A bsolute M easurable S paces Absolutemeasurablespaceandabsolutenull spaceareveryold topological notions, developed from descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition aretheaction of thegroup of homeomorphismsof a spaceon Borelmeasures,the Oxtoby–Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem, and recent advances on homeomorphic Borel probability measures on the Cantor space are among the many topics discussed. A brief discussion of set-theoretic results on absolutenull spaceis also given. A four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. The exposition will suit researchers and graduate students of real analysis, set theory and measure theory. Togo Nishiurais Professor Emeritus at Wayne State University, Detroit, andAsso- ciate Fell in Mathematics at Dickinson College, Pennsylvania. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed bel can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=EOM! 63. A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65. K. Engel Sperner Theory 66. D. Cvetkovic, P. Rlinson, and S. Simic Eigenspaces of Graphs 67. F. Bergeron, G. Labelle, and P. Leroux Combinational Species and Tree-Like Structures 68. R. Goodman and N. Wallach Representations and Invariants of the Classical Groups 69. T. Beth, D. Jungnickel, and H. Lenz Design Theory 1, 2nd edn 70. A. Pietsch and J. 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L. W. Beineke, R. J. Wilson, and P. J. Cameron. (eds.) Topics in Algebraic Graph Theory 103. O. Staffans Well-Posed Linear Systems 104. J. M. Lewis, S. Lakshmivarahan, and S. Dhall Dynamic Data Assimilation 105. M. Lothaire Applied Combinatorics on Words 106. A. Markoe Analytic Tomography 107. P. A. Martin Multiple Scattering 108. R. A. Brualdi Combinatorial Matrix Classes 110. M.-J. Lai and L. L. Schumaker Spline Functions on Triangulations 111. R. T. Curtis Symetric Generation of Groups 112. H. Salzmann, T. Grundhöfer, H. Hähl, and R. Löwen The Classical Fields 113. S. Peszat and J. Zabczyk Stochastic Partial Differential Equations with Lévy Noise 114. J. Beck Combinatorial Games 115. L. Barreira and Y. Pesin Nonuniform Hyperbolicity 116. D. Z. Arov and H. Dym J-Contractive Matrix Valued Functions and Related Topics 117. R. Glinski, J.-L. Lions, and J. He Exact and Approximate Controllability for Distributed Parameter Systems 118. A. A. Borovkov and K. A. Borovkov Asympotic Analysis of Random Walks 119. M. Deza and M. Dutour Sikiri´c Geometry of Chemical Graphs 120. T. Nishiura Absolute Measurable Spaces ENCYCLOPEDIAOF MATHEMATICSAND ITS APPLICATIONS AbsoluteMeasurableSpaces TOGO NISHIURA cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Tn, Singapore, São Paulo, Delhi Cambridge University Press TheEdinburgh Building, CambridgeCB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521875561 © T. Nishiura 2008 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproductionof any part may takeplacewithout the written permission of Cambridge University Press. 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Contents Preface page ix Acknowledgements xiii 1 The absolute property 1 1.1 Absolutemeasurablespaces 1 1.2 Absolutenull spaces 7 1.3 Existenceofabsolutenull spaces 10 1.4 Grzegorek’s cardinal number κG 18 1.5 Moreon existenceofabsolutenull spaces 24 1.6 Comments 26 Exercises 28 2 The universally measurable property 30 2.1 Universally measurable sets 31 2.2 Positive measures 35 2.3 Universally measurable maps 37 2.4 Symmetric difference of Borel and null sets 39 2.5 Early results 42 2.6 Thehomeomorphismgroup of [0, 1] 43 2.7 Thegroup of B-homeomorphisms 46 2.8 Comments 49 Exercises 52 3 The homeomorphism group of X 53 3.1 A metric for HOMEO(X ) 54 3.2 General properties 56 3.3 One-dimensional spaces 57 3.4 The Oxtoby–Ulam theorem 61 3.5 n-dimensional manifolds 73 3.6 TheHilbertcube 76 3.7 Zero-dimensional spaces 82 vi Contents 3.8 Other examples 88 3.9 Comments 90 Exercises 97 4 Real-valued functions 99 4.1 A solution to Goldman’s problem 100 4.2 Differentiability and B-maps 103 4.3 Radon–Nikodym derivative and Oxtoby–Ulam theorem 105 4.4 Zahorski spaces 112 4.5 Bruckner–Davies–Goffman theorem 115 4.6 Changeof variable 126 4.7 Images of Lusin sets 128 4.8 Comments 130 Exercises 134 5 Hausdorff measure and dimension 136 5.1 Universally null sets in metric spaces 136 5.2 A summary of Hausdorff dimension theory 137 5.3 Cantor cubes 139 5.4 Zindulka’s theorem 143 5.5 Analytic sets in Rn 146 5.6 Zindulka’s opaquesets 151 5.7 Comments 154 Exercises 156 6 Martin axiom 157 6.1 CH and universally null sets: a historical tour 157 6.2 Absolutenull spaceand cardinal numbers 165 6.3 Consequences of the Martin axiom 168 6.4 Topological dimension and MA 171 6.5 Comments 173 Exercises 178 AppendixA Preliminary material 179 A.1 Complete metric spaces 179 A.2 Borel measurable maps 182 A.3 Totally imperfect spaces 185 A.4 Complete Borel measure spaces 186 A.5 The sum of Borel measures 192 A.6 Zahorski spaces 193 A.7 Purves’s theorem 194 A.8 Comments 203 Exercises 203 Contents vii AppendixB Probability theoretic approach 204 B.1 Basic definitions 204 B.2 Separable metrizability 206 B.3 Shortt’s observation 208 B.4 Lusin measurable space 210 B.5 Comments 212 Exercises 213 AppendixC Cantor spaces 214 C.1 Closed and open sets 215 C.2 A metric for k N 217 C.3 Bernoulli measures 219 C.4 Uniform Bernoulli distribution 220 C.5 Binomial Bernoulli distribution 221 C.6 Linear ordering of {0, 1}N and good measures 230 C.7 Refinable numbers 233 C.8 Refinable numbers and good measures 239 C.9 Comments 240 Exercises 242 AppendixD Dimensions and measures 244 D.1 Topological dimension 244 D.2 Measure theoretical dimension 246 D.3 Zindulka’s dimension theorem 249 D.4 Geometric measure theory 253 D.5 Marstrand’s theorem 255 Exercises 257 Bibliography 258 Notation index 267 Author index 270 Subject index 272 Preface This book is about absolutemeasurablespaces.What is an absolutemeasurablespace and why study them? To aner the first question, an absolute measurable space, simply put, is a sep- arablemetrizablespaceX with the property that every topological embedding of X into any separable metrizable space Y results in a set that is µ-measurable for every continuous, complete, finite Borel measure µ on Y . Of course, only Borel measures areconsideredsincethetopology of Y must play a rolein thedefinition. For an aner to the second question, observe that the notion of absolute measurable spaceis a topological onein thespirit of many othernotions of “absolute”such as absolute Borel space, absolute Gδ space, absolute retract and many more. As the definition is topological, one is led to many topological questions about such spaces.