STROMBERG Real and Abstract Analysis 26 MANES Algebraic Theories 27 KELLEY General Topology 28 ZARISKI/SAMUEL Commutative Algebra Vol.1 29 ZARISKI/SAMUEL Commutative Algebra Vol.11 30 JACOBSON Lectures in Abstract Algebra I, Basic Concepts 31 JACOBSON Lectures in Abstract Algebra II Linear Algebra 32 JACOBSON Lectures in Abstract Algebra IH Theory of Fields and Galois Theory 36 KELLEY/NAMIOKA et al Linear KEMENY/SNELL/KNAPP Denumerable continued after index 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 Department of Mathematics San Francisco State University San Francisco, CA 94132 USA F.W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720 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) 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 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 appreciation 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 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 Srivastava Contents Acknowledgments vii Introduction xi About This Book xv Cardinal and Ordinal Numbers 1.1 Countable Sets 1.2 Order of Infinity 1.3 The Axiom of Choice 1.4 More on Equinumerosity 1.5 Arithmetic of Cardinal Numbers 1.6 Well-Ordered Sets 1.7 Transfinite Induction 1.8 Ordinal Numbers 1.9 Alephs 1.10 Trees 1.11 Induction on Trees 1.12 The Souslin Operation 1.13 Idempotence of the Souslin Operation 1 11 13 15 18 21 24 26 29 31 34 Topological Preliminaries 2.1 Metric Spaces 2.2 Polish Spaces 2.3 Compact Metric Spaces 2.4 More Examples 39 39 52 57 63 x Contents 2.5 2.6 The Baire Category Theorem Transfer Theorems Standard Borel Spaces 3.1 Measurable Sets and Functions 3.2 Borel-Generated Topologies 3.3 The Borel Isomorphism Theorem 3.4 Measures 3.5 Category 3.6 Borel Pointclasses 69 74 81 81 91 94 100 107 115 Analytic and Coanalytic Sets 4.1 Projective Sets 4.2 Σ11 and Π11 Complete Sets 4.3 Regularity Properties 4.4 The First Separation Theorem 4.5 One-to-One Borel Functions 4.6 The Generalized First Separation Theorem 4.7 Borel Sets with Compact Sections 4.8 Polish Groups 4.9 Reduction Theorems 4.10 Choquet Capacitability Theorem 4.11 The Second Separation Theorem 4.12 Countable-to-One Borel Functions 127 127 135 141 147 150 155 157 160 164 172 175 178 Selection and Uniformization Theorems 5.1 Preliminaries 5.2 Kuratowski and Ryll-Nardzewski’s Theorem 5.3 Dubins – Savage Selection Theorems 5.4 Partitions into Closed Sets 5.5 Von Neumann’s Theorem 5.6 A Selection Theorem for Group Actions 5.7 Borel Sets with Small Sections 5.8 Borel Sets with Large Sections 5.9 Partitions into Gδ Sets 5.10 Reflection Phenomenon 5.11 Complementation in Borel Structures 5.12 Borel Sets with σ-Compact Sections 5.13 Topological Vaught Conjecture 5.14 Uniformizing Coanalytic Sets 183 184 189 194 195 198 200 204 206 212 216 218 219 227 236 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 Dirichlet 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 continuous functions and iterating the operation of pointwise limit on a sequence 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 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 f : R2 −→ R is a Baire function such that for every x, the equation f (x, y) = 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 operations of union and intersection on intervals He showed that there are 248 References [103] M Schă al A selection theorem for optimization problems Arch Math., 25 (1974), 219 224 [104] M Schă al Conditions for optimality in dynamic programming and for the limit of n-state optimal policies to be optimal Z Wahrscheinlichkeitstheorie and verw Gebiete, 32 (1975), 179 – 196 [105] W Sierpi´ nski Sur une classe d’ensembles Fund Math., (1925), 237 – 243 [106] J H Silver Counting the number of equivalence classes of Borel and coanalytic equivalence relations Ann Math Logic, 18 (1980), – 28 [107] M Sion On capacitability and measurability Ann Inst Fourier, Grenoble, 13 (1963), 88 – 99 [108] S Solecki Equivalence relations induced by actions of Polish groups Trans Amer Math Soc., 347 (1995), 4765 – 4777 [109] S Solecki and S M Srivastava Automatic continuity of group operations Topology and its applications, 77 (1997), 65 – 75 [110] R M Solovay A model of set theory in which every set of reals is Lebesgue measurable Annals of Mathematics, 92 (1970), – 56 [111] M Souslin Sur une definition des ensembles B sans nombres tranfinis C R Acad Sciences, Paris, 164 (1917), 88 – 91 [112] J R Steel On Vaught’s conjecture Cabal Seminar 1976 – 1977, Lecture Notes in Mathematics, 689, Springer-Verlag, Berlin, 1978 [113] E Szpilrajn-Marczewski O mierzalno´sci i warunku Baire’a C R du I congr`es des Math des Pays Slaves, Varsovie 1929, p 209 [114] S M Srivastava Selection theorems for Gδ -valued multifunctions Trans Amer Math Soc., 254 (1979), 283 – 293 [115] S M Srivastava A representation theorem for closed valued multifunctions Bull Polish Acad des Sciences, 27 (1979), 511 – 514 [116] S M Srivastava A representation theorem for Gδ -valued multifunctions American J Math., 102 (1980), 165 – 178 [117] S M Srivastava Transfinite Numbers Resonance, (3), 1997, 58 – 68 [118] J Stern Effective partitions of the real line into Borel sets of bounded rank Ann Math Logic, 18 (1980), 29 – 60 [119] J Stern On Lusin’s restricted continuum hypothesis Annals of Mathematics, 120 (1984), – 37 References 249 [120] A H Stone Non-separable Borel sets Dissertationes Mathematicae (Rozprawy Matematyczne), 28 (1962) [121] S M Ulam A collection of mathematical problems Interscience, New York 1960 [122] R L Vaught Denumerable models of complete theories Infinitistic Methods: Proceedings of the symposium on foundations of mathematics, PWN, Warsaw, 1961, 303 – 321 [123] R L Vaught Invariant sets in topology and logic Fund Math., 82 (1974), 269 – 293 [124] J Von Neumann On rings of operators: Reduction Theory Annals of Mathematics, 50 (1949), 401 – 485 Glossary ≡ ω , ℵ1 , ω α , ℵα 24 N κ+ , 25 Q, X k , X