Measure and Probability

Description

This book covers the fundamentals of measure theory and probability theory. It begins with the construction of Lebesgue measure via Caratheodory’s outer measure approach and goes on to discuss integration and standard convergence theorems and contains an entire chapter devoted to complex measures, Lp spaces, Radon–Nikodym theorem, and the Riesz representation theorem. It presents the elements of probability theory, the law of large numbers, and central limit theorem. The book then discusses discrete time Markov chains, stationary distributions and limit theorems. The appendix covers many basic topics such as metric spaces, topological spaces and the Stone–Weierstrass theorem. Probabilities and Measures Introduction σ-algebras as events Algebras, monotone classes, etc. Preliminaries on measures Outer measures and Caratheodory extension Lebesgue measure Regularity Bernoulli trials Integration Measurable functions Integration a.e. considerations Random Variables Distribution and expectation Independent events and tail σ-algebra Some distributions Conditional expectation Probability Measures on Product Spaces Product measures Joint distribution and independence Probability measures on infinite product spaces Kolmogorov consistency theorem Characteristics and Convergences Characteristic functions Modes of convergence Central limit theorem Law of large numbers Markov Chains Discrete time MC Examples Classification of states Strong Markov property Stationary distribution Limit theorems Some Analysis Complex measures Lp spaces Radon–Nikodym theorem Change of variables Differentiation The Riesz representation theorem Appendix Metric spaces Topological spaces Compactness The Stone–Weierstrass theorem Tables References Index