The Theory of Measures and Integration
By addebook • Sep 8th, 2008 • Category: Mathematics •
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The Theory of Measures and Integration (Wiley Series in Probability and Statistics)
The Theory of Measures and Integration (Wiley Series in Probability and Statistics)
By Eric M. Vestrup
Publisher: Wiley-Interscience
Number Of Pages: 594
Publication Date: 2003-09-18
ISBN-10 / ASIN: 0471249777
ISBN-13 / EAN: 9780471249771
Binding: Hardcover
An accessible, clearly organized survey of the basic topics of measure theory for students and researchers in mathematics, statistics, and physics
In order to fully understand and appreciate advanced probability, analysis, and advanced mathematical statistics, a rudimentary knowledge of measure theory and like subjects must first be obtained. The Theory of Measures and Integration illuminates the fundamental ideas of the subject-fascinating in their own right-for both students and researchers, providing a useful theoretical background as well as a solid foundation for further inquiry.
Eric Vestrup’s patient and measured text presents the major results of classical measure and integration theory in a clear and rigorous fashion. Besides offering the mainstream fare, the author also offers detailed discussions of extensions, the structure of Borel and Lebesgue sets, set-theoretic considerations, the Riesz representation theorem, and the Hardy-Littlewood theorem, among other topics, employing a clear presentation style that is both evenly paced and user-friendly. Chapters include:
* Measurable Functions
* The Lp Spaces
* The Radon-Nikodym Theorem
* Products of Two Measure Spaces
* Arbitrary Products of Measure Spaces
Sections conclude with exercises that range in difficulty between easy “finger exercises”and substantial and independent points of interest. These more difficult exercises are accompanied by detailed hints and outlines. They demonstrate optional side paths in the subject as well as alternative ways of presenting the mainstream topics.
In writing his proofs and notation, Vestrup targets the person who wants all of the details shown up front. Ideal for graduate students in mathematics, statistics, and physics, as well as strong undergraduates in these disciplines and practicing researchers, The Theory of Measures and Integration proves both an able primary text for a real analysis sequence with a focus on measure theory and a helpful background text for advanced courses in probability and statistics.
Summary: The New Standard for Measure Theory Books
Rating: 5
This is a fantastic book on measure theory. The focus is on measure theory on its own right and not on probability. I was lucky to come across this book while canvassing the measure theory books at our library. I looked at the books by Billingsley, Halmos, Chung, Resnick, Rao, Rudin, Pollard, Dudley, Nielson, Stroock, Williams, Pitt, and many others. Hand-down, Vestrup is the best.
I believe after scrutinizing so many books, I have a very good baseline to judge Vestrup’s work. Here are a few specific reasons:
(1) If you don’t like detail and revel in banging your head against the walls to figure out the skipped details in Billingsley, this is not the book for you. But If you are a first timer to measure theory, this is as good as it will get; All the major results of measure theory are presented in detailed and clear manner with few skipped details and few not-so-obvious “it is obvious” remarks.
(2) Vestrup has a lot of exercises with lots of helpful hints. Some problems at first appear to be long and intimidating till you look closely and discover that Vestrup leads you through the problems with his hints.
(3) Certain topics central to understanding of measure theory were given cursory coverage by most of the books mentioned above. Not Vestrup. For example, Vestrup devotes a whole chapter to extensions. This is just one example of many central ideas Vestrup develops meticulously and painstakingly.
This book is fairly new and I think its popularity will grow as more students and professionals discover it. I suppose the only criticism I have is that the typesetting can be improved (second edition maybe?)
There are a few other good books (Ash, Bartle, and Royden) that are out there that you may consider but again Vestrup trumps them all. Whatever you decide on, I strongly warn against using Billingsley.
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