Real Analysis and ProbabilityWritten by one of the best-known probabilists in the world this text offers a clear and modern presentation of modern probability theory and an exposition of the interplay between the properties of metric spaces and those of probability measures. This text is the first at this level to include discussions of the subadditive ergodic theorems, metrics for convergence in laws and the Borel isomorphism theory. The proofs for the theorems are consistently brief and clear and each chapter concludes with a set of historical notes and references. This book should be of interest to students taking degree courses in real analysis and/or probability theory. |
Contents
| 1 | |
CHAPTER 2 General Topology | 19 |
CHAPTER 3 Measures | 63 |
CHAPTER 4 Integration | 86 |
CHAPTER 5 LsupP Spaces Introduction to Functional Analysis | 116 |
CHAPTER 6 Convex Sets and Duality of Normed Spaces | 145 |
CHAPTER 7 Measure Topology and Differentiation | 173 |
CHAPTER 8 Introduction to Probability Theory | 195 |
CHAPTER 10 Conditional Expectations and Martingales | 264 |
CHAPTER 11 Convergence of Laws on Separable Metric Spaces | 302 |
CHAPTER 12 Stochastic Processes | 346 |
Borel Isomorphism and Analytic Sets | 383 |
APPENDIXES | 396 |
Author Index | 428 |
| 431 | |
CHAPTER 9 Convergence of Laws and Central Limit Theorems | 221 |
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Common terms and phrases
algebra apply assume axiom Banach space base Borel set bounded called characteristic function clearly collection compact complete conditional consider constant containing convergence convex countable countably additive defined definition dense disjoint distribution function element equal equicontinuous equivalent example exists extended fact Find finite follows function f given gives holds implies included independent inequality infinite integrable intersection interval least Lebesgue measure Lemma limit linear martingale Math mean measurable function measure space nonnegative norm normal Note o-algebra open sets ordered probability probability space Problem PROOF properties Proposition proved random variables range real-valued Recall regular relation respect separable metric space sequence Show shown smallest stopping subsets Suppose Theorem theory topological space topology uniform uniformly union unique usual values vector