Written 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.
Table of Contents
1. Foundations; Set Theory 2. General Topology 3. Measures 4. Integration 5. Lp Spaces; Introduction to Functional Analysis 6. Convex Sets and Duality of Normed Spaces 7. Measure, Topology, and Differentiation 8. Introduction to Probability Theory 9. Convergence of Laws and Central Limit Theorems 10. Conditional Expectations and Martingales 11. Convergence of Laws on Separable Metric Spaces 12. Stochastic Processes 13. Measurability: Borel Isomorphism and Analytic Sets