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Measures, Integrals and Martingales

René L. Schilling
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Ita Cirovic Donev
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The first thing that comes to mind when considering measure theory, at least from a student's perspective, is that it is "extremely hard." Many books published in this field really do not help students to think otherwise. Measures, Integral and Martingales can potentially change this bias.

So what is so different about this book? Well, first thing one would notice is the range of topics. The author truly covers a wide range of topics going from measure theory to integration and to the probabilistic point of view. Overall it gives a nice overview of the connections among several fields.

As the author does not assume any previous knowledge of measure theory, the book starts out very lightly, explaining the need to measure objects and the potential problems associated with such measurements. The book develops in a natural way, as it can be seen from the table of contents.

The presentation of the material is very detailed, with plenty of examples to support the theory. There are more examples as we proceed further into the book. What I like the most about the book is the presentation of proofs within chapters. Many books on measure theory do a lot of hand-waving when it comes to proofs. This one doesn't. Proofs are written in a very organized and detailed manner.

The most important part of the book are the exercises. There are really quite a number of them for each chapter. An interesting feature of the exercises is that they do not involve completing proofs left incomplete in the text. This is really a very different approach, which should be greatly appreciated by students, as it is frustrating — and, alas, common — for authors to use problems to deal with things that he/she should actually explain in the book. Schilling provides hints and solutions for the problems. Additional information on the book and solutions/hints to exercises are provided on author's website.

The prerequisites are the usual ones; namely, calculus and a serious undergraduate course in mathematical analysis in R. Overall, I believe this to be a great book for self-study as well as for course use. The book is ideal for future probabilists as well as statisticians, and can serve as a good introduction for mathematicians interested in measure theory.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She holds a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

 Prelude; Dependence chart; Prologue; 1. The pleasures of counting; 2. o-algebras; 3. Measures; 4. Uniqueness of measures; 5. Existance of measures; 6. Measurable mappings; 7. Measurable functions; 8. Integration of positive functions; 9. Integrals of measurable functions and null sets; 10. Convergence theroems and their applications; 11. The function spaces; 12. Product measures and Fubini’s theorem; 13. Integrals with respect to image measures; 14. Integrals of images and Jacobi’s transformation rule; 15. Uniform integrability and Vitali’s convergence theorem; 16. Martingales; 17. Martingale convergence theorems; 18. The Radon-Nikodym theorem and other applications of martingales; 19. Inner product spaces; 20. Hilbert space; 21. Conditional expectations in ; 22. Conditional expectations in ; 23. Orthonormal systems and their convergence behaviour; Appendix a: Lim inf and lim supp; Appendix b: Some facts from point-set topology; Appendix c: The volume of a parallelepiped; Appendix d: Non-measurable sets; Appendix e: A summary of the Riemann integral; Further reading; Bibliography; Notation index; Name and subject index.