Real analysis and probability.

*(English)*Zbl 0686.60001
Wadsworth & Brooks/Cole Mathematics Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. xi, 436 p. $ 52.95 (1989).

This book gives an exposition of the main parts of real analysis which may become important at some stage for probabilists. Probability itself does not cover a large proportion of the text even though central limit theorem and laws of large numbers are included. This monograph is comparable in spirit to the textbooks of H. Bauer [Probability theory and elements of measure theory. (1981; Zbl 0466.60001)] and J. Neveu [Bases mathématiques du calcul des probabilités. (1964; Zbl 0137.112)] but it contains much more material than either of them and is more ambitious in providing background information. In the reviewers opinion it will serve for a long time as a standard reference for the teacher of measure theoretic probability.

The different sections of the text provide excellent short introductions to the respective areas. Whether or not a modern student will be patient enough to learn all these tools before coming to the ‘real thing’, viz. concrete and computable applications in probability and statistics or in classical analysis, may depend on the circumstances.

In addition to the main text there are many exercises and interesting historical notes. In the search of the origin of the ideas the author has traced quite a few pioneering articles which are not widely acknowledged. Nevertheless the author is always looking for brevity. So really a very large area of subjects is covered. A couple of classical results are given less known short proofs. (In the reviewers view, sometimes the standard arguments would have been clearer, e.g. in the completion of metric spaces or in the strong law of large numbers. Also the brevity may be misleading, e.g. in the statement of the ergodicity of the shift it is not clear that the underlying measure is a product, and in the proof of theorem 13.2.7 the function g needs to be composed with a projection.)

Here are some important items which seem to appear for the first time in a general textbook: The Brunn-Minkowski-inequality, the subadditive ergodic theorem, the Kantorovich-Rubinstein theorem. (It is not surprising that the chapter on convergence of laws is particularly enlightening in a book by this author.)

The chapters are: 1. Foundations; Set theory. 2. General topology. 3. Measures. 4. Integration. 5. \(L^ p\)-spaces; Introduction to functional analysis. 6. Convex sets and duality of normed spaces. 7. Measure, topology, and differentation. 8. Introduction to probability. 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. Five Appendices (beginning with A: Axiomatic set theory).

The different sections of the text provide excellent short introductions to the respective areas. Whether or not a modern student will be patient enough to learn all these tools before coming to the ‘real thing’, viz. concrete and computable applications in probability and statistics or in classical analysis, may depend on the circumstances.

In addition to the main text there are many exercises and interesting historical notes. In the search of the origin of the ideas the author has traced quite a few pioneering articles which are not widely acknowledged. Nevertheless the author is always looking for brevity. So really a very large area of subjects is covered. A couple of classical results are given less known short proofs. (In the reviewers view, sometimes the standard arguments would have been clearer, e.g. in the completion of metric spaces or in the strong law of large numbers. Also the brevity may be misleading, e.g. in the statement of the ergodicity of the shift it is not clear that the underlying measure is a product, and in the proof of theorem 13.2.7 the function g needs to be composed with a projection.)

Here are some important items which seem to appear for the first time in a general textbook: The Brunn-Minkowski-inequality, the subadditive ergodic theorem, the Kantorovich-Rubinstein theorem. (It is not surprising that the chapter on convergence of laws is particularly enlightening in a book by this author.)

The chapters are: 1. Foundations; Set theory. 2. General topology. 3. Measures. 4. Integration. 5. \(L^ p\)-spaces; Introduction to functional analysis. 6. Convex sets and duality of normed spaces. 7. Measure, topology, and differentation. 8. Introduction to probability. 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. Five Appendices (beginning with A: Axiomatic set theory).

Reviewer: H.v.Weizsäcker