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Measures, Integrals and Martingales

René L. Schilling
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

See also our review of the first edition. This is a good introduction to measure and integration, that covers the general theory but also manages to stick close to the the important special cases of Lebesgue measure on the real line and the plane.

The author says on p. xiv, “Unlike many textbooks this is not an introduction to integration for analysts or a probabilistic measure theory. I want to reach both (future) analysts and (future) probabilists, and to provide a foundation which will be useful for both communities and for further, more specialized, studies.” He also states (p. xiii) that the book is intended to be “within the grasp of second- or third-year undergraduates.” I think by these statements he means that he wants to teach the rudiments of measure and integration early, as soon as the student has enough calculus and real analysis to handle it, rather than doing an in-depth course at the graduate level. I think the book works well for this. I particular, it manages to avoid most of topology, and gives helpful hints such as that you can think of the underlying space as the real line if you don’t know about topology or metric spaces.

The first 40% of the book is a development of the theory of measure and integration, the middle 25% is special topics, and remaining 35% is about half martingales and half functional analysis and probability topics (that use martingales in their proofs). The main difference from the first edition is in the middle section, which has five new chapters. I especially like the middle section, which covers a wide variety of topics without requiring too much preparation, and should generate some excitement from seeing actual uses of this material. The chapter on the Fourier transform is especially charming; it manages to give a good cross-section of the material in 15 pages, and even dips into the Wiener algebra.

An unusual feature of this book is the inclusion of martingales. They are treated from an analytic viewpoint rather than a probability theory viewpoint. There’s a straightforward development of the theory, with lots of concrete examples. Then they are used to prove several classic theorems: the Radon–Nikodym theorem, the Hardy–Littlewood maximal theorem, Lebesgue’s differentiation theorem, and the Calderón–Zygmund decomposition. The chapters on functional analysis and on conditional probability are developed from scratch, and use martingales in some of the convergence proofs.

The book is well-produced, with a clear layout, a thorough index, and an especially-useful two-page List of Symbols, that gives both brief definitions and page references to the definitions. Each chapter has an extensive problem section, that covers a mix of concrete examples and supplemental theorems. The author’s web site,, has a set of errata and a complete 330-page solutions manual.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

List of symbols
Dependence chart
1. Prologue
2. The pleasures of counting
3. σ-algebras
4. Measures
5. Uniqueness of measures
6. Existence of measures
7. Measurable mappings
8. Measurable functions
9. Integration of positive functions
10. Integrals of measurable functions
11. Null sets and the 'almost everywhere'
12. Convergence theorems and their applications
13. The function spaces Lp
14. Product measures and Fubini's theorem
15. Integrals with respect to image measures
16. Jacobi's transformation theorem
17. Dense and determining sets
18. Hausdorff measure
19. The Fourier transform
20. The Radon–Nikodym theorem
21. Riesz representation theorems
22. Uniform integrability and Vitali's convergence theorem
23. Martingales
24. Martingale convergence theorems
25. Martingales in action
26. Abstract Hilbert spaces
27. Conditional expectations
28. Orthonormal systems and their convergence behaviour
Appendix A. Lim inf and lim sup
Appendix B. Some facts from topology
Appendix C. The volume of a parallelepiped
Appendix D. The integral of complex valued functions
Appendix E. Measurability of the continuity points of a function
Appendix F. Vitali's covering theorem
Appendix G. Non-measurable sets
Appendix H. Regularity of measures
Appendix I. A summary of the Riemann integral