A course on borel sets, s m srivastava

Graduate Texts in Mathematics 180 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 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 Berkelq, 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 978-3-642-85475-0 ISBN 978-3-642-85473-6 (eBook) DOI 10.1007/978-3-642-85473-6 Borel sets J Title 11 Series QA248.S741998 97-43726 511,3'2-dc21 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, eJecttonic 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 978-3-642-85475-0 Srivastava: A Course on Borel Sets e 1998 Springer-Verlag New York, Inc All rights reserved No part of this publication may be reproduced, stored in any electronic or mechanical form, including photocopy, recording or otherwise, without the prior written permission of the publisher First Indian Reprint 2010 "This edition is licensed for sale only in India, Pakistan, Bangladesh, Sri Lanka and Nepal Circulation of this edition outside of these territories is UNAUTHORIL:ED AND STRICTLY PROHIBITED." This edition is published by Springer (India) Private Limited, (a part of Springer Science+Business Media), Registered Office: 3'J Floor, Gandharva Mahavidyalaya, 212, Deen Dayal Upadhyaya Marg, New Delhi - 110 002, India 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 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 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 Introduction vii 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 Haire 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 Projective Sets E~ and nf Complete Sets Regularity Properties The First Separation Theorem One-to-One Borel Functions The Generalized First Separation Theorem Borel Sets with Compact Sections Polish Groups Reduction Theorems Choquet Capacitability Theorem The Second Separation Theorem Countable-to-One Borel Functions 127 127 Selection and U niformization 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 G6 Sets 5.10 Reflection Phenomenon 5.11 Complementation in Borel Structures 5.12 Borel Sets with u-Compact Sections 5.13 Topological Vaught Conjecture 5.14 Uniformizing Coanalytic Sets 183 References 241 Glossary 251 Index 253 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 135 141 147 150 155 157 160 164 172 175 178 184 189 194 195 198 200 204206 212 216 218 219 227 236 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 0/ pointwise limit on a sequence 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) = 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 hypothesis of Cantor: Every uncountable set of real numbers is in one-to-one correspondence with the real line Alexandrov showed that every uncountable 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 uniformization? By Lusin's theorem stated earlier, if B admits a Borel uniformization, 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 246 References [71] N Lusin Sur Ie probl~me de M J Hadamard d'unifonnisation des ensembles C R Acad des Sci., Paris, 190 (1930), 349 - 351 [72] G W Mackey Borel structures in groups and their duals Trans Amer Math Soc., 85 (1957), 134 - 165 [73] G W Mackey The Theory 0/ Unitary Group Representations Univ of Chicago Press, Chicago, 1976 [74] A Maitra Discounted dynamic programming on compact metric spaces Sankhya A, 30 (1968), 211 - 216 [75] A Maitra Selectors for Borel sets with large sections Proc Amer Math Soc., 89 (1983), 705 - 708 [76] A Maitra and C Ryll-Nardzewski On the Existence of Two Analytic Non-Borel Sets Which Are Not Isomorphic Bull Polish Acad des Sciences, 28 (1970), 177 - 178 [77] A Maitra and B V Rao Generalizations of Castaing's theorem on selectors ColI Math., 42 (1979), 295 - 300 [78] A Maitra and W Sudderth Discrete Gambling and Stochastic Games Applications in Mathematics 32, Springer-Verlag, New York, 1996 (79) R Mansfield and G Weitkamp Recursive Aspects 0/ Descriptive Set Theory Oxford Logic Group, Oxford University Press, New York, Oxford, 1985 [80] D A Martin Borel determinacy Annals of Mathematics, 102 (1975), 363 - 371 (81) R D Mauldin Bimeasurable Functions Proc Amero' Math Soc., 29 (1980), 161 - 165 [82] E Michael Continuous selections I Annals of Mathematics, 63 (1956), 361 - 382 [83] A W Miller Descriptive Set Theory and Forcing: How to protIe theorems about Borel sets in a hard way Lecture notes in Logic 4, SpringerVerlag, New York, 1995 [84] D E Miller On the measurability of orbits in Borel actions Proc Amer Math Soc., 63 (1977), 165 - 170 [85] D E Miller A selector for equivalence relations with G" orbits Proc Amer Math Soc., 72 (1978), 365 - 369 [86] G Mokobodzki Demonstration elementaire d'un theoreme de Novikov Sem de Prob X, Lecture Notes in Math 511, Springer-Vrlag, Heidelberg, 1976, 539 - 543 References 247 [87) C C Moore Appendix to "Polarization and unitary representations of solvable Lie groups" Invent Math., 14 (1971), 351 - 354 [88) Y N Moschovakis Descriptive Set Theory Studies in Logic and the Foundations of Mathematics, 100, North-Holland Publishing Company, 1980 [a9) M G Nadkarni Basic Ergodic Theory Texts and Readings in Mathematics 7, Hindustan Book Agency, New Delhi, 1995 [90) P Novikoff Sur les fonctions implicites mesurables B Fund Math., 17(1931), 8-25 [91) C Olech Existence theorems for optimal problems with vector-valued cost function Trans Amer Math Soc., 136 (1969), 159 - 180 [92] D Preiss The convex generation of convex Borel sets in Banach spaces Mathematika, 20 (1973), - [93] R Purves On Bimeasurable Functions Fund Math., 58 (1966), 149 - 157 [94] B V Rao On discrete Borel spaces and projective sets Bull Amer Math Soc., 75 (1969), 614 - 617 [95) B V Rao Remarks on analytic sets Fund Math., 66 (1970), 237 239 [96) B V Rao and S M Srivastava An Elementary Proof of the Borel Isomorphism Theorem Real Analysis Exchange, 20 (1), 1994/95 - [97) J Saint Raymond Boreliens A coupes K eT • Bull Soc Math France, (2) 100 (1978), 141 - 147 [98) R Sami Polish group actions and the Vaught conjecture Trans Amer Math Soc., 341 (1994), 335 - 353 [99) H Sarbadhikari A note on some properties of A-functions Proc Amer Math Soc., 56 (1976), 321 - 324 [100] H Sarbadhikari Some uniformization results Fund Math., 97 (1977), 209 - 214 [101) H Sarbadhikari and S M Srivastava Parametrizations of G.s-valued multifunctions Trans Amer Math Soc., 258 (1980), 165 - 178 (102) H Sarbadhikari and S M Srivastava Random theorems in topology Fund Math., 136 (1990), 65 - 72 248 References [103] M Schil A selection theorem for optimi38tion problems Arch Math., 25 (1974), 219 - 224 [104] M Schi1 Conditions for optimality in dynamic programming and for the limit of n-state optimal policies to be optimal Z Wabrscheinlichkeitstheorie and verw Gebiete, 32 (1975), 179 - 196 [105] W Sierpinski Sur une c1asse 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 [Ill] 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 mierzalno8ci i warunku Baire'a C R du I congres des Math des Pays Slaves, Varsovie 1929, p 209 [114] S M Srivastava Selection theorems for G6-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 G6-valued multifunctions American J Math., 102 (198O), 165 - 178 (117) S M Srivastava Transfinite Numbers Resonance, (3), 1997, 58 - 68 [118] J Stem Effective partitions of the real line into Borel sets of bounded rank Ann Math Logic, 18 (1980), 29 - 60 [119] J Stem 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 - Wt Nt N 1I:+,::la 25 A