| Preface | p. xi |
| Motivation | |
| Why bother with measure theory? | p. 1 |
| The cost and benefit of rigor | p. 3 |
| Where to start: probabilities or expectations? | p. 5 |
| The de Finetti notation | p. 7 |
| Fair prices | p. 11 |
| Problems | p. 13 |
| Notes | p. 14 |
| A modicum of measure theory | |
| Measures and sigma-fields | p. 17 |
| Measurable functions | p. 22 |
| Integrals | p. 26 |
| Construction of integrals from measures | p. 29 |
| Limit theorems | p. 31 |
| Negligible sets | p. 33 |
| L[superscript p] spaces | p. 36 |
| Uniform integrability | p. 37 |
| Image measures and distributions | p. 39 |
| Generating classes of sets | p. 41 |
| Generating classes of functions | p. 43 |
| Problems | p. 45 |
| Notes | p. 51 |
| Densities and derivatives | |
| Densities and absolute continuity | p. 53 |
| The Lebesgue decomposition | p. 58 |
| Distances and affinities between measures | p. 59 |
| The classical concept of absolute continuity | p. 65 |
| Vitali covering lemma | p. 68 |
| Densities as almost sure derivatives | p. 70 |
| Problems | p. 71 |
| Notes | p. 75 |
| Product spaces and independence | |
| Independence | p. 77 |
| Independence of sigma-fields | p. 80 |
| Construction of measures on a product space | p. 83 |
| Product measures | p. 88 |
| Beyond sigma-finiteness | p. 93 |
| SLLN via blocking | p. 95 |
| SLLN for identically distributed summands | p. 97 |
| Infinite product spaces | p. 99 |
| Problems | p. 102 |
| Notes | p. 108 |
| Conditioning | |
| Conditional distributions: the elementary case | p. 111 |
| Conditional distributions: the general case | p. 113 |
| Integration and disintegration | p. 116 |
| Conditional densities | p. 118 |
| Invariance | p. 121 |
| Kolgomorov's abstract conditional expectation | p. 123 |
| Sufficiency | p. 128 |
| Problems | p. 131 |
| Notes | p. 135 |
| Martingale et al. | |
| What are they? | p. 138 |
| Stopping times | p. 142 |
| Convergence of positive supermartingales | p. 147 |
| Convergence of submartingales | p. 151 |
| Proof of the Krickeberg decomposition | p. 152 |
| Uniform integrability | p. 153 |
| Reversed martingales | p. 155 |
| Symmetry and exchangeability | p. 159 |
| Problems | p. 162 |
| Notes | p. 166 |
| Convergence in distribution | |
| Definition and consequences | p. 169 |
| Lindeberg's method for the central limit theorem | p. 176 |
| Multivariate limit theorems | p. 181 |
| Stochastic order symbols | p. 182 |
| Weakly convergent subsequences | p. 184 |
| Problems | p. 186 |
| Notes | p. 190 |
| Fourier transforms | |
| Definitions and basic properties | p. 193 |
| Inversion formula | p. 195 |
| A mystery? | p. 198 |
| Convergence in distribution | p. 198 |
| A martingale central limit theorem | p. 200 |
| Multivariate Fourier transforms | p. 202 |
| Cramer-Wold without Fourier transforms | p. 203 |
| The Levy-Cramer theorem | p. 205 |
| Problems | p. 206 |
| Notes | p. 208 |
| Brownian motion | |
| Prerequisites | p. 211 |
| Brownian motion and Wiener measure | p. 213 |
| Existence of Brownian motion | p. 215 |
| Finer properties of sample paths | p. 217 |
| Strong Markov property | p. 219 |
| Martingale characterizations of Brownian motion | p. 222 |
| Functionals of Brownian motion | p. 226 |
| Option pricing | p. 228 |
| Problems | p. 230 |
| Notes | p. 234 |
| Representations and couplings | |
| What is coupling? | p. 237 |
| Almost sure representations | p. 239 |
| Strassen's Theorem | p. 242 |
| The Yurinskii coupling | p. 244 |
| Quantile coupling of Binomial with normal | p. 248 |
| Haar coupling--the Hungarian construction | p. 249 |
| The Komlos-Major-Tusnady coupling | p. 252 |
| Problems | p. 256 |
| Notes | p. 258 |
| Exponential tails and the law of the iterated logarithm | |
| LIL for normal summands | p. 261 |
| LIL for bounded summands | p. 264 |
| Kolmogorov's exponential lower bound | p. 266 |
| Identically distributed summands | p. 268 |
| Problems | p. 271 |
| Notes | p. 272 |
| Multivariate normal distributions | |
| Introduction | p. 274 |
| Fernique's inequality | p. 275 |
| Proof of Fernique's inequality | p. 276 |
| Gaussian isoperimetric inequality | p. 278 |
| Proof of the isoperimetric inequality | p. 280 |
| Problems | p. 285 |
| Notes | p. 287 |
| Measures and integrals | |
| Measures and inner measure | p. 289 |
| Tightness | p. 291 |
| Countable additivity | p. 292 |
| Extension to the [intersection]c-closure | p. 294 |
| Lebesgue measure | p. 295 |
| Integral representations | p. 296 |
| Problems | p. 300 |
| Notes | p. 300 |
| Hilbert spaces | |
| Definitions | p. 301 |
| Orthogonal projections | p. 302 |
| Orthonormal bases | p. 303 |
| Series expansions of random processes | p. 305 |
| Problems | p. 306 |
| Notes | p. 306 |
| Convexity | |
| Convex sets and functions | p. 307 |
| One-sided derivatives | p. 308 |
| Integral representations | p. 310 |
| Relative interior of a convex set | p. 312 |
| Separation of convex sets by linear functionals | p. 313 |
| Problems | p. 315 |
| Notes | p. 316 |
| Binomial and normal distributions | |
| Tails of the normal distributions | p. 317 |
| Quantile coupling of Binomial with normal | p. 320 |
| Proof of the approximation theorem | p. 324 |
| Notes | p. 328 |
| Martingales in continuous time | |
| Filtrations, sample paths, and stopping times | p. 329 |
| Preservation of martingale properties at stopping times | p. 332 |
| Supermartingales from their rational skeletons | p. 334 |
| The Brownian filtration | p. 336 |
| Problems | p. 338 |
| Notes | p. 338 |
| Disintegration of measures | |
| Representation of measures on product spaces | p. 339 |
| Disintegrations with respect to a measurable map | p. 342 |
| Problems | p. 343 |
| Notes | p. 345 |
| Index | p. 347 |
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