Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures

Schwartz, L.Radon measures on arbitrary topological spaces and cylindrical measures(Tata Institute Monographs on Mathematics & Physics) (OUP,1973) ......Page 4
Copyright ......Page 5
CONTENTS ......Page 9
Introduction v ......Page 6
Remerciement vii   ......Page 8
PART I......Page 14
Summary of Chapter I 3 ......Page 16
1.Preliminaries 7 ......Page 20
2.Definition of Radon measure on general topological spaces 12 ......Page 25
3.Induced measures 20 ......Page 33
4.Product of a measure by a locally integrable function 23 ......Page 36
5.Measurable mappings and image measures 25 ......Page 38
6.Other properties of Radon measures 40 ......Page 53
7.Real and complex measures 53 ......Page 66
8.Different ways of defining Radon measures 58 ......Page 71
9.Tensor product of measures 63 ......Page 76
10.Projective limit of measures 74 ......Page 87
11.Non-Hausdorff spaces 82 ......Page 95
Summary of Chapter II 89 ......Page 102
1.Definitions and topological properties 92 ......Page 105
2.Examples of Lusin spaces 112 ......Page 125
3.Radon spaces 117 ......Page 130
4.Lifting of L to K 130 ......Page 143
5.Spaces of regulated functions 136 ......Page 149
Terminological Index to Part I 144 ......Page 157
References 146 ......Page 159
PART II  149 ......Page 162
Introduction to Part II 151  ......Page 164
Summary of Chapter I 152 ......Page 165
2.Equivalent topologies 155 ......Page 166
1.Equivalent measures 153 ......Page 168
3.A Borel-graph theorem 160 ......Page 173
4.Equivalence of the strong and weak topologies 161 ......Page 174
5.Equivalence of the strong topology with coarser topologies 166 ......Page 179
Summary of Chapter II 171 ......Page 184
1.Definition of cylindrical measures 172 ......Page 185
2.Operations with cylindrical measures 180 ......Page 193
3.Concentrations 188 ......Page 201
Summary of Chapter III 206 ......Page 219
1.Hilbert-Schmidt operators 207 ......Page 220
2.The main lemma 212 ......Page 225
3.The theorem of Sazonov 214 ......Page 227
Summary of Chapter IV 218 ......Page 231
1.Nuclear operators 219 ......Page 232
2.Nuclear spaces 226 ......Page 239
3.The theorem of Minlos 233 ......Page 246
4.The theorem of Sazonov-Badrikian 237 ......Page 250
Summary of Chapter V 242 ......Page 255
1.Basic concepts of probability theory 244 ......Page 257
2.Randon functions 254 ......Page 267
3.Examples and applications 269 ......Page 282
4.An analogue of Prokhorov’s theorem for random variables with values in a Suslin space 288 ......Page 301
5.Categories of random functions 305 ......Page 318
Summary of Chapter VI 324 ......Page 337
1.The definition of Gauss measures 325 ......Page 338
2.The Hilbert subspaces of a locally convex space 330 ......Page 343
3.The converse of the theorems of Sazonov and Minlos 341 ......Page 354
4.The analogue of Sazonov’s theorem for l^p spaces (0 < p < + oo) and applications 347 ......Page 360
5.An application to Brownian motion 350 ......Page 363
Summary of Appendix 368 ......Page 381
1.Narrow convergence on an arbitrary topological space 369 ......Page 382
2.Riemann-integrable functions 374 ......Page 387
3.A compactness criterion for the narrow topology 379 ......Page 392
4.A theorem of Paul Levy 382 ......Page 395
5.The space of finite measures on a Suslin space 385 ......Page 398
Terminological Index to Part II 390 ......Page 403
References 392 ......Page 405
cover......Page 1