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| Preface | |
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| Motivation | |
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| Why bother with measure theory? | |
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| The cost and benefit of rigor | |
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| Where to start: probabilities or expectations? | |
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| The de Finetti notation | |
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| Fair prices | |
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| Problems | |
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| Notes | |
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| A modicum of measure theory | |
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| Measures and sigma-fields | |
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| Measurable functions | |
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| Integrals | |
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| Construction of integrals from measures | |
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| Limit theorems | |
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| Negligible sets | |
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| L[superscript p] spaces | |
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| Uniform integrability | |
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| Image measures and distributions | |
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| Generating classes of sets | |
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| Generating classes of functions | |
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| Problems | |
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| Notes | |
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| Densities and derivatives | |
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| Densities and absolute continuity | |
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| The Lebesgue decomposition | |
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| Distances and affinities between measures | |
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| The classical concept of absolute continuity | |
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| Vitali covering lemma | |
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| Densities as almost sure derivatives | |
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| Problems | |
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| Notes | |
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| Product spaces and independence | |
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| Independence | |
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| Independence of sigma-fields | |
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| Construction of measures on a product space | |
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| Product measures | |
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| Beyond sigma-finiteness | |
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| SLLN via blocking | |
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| SLLN for identically distributed summands | |
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| Infinite product spaces | |
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| Problems | |
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| Notes | |
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| Conditioning | |
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| Conditional distributions: the elementary case | |
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| Conditional distributions: the general case | |
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| Integration and disintegration | |
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| Conditional densities | |
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| Invariance | |
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| Kolgomorov's abstract conditional expectation | |
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| Sufficiency | |
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| Problems | |
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| Notes | |
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| Martingale et al. | |
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| What are they? | |
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| Stopping times | |
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| Convergence of positive supermartingales | |
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| Convergence of submartingales | |
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| Proof of the Krickeberg decomposition | |
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| Uniform integrability | |
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| Reversed martingales | |
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| Symmetry and exchangeability | |
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| Problems | |
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| Notes | |
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| Convergence in distribution | |
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| Definition and consequences | |
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| Lindeberg's method for the central limit theorem | |
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| Multivariate limit theorems | |
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| Stochastic order symbols | |
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| Weakly convergent subsequences | |
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| Problems | |
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| Notes | |
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| Fourier transforms | |
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| Definitions and basic properties | |
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| Inversion formula | |
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| A mystery? | |
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| Convergence in distribution | |
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| A martingale central limit theorem | |
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| Multivariate Fourier transforms | |
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| Cramer-Wold without Fourier transforms | |
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| The Levy-Cramer theorem | |
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| Problems | |
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| Notes | |
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| Brownian motion | |
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| Prerequisites | |
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| Brownian motion and Wiener measure | |
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| Existence of Brownian motion | |
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| Finer properties of sample paths | |
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| Strong Markov property | |
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| Martingale characterizations of Brownian motion | |
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| Functionals of Brownian motion | |
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| Option pricing | |
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| Problems | |
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| Notes | |
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| Representations and couplings | |
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| What is coupling? | |
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| Almost sure representations | |
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| Strassen's Theorem | |
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| The Yurinskii coupling | |
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| Quantile coupling of Binomial with normal | |
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| Haar coupling--the Hungarian construction | |
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| The Komlos-Major-Tusnady coupling | |
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| Problems | |
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| Notes | |
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| Exponential tails and the law of the iterated logarithm | |
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| LIL for normal summands | |
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| LIL for bounded summands | |
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| Kolmogorov's exponential lower bound | |
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| Identically distributed summands | |
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| Problems | |
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| Notes | |
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| Multivariate normal distributions | |
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| Introduction | |
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| Fernique's inequality | |
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| Proof of Fernique's inequality | |
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| Gaussian isoperimetric inequality | |
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| Proof of the isoperimetric inequality | |
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| Problems | |
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| Notes | |
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| Measures and integrals | |
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| Measures and inner measure | |
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| Tightness | |
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| Countable additivity | |
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| Extension to the [intersection]c-closure | |
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| Lebesgue measure | |
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| Integral representations | |
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| Problems | |
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| Notes | |
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| Hilbert spaces | |
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| Definitions | |
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| Orthogonal projections | |
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| Orthonormal bases | |
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| Series expansions of random processes | |
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| Problems | |
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| Notes | |
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| Convexity | |
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| Convex sets and functions | |
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| One-sided derivatives | |
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| Integral representations | |
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| Relative interior of a convex set | |
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| Separation of convex sets by linear functionals | |
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| Problems | |
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| Notes | |
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| Binomial and normal distributions | |
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| Tails of the normal distributions | |
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| Quantile coupling of Binomial with normal | |
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| Proof of the approximation theorem | |
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| Notes | |
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| Martingales in continuous time | |
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| Filtrations, sample paths, and stopping times | |
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| Preservation of martingale properties at stopping times | |
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| Supermartingales from their rational skeletons | |
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| The Brownian filtration | |
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| Problems | |
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| Notes | |
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| Disintegration of measures | |
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| Representation of measures on product spaces | |
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| Disintegrations with respect to a measurable map | |
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| Problems | |
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| Notes | |
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| Index | |