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Eric M. Vestrup’s book on measure theory and integration is an excellent read. It is carefully written and gives a rigorous, detailed treatment and a particularly clear presentation of the classical and some slightly less standard results in measure theory and integration. The book starts with a thorough introduction to set systems, measures, extensions of measures, Lebesgue measures, and measurable functions. It then continues to construct the Lebesgue integral, integrals relative to Lebesgue measure, and finally introduces L spaces, the Radon–Nikodym theorem, and products of measure spaces. Among the numerous additions to this standard program are results on the structure of Borel sets, Lebesgue sets, and the Hardy–Littlewood theorems. Of course, there is a host of excellent books already available that cover a very similar scope: Halmos (1974), Rao (1987), and Billingsley (1995) to name just a few. Nonetheless, I believe that Vestrup’s book is a valuable addition to the field and will be a prized possession for many researchers and graduate students. The author’s own hope is to provide a more modern and expository version of Halmos’ classic text, which has roots in the 1950s and thus has a somewhat different approach and choice of topics than one might look for today. In my opinion, it is a quite felicitous attempt at that. The book is a bit less general and less abstract but more detailed than the more recent text by Rao (1987), since it is intended to be accessible to graduate students just beginning to learn measure and integration theory. And finally, unlike Billingsley (1995) and a number of other classics, it focuses just on measure and integration, and does not solely put this topic in a probabilistic context, although of course, many examples, exercises, and applications are drawn from probability. So who might want to read this book? The author suggests to use it as a textbook or background reading for a semester or even year-long graduate level course on the topic addressed to students of mathematics, statistics, or even physics. In my opinion, it would be a good if ambitious choice as a textbook; it will certainly make for great background reading. The first-timer and even the more advanced researcher will no doubt appreciate that the material is presented in much depth and detail. No steps in the proofs are skipped or claimed to be trivial; everything is laid out for the reader in a complete, precise, and patient way. Each section comes with a wealth of exercises ranging from easy “finger exercises” to much more challenging topical excursions. Whereas no explicit solutions are given to exercises, they are accompanied by many comments that put them into context, as well as by quite detailed hints and proof outlines for many of the more difficult problems. All of this makes the book well suited for anybody wanting to learn measure and integration theory via self-study provided that he or she is highly motivated and interested to learn the subject in some depth. This is because some beginners with no other guidance might feel a bit overwhelmed with the amount of information and detail provided. The book gives some suggestions of which sections are less essential and may be skipped. Even if one skipped all of those sections, it would still be a long book. To give a sense of this, I will just say that the book has close to 600 pages which are densely packed with small (and for exercises and examples even smaller) print. This may not impede its usefulness as a textbook since lectures can provide the outline and the big picture and must necessarily leave out detail that Vestrup will then provide in impeccable fashion. Any graduate student or researcher who has internalized the basic concepts already will be thankful for this book as an excellent and meticulous reference, and this may become the book’s main use ultimately. Having said that Vestrup’s book will make a valued reference book, it is unfortunate that the index is not very extensive. I should also mention that the layout could be improved. In general, the text should have been a bit better structured, with slightly more space between text units. I suspect that given the length of the text, saving page space has been a priority to the (slight) detriment of readability. Section numbers in the header would aid cross referencing, particularly since “claims” and theorems do not carry section numbers. There are also some occasional typos in the text. All of these relatively minor points of criticism may be addressed quite easily in a second edition. Overall, I enjoyed reading this book very much. I liked Vestrup’s intuitive explanations and nice, if brief, historical accounts and insights into why one path or approach is taken over another. I appreciated his frequent references to alternative terminology and notation, which is also commonly used. And most of all, I was impressed with the wealth of information and the amount of flawless detail.

Authors

Jun Yan

Journal

Journal of the American Statistical Association

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