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**Measure theory and probability theory.**
*(English)*
Zbl 1125.60001

Springer Texts in Statistics. New York, NY: Springer (ISBN 0-387-32903-X/hbk; 978-1-4419-2191-8/pbk; 978-0-387-35434-7/ebook). xviii, 618 p. (2006).

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).

The first two chapters contain the classical basic theory of measure and integration. In particular the Caratheodory extension procedure and the Lebesgue type limit theorems.

Chapter 3 is devoted to the basic theory of Banach spaces (mainly \(L_p\) and Hilbert spaces). In particular, the Riesz representation theorem is proved.

The Lebesgue-Radon-Nikodym theorem, Lebesgue and Jordan decompositions and functions of bounded variation on the real line are the main topics of chapter 4.

Chapter 5 starts with the product measure and the Fubini theorem. Then the authors describe the convolution of measures on the real line and the transforms of Laplace, Fourier and Plancherel.

The actual probability theory begins with chapter 6, where the Kolmogorov consistency theorem and its application to the existence of stochastic processes are presented.

The notion of independence of random variables is carefully developed in chapter 7.

Weak and strong laws of large numbers are described in chapter 8 (including the Marcinkiewicz-Zygmund strong law of large numbers).

The weak convergence of probabilities on the real line and on metric spaces is treated in chapter 9.

In chapter 10 the characteristic function and probability of a random variable are introduced and the Levy-Cramer continuity theorem is proved. Also finite dimensional random variables are considered.

Chapter 11 is devoted to the central limit theorem and its extensions to stable and infinitely divisible probabilities.

Chapter 12 develops the theory of conditional expectations, introduced via projections in \(L^2\) rather, than by a direct application of the Lebesgue-Radon-Nikodym theorem.

Discrete parameter martingales are the main topic of chapter 13. Among several applications there are also martingale proofs of the Lebesgue decomposition theorem for measures and then the Kakutani theorem for probabilities on a countable product of the real lines.

Chapters 14-18 are devoted to more specialized topics. A comprehensive theory of Markov chains (including Markov chain Monte Carlo) is presented in chapter 14. The continuous time Markov chains and an elementary treatment of Brownian motion are presented in chapter 15.

The last three chapters present outlines of bootstrap theory, mixing processes and branching processes.

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. I would suggest first of all a one-two semester course on advanced probability theory for those with the necessary measure theory background.

The first two chapters contain the classical basic theory of measure and integration. In particular the Caratheodory extension procedure and the Lebesgue type limit theorems.

Chapter 3 is devoted to the basic theory of Banach spaces (mainly \(L_p\) and Hilbert spaces). In particular, the Riesz representation theorem is proved.

The Lebesgue-Radon-Nikodym theorem, Lebesgue and Jordan decompositions and functions of bounded variation on the real line are the main topics of chapter 4.

Chapter 5 starts with the product measure and the Fubini theorem. Then the authors describe the convolution of measures on the real line and the transforms of Laplace, Fourier and Plancherel.

The actual probability theory begins with chapter 6, where the Kolmogorov consistency theorem and its application to the existence of stochastic processes are presented.

The notion of independence of random variables is carefully developed in chapter 7.

Weak and strong laws of large numbers are described in chapter 8 (including the Marcinkiewicz-Zygmund strong law of large numbers).

The weak convergence of probabilities on the real line and on metric spaces is treated in chapter 9.

In chapter 10 the characteristic function and probability of a random variable are introduced and the Levy-Cramer continuity theorem is proved. Also finite dimensional random variables are considered.

Chapter 11 is devoted to the central limit theorem and its extensions to stable and infinitely divisible probabilities.

Chapter 12 develops the theory of conditional expectations, introduced via projections in \(L^2\) rather, than by a direct application of the Lebesgue-Radon-Nikodym theorem.

Discrete parameter martingales are the main topic of chapter 13. Among several applications there are also martingale proofs of the Lebesgue decomposition theorem for measures and then the Kakutani theorem for probabilities on a countable product of the real lines.

Chapters 14-18 are devoted to more specialized topics. A comprehensive theory of Markov chains (including Markov chain Monte Carlo) is presented in chapter 14. The continuous time Markov chains and an elementary treatment of Brownian motion are presented in chapter 15.

The last three chapters present outlines of bootstrap theory, mixing processes and branching processes.

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. I would suggest first of all a one-two semester course on advanced probability theory for those with the necessary measure theory background.

Reviewer: Kazimierz Musiał (Wrocław)