Measure Theory and Probability Theory

• Measures and Integration: An Informal Introduction.- Measures.- Integration.- Lp-Spaces.- Differentiation.- Product Measures, Convolutions, and Transforms.- Probability Spaces.- Independence.- Laws of Large Numbers.- Convergence in Distribution.- Characteristic Functions.- Central Limit Theorems.- Conditional Expectation and Conditional Probability.- Discrete Parameter Martingales.- Markov Chains and MCMC.- Stochastic Processes.- Limit Theorems for Dependent Processes.- The Bootstrap.- Branching Processes.

From the reviews: "...There are interesting and non-standard topics that are not usually included in a first course in measture-theoretic probability including Markov Chains and MCMC, the bootstrap, limit theorems for martingales and mixing sequences, Brownian motion and Markov processes. The material is well-suported with many end-of-chapter problems." D.L. McLeish for Short Book Reviews of the ISI, December 2006 "The reader sees not only how measure theory is used to develop probability theory, but also how probability theory is used in applications. ... The discourse is delivered in a theorem proof format and thus is better suited for classroom ... . The authors prose is generally well thought out ... . will make an attractive choice for a two-semester course on measure and probability, or as a second course for students with a semester of measure or probability theory under their belt." (Peter C. Kiessler, Journal of the American Statistical Association, Vol. 102 (479), 2007) "The book is a well written self-contained textbook on measure and probability theory. It consists of 18 chapters. Every chapter contains many well chosen examples and ends with several problems related to the earlier developed theory (some with hints). ... At the very end of the book there is an appendix collecting necessary facts from set theory, calculus and metric spaces. The authors suggest a few possibilities on how to use their book." (Kazimierz Musial, Zentralblatt MATH, Vol. 1125 (2), 2008) "The title of the book consists of the names of its two basic parts. The book's third part is comprised of some special topics from probability theory. ... The authors suggest using the book in two-semester graduate programs in statistics or a one-semester seminar on special topics. The material of the book is standard ... is clear, comprehensive and 'without being intimidating'." (Rimas Norvaisa, Mathematical Reviews, Issue 2007 f) "Probabilists have a special relationship to measure theory. ... The style of writing is clear and precise ... . Its wide range of topics and results makes Measure Theory and Probability Theory not only a splendid textbook but also a nice addition to any probabilist's reference library. ... a researcher in need of a reference work, or just somebody who wants to learn some measure theory to lighten up your life, Measure Theory and Probability Theory is an excellent text that I highly recommend." (Peter Olofsson, SIAM Review, Vol. 49 (3), 2007)