Absolute Measurable Spaces (Encyclopedia of Mathematics and its Applications, Series Number 120)

Absolute measurable space and absolute null space are very old topological notions, developed from well-known facts of 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 are the action of the group of homeomorphisms of a space on Borel measures, 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 absolute null space is given, and a four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 10 Acknowledgements......Page 14 1.1 Absoluteme asurablespace s......Page 16 1.2 Absolutenull spaces......Page 22 1.3 Existenceof absolutenull spaces......Page 25 1.4 Grzegorek?ˉs cardinal number |êG......Page 33 1.5 Moreon existenceof absolutenull spaces......Page 39 1.6 Comments......Page 41 Exercises......Page 43 2 The universally measurable property......Page 45 2.1 Universally measurable sets......Page 46 2.2 Positive measures......Page 50 2.3 Universally measurable maps......Page 52 2.4 Symmetric difference of Borel and null sets......Page 54 2.5 Early results......Page 57 2.6 Thehome omorphism group of [0, 1]......Page 58 2.7 Thegroup of B-homeomorphisms......Page 61 2.8 Comments......Page 64 Exercises......Page 67 3 The homeomorphism group of X......Page 68 3.1 A metric for HOMEO(X )......Page 69 3.2 General properties......Page 71 3.3 One-dimensional spaces......Page 72 3.4 The Oxtoby ̈CUlam theorem......Page 76 3.5 n-dimensional manifolds......Page 88 3.6 TheHilbe rt cube......Page 91 3.7 Zero-dimensional spaces......Page 97 3.8 Other examples......Page 103 3.9 Comments......Page 105 Exercises......Page 112 4 Real-valued functions......Page 114 4.1 A solution to Goldman?ˉs problem......Page 115 4.2 Differentiability and B-maps......Page 118 4.3 Radon ̈CNikodym derivative and Oxtoby ̈CUlam theorem......Page 120 4.4 Zahorski spaces......Page 127 4.5 Bruckner ̈CDavies ̈CGoffman theorem......Page 130 4.6 Changeof variable......Page 141 4.7 Images of Lusin sets......Page 143 4.8 Comments......Page 145 Exercises......Page 149 5.1 Universally null sets in metric spaces......Page 151 5.2 A summary of Hausdorff dimension theory......Page 152 5.3 Cantor cubes......Page 154 5.4 Zindulka?ˉs theorem......Page 158 5.5 Analytic sets in Rn......Page 161 5.6 Zindulka?ˉs opaquese ts......Page 166 5.7 Comments......Page 169 Exercises......Page 171 6.1 CH and universally null sets: a historical tour......Page 172 6.2 Absolutenull spaceand cardinal numbers......Page 180 6.3 Consequences of the Martin axiom......Page 183 6.4 Topological dimension and MA......Page 186 6.5 Comments......Page 188 Exercises......Page 193 A.1 Complete metric spaces......Page 194 A.2 Borel measurable maps......Page 197 A.3 Totally imperfect spaces......Page 200 A.4 Complete Borel measure spaces......Page 201 A.5 The sum of Borel measures......Page 207 A.6 Zahorski spaces......Page 208 A.7 Purves?ˉs theorem......Page 209 Exercises......Page 218 B.1 Basic definitions......Page 219 B.2 Separable metrizability......Page 221 B.3 Shortt?ˉs observation......Page 223 B.4 Lusin measurable space......Page 225 B.5 Comments......Page 227 Exercises......Page 228 Appendix C Cantor spaces......Page 229 C.1 Closed and open sets......Page 230 C.2 A metric for k N......Page 232 C.3 Bernoulli measures......Page 234 C.4 Uniform Bernoulli distribution......Page 235 C.5 Binomial Bernoulli distribution......Page 236 C.6 Linear ordering of {0, 1}N and good measures......Page 245 C.7 Refinable numbers......Page 248 C.8 Refinable numbers and good measures......Page 254 C.9 Comments......Page 255 Exercises......Page 257 D.1 Topological dimension......Page 259 D.2 Measure theoretical dimension......Page 261 D.3 Zindulka?ˉs dimension theorem......Page 264 D.4 Geometric measure theory......Page 268 D.5 Marstrand?ˉs theorem......Page 270 Exercises......Page 272 Bibliography......Page 273 Notation index......Page 282 Author index......Page 285 Subject index......Page 287 "Absolute measurable space and absolute null space are very old 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 the notions."--Jacket