A User’s Guide to Measure Theoretic Probability

Contents ... 8
Preface ... 12
Chapter 1 Motivation ... 16
	1. Why bother with measure  theory? ... 16
	2. The cost and benefit of rigor ... 18
	3. Where to start: probabilities or expectations? ... 20
	4. The de  Finetti notation ... 22
	5. Fair prices ... 26
	6. Problems ... 28
	7. Notes ... 29
Chapter 2 A modicum of measure theory ... 32
	1. Measures and sigma-fields ... 32
	2. Measurable functions ... 37
	3. Integrals ... 41
	4. Construction of integrals from measures ... 44
	5. Limit theorems ... 46
	6. Negligible sets ... 48
	7. LP spaces ... 51
	8. Uniform integrability ... 52
	9. Image measures and distributions ... 54
	10. Generating classes  of sets ... 56
	11. Generating classes of functions ... 58
	12. Problems ... 60
Chapter 3 Densities and derivatives ... 68
	1. Densities and absolute continuity ... 68
	2. The Lebesgue  decomposition ... 73
	3. Distances and affinities between  measures ... 74
	4. The classical concept of absolute continuity ... 80
	5. Vitali covering lemma ... 83
	6. Densities as  almost sure  derivatives ... 85
	7. Problems ... 86
Chapter 4 Product spaces and independence ... 92
	1. Independence ... 92
	2. Independence of sigma-fields ... 95
	3. Construction of measures on a product  space ... 98
	4. Product measures ... 103
	5. Beyond sigma-fi niteness ... 108
	6. SLLN via blocking ... 110
	7. SLLN for identically distributed summands ... 112
	8. Infinite product  spaces ... 114
	9. Problems ... 117
	10. Notes ... 123
Chapter 5 Conditioning ... 126
	1. Conditional distributions: the elementary case ... 126
	2. Conditional distributions: the general case ... 128
	3. Integration and disintegration ... 131
	4. Conditional densities ... 133
	5. Invariance ... 136
	6. Kolmogorov\'s abstract conditional expectation ... 138
	7. Sufficiency ... 143
	8. Problems ... 146
	9. Notes ... 150
Chapter 6 Martingale et al. ... 153
	1. What are they? ... 153
	2. Stopping times ... 157
	3. Convergence of positive supermartingales ... 162
	4. Convergence of submartingales ... 166
	5. Proof of the Krickeberg  decomposition ... 167
	6. Uniform integrability ... 168
	7. Reversed martingales ... 170
	8. Symmetry and exchangeability ... 174
	9. Problems ... 177
	10. Notes ... 181
Chapter 7 Convergence in distribution ... 184
	1. Definition and  consequences ... 184
	2. Lindeberg\'s method  for the  central limit theorem ... 191
	3. Multivariate limit theorems ... 196
	4. Stochastic order symbols ... 197
	5. Weakly convergent  subsequences ... 199
	6. Problems ... 201
Chapter 8 Fourier transforms ... 208
	1. Definitions and basic  properties ... 208
	2. Inversion formula ... 210
	3. A mystery? ... 213
	4. Convergence in distribution ... 213
	5. A martingale central limit theorem ... 215
	6. Multivariate Fourier transforms ... 217
	7. Cramer-Wold without Fourier transforms ... 218
	8. The Levy-Cramer theorem ... 220
	9. Problems ... 221
Chapter 9 Brownian motion ... 226
	1. Prerequisites ... 226
	2. Brownian motion and Wiener measure ... 228
	3. Existence of Brownian motion ... 230
	4. Finer properties of sample paths ... 232
	5. Strong Markov property ... 234
	6. Martingale characterizations  of Brownian motion ... 237
	7. Functionals of Brownian motion ... 241
	8. Option pricing ... 243
	9. Problems ... 245
Chapter 10 Representations and couplings ... 252
	1. What is  coupling? ... 252
	2. Almost sure representations ... 254
	3. Strassen\'s Theorem ... 257
	4. The Yurinskii  coupling ... 259
	5. Quantile coupling of Binomial with normal ... 263
	6. Haar coupling-the Hungarian  construction ... 264
	7. The Komlos-Major-Tusnady coupling ... 267
	8. Problems ... 271
Chapter 11 Exponential tails and the law of the iterated logarithm ... 276
	1. LIL for normal summands ... 276
	2. LIL for bounded summands ... 279
	3. Kolmogorov\'s exponential lower bound ... 281
	4. Identically distributed summands ... 283
	5. Problems ... 286
	6. Notes ... 287
Chapter 12 Multivariate normal distributions ... 289
	1. Introduction ... 289
	2. Fernique\'s inequality ... 290
	3. Proof of Fernique\'s inequality ... 291
	4. Gaussian isoperimetric inequality ... 293
	5. Proof of the isoperimetric inequality ... 295
	6. Problems ... 300
	7. Notes ... 302
Appendix A Measures and integrals ... 304
	1. Measures and  inner measure ... 304
	2. Tightness ... 306
	3. Countable additivity ... 307
	4. Extension to the nc-closure ... 309
	5. Lebesgue measure ... 310
	6. Integral representations ... 311
	7. Problems ... 315
	8. Notes ... 315
Appendix B Hilbert spaces ... 316
	1. Definitions ... 316
	2. Orthogonal projections ... 317
	3. Orthonormal bases ... 318
	4. Series expansions of random processes ... 320
	5. Problems ... 321
	6. Notes ... 321
Appendix C Convexity ... 322
	1. Convex sets and functions ... 322
	2. One-sided derivatives ... 323
	3. Integral representations ... 325
	4. Relative interior of a convex set ... 327
	5. Separation of convex sets by linear functionals ... 328
	6. Problems ... 330
	7. Notes ... 331
Appendix D Binomial and normal distributions ... 332
Appendix E Martingales in continuous time ... 344
	1. Filtrations, sample paths, and stopping times ... 344
	2. Preservation of martingale properties at stopping times ... 347
	3. Supermartingales from  their rational skeletons ... 349
	4. The Brownian filtration ... 351
	5. Problems ... 353
	6. Notes ... 353
Appendix F Disintegration of measures ... 354
	1. Representation of measures on product spaces ... 354
	2. Disintegrations with respect to a  measurable map ... 357
	3. Problems ... 358
	4. Notes ... 360
Index ... 362