A user’s guide to measure theoretic probability.

*(English)*Zbl 0992.60001
Cambridge Series in Statistical and Probabilistic Mathematics. 8. Cambridge: Cambridge University Press. xiv, 351 p. (2002).

The book is based on lectures given to students at the advanced undergraduate, introductory graduate level, where the students have an inhomogeneous mathematical background and different intentions of study. The book is divided into twelve chapters and has six appendices. The chapter headings are: 1. Motivation, 2. A modicum of measure theory, 3. Densities and derivatives, 4. Product spaces and independence, 5. Conditioning, 6. Martingale et al., 7. Convergence in distribution, 8. Fourier transforms, 9. Brownian motion, 10. Representation and couplings, 11. Exponential tails and the law of the iterated logarithm, 12. Multivariate normal distributions; Appendices: A. Measures and integrals, B. Hilbert spaces, C. Convexity, D. Binomial and normal distribution, E. Martingales in continuous time, F. Disintegration of measures.

The first chapter provides some motivation (the first section is called “Why bother with measure theory?”) and explains the virtues of the use of linear functional notation for expectations (de Finetti notation). The same symbol for a set and its indicator function is used, and, as \(P(A) = E(1_A)\), consequently the same symbol, i.e. \(P\), is used for probability measure and expectation. According to the author this represents a departure from tradition. In other words this may be a matter of personal taste. All chapters (except Appendix D) contain problems (between 2 and 31). Each chapter ends with notes concerning the origin of the material and the corresponding references. The text contains illustrating examples and remarks and the style of exposition is explanatory. The historical development of ideas and the interrelations of concepts are also discussed. Summarising the book is very enjoyable and provides a careful well-motivated presentation of the amount of material covered.

The first chapter provides some motivation (the first section is called “Why bother with measure theory?”) and explains the virtues of the use of linear functional notation for expectations (de Finetti notation). The same symbol for a set and its indicator function is used, and, as \(P(A) = E(1_A)\), consequently the same symbol, i.e. \(P\), is used for probability measure and expectation. According to the author this represents a departure from tradition. In other words this may be a matter of personal taste. All chapters (except Appendix D) contain problems (between 2 and 31). Each chapter ends with notes concerning the origin of the material and the corresponding references. The text contains illustrating examples and remarks and the style of exposition is explanatory. The historical development of ideas and the interrelations of concepts are also discussed. Summarising the book is very enjoyable and provides a careful well-motivated presentation of the amount of material covered.

Reviewer: Evelyn Buckwar (Berlin)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60Axx | Foundations of probability theory |

60Fxx | Limit theorems in probability theory |

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |