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Title: Measure Theory (Mechanique Des Roches. Supplementum,) by P.R. Halmos ISBN: 0-387-90088-8 Publisher: Springer-Verlag Pub. Date: 01 June, 1974 Format: Hardcover Volumes: 1 List Price(USD): $54.95 |
Average Customer Rating: 4.5 (2 reviews)
Rating: 4
Summary: A classic in the field
Comment: This book is an overview of measure theory that is somewhat dated in terms of the presentation, but could still be read profitably by someone interested in studying the subject with greater generality than more modern texts. Measure theory has abundant applications, and has even gained importance in recent years in such areas as financial engineering. Those interested in the applications of measure theory to financial engineering should choose another book however, since this one does not even mention the word martingale. After a review of elementary topology and set theory in chapter 1, the author begins to define the elementary notions of measure theory in chapter 2. His approach is more general than other texts, since he works over a ring instead of an algebra. Measures on intervals of real numbers is given as an example. Measures and outer measures are defined, and it is shown how a measure induces an outer measure and how an outer measure induces a measure.
The next chapter explores more carefully the relation between measures and outer measures. It is also shown in this chapter to what extent a measure on a ring can be extended to the generated sigma-ring. The all-important Lebesgue measure is developed here also, and the author exhibits an example of a non-measurable set.
In order to develop an integration theory, one must first characterize the collection of measurable functions, and the author does this in chapter 4. The convergence properties of measurable functions are carefully outlined by the author.
The theory of integration begins in chapter 5, wherein the author follows the standard construction of an integral by first defining integrals over simple functions. Then in chapter 6, signed measures are defined, and the Lebesgue bounded convergence theorem is proven and the Hahn and Jordan decompositions of these measures are discussed. The all-important Radon-Nikodym theorem, which gives an integral representation of an absolutely continuous sigma-finite signed measure, is proven in detail.
One can of course take the Cartesian product of two measurable spaces, and the author shows how to define measures on these products in chapter 7, including infinite products. The physicist reader may want to pay attention to the section on infinite dimensional product measures, as it does have applications to functional integration in quantum field theory (although somewhat weakly).
The author treats measurable transformations in chapter 8, but interestingly, the word "ergodic" is never mentioned. He also introduces briefly the L-p spaces, so very important in many areas of mathematics, and proves the Holder and Minkowski inequalities.
The next chapter is the most important in the book, for it covers the notion of probability on measure spaces. After an brief motivation in the first section of the chapter, probability spaces are defined, and Bayes' theorem is discussed as an exercise. Both the weak and strong law of large numbers is proven in detail.
Things get more abstract in chapter 10, which discusses measure theory on locally compact spaces. Borel and Baire sets on these kinds of spaces are defined, and the author gives detailed arguments on what must be changed when doing measure theory in this more general kind of space.
The book ends with a discussion of measure theory on topological groups via the Haar measure. This chapter also has connections to physics, such as in the Faddeev-Popov volume measure over gauge equivalent classes in quantum field theory. The author does a fine job of characterizing the important properties of the Haar measure.
Rating: 5
Summary: an excellent book
Comment: If you want to stydy measure theory from scratch, I do recommend this book. This book is based on a ring, not an algebra, and is a little old-fashioned. So some people feel uncomfortable. But in particular, product spaces, the Fubini theorem and extension theorems are written very clearly. I'm convinced this book will facilitate your learning in measure theory and probability theory.
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Title: Functional Analysis by Walter Rudin ISBN: 0070542368 Publisher: McGraw-Hill Science/Engineering/Math Pub. Date: 01 January, 1991 List Price(USD): $140.60 |
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Title: Real and Complex Analysis by Walter Rudin ISBN: 0070542341 Publisher: McGraw-Hill Science/Engineering/Math Pub. Date: 01 May, 1986 List Price(USD): $150.55 |
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Title: Counterexamples in Analysis by Bernard R. Gelbaum, John M. H. Olmsted ISBN: 0486428753 Publisher: Dover Publications Pub. Date: 01 June, 2003 List Price(USD): $14.95 |
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Title: Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov, S. V. Fomin ISBN: 0486406830 Publisher: Dover Publications Pub. Date: 01 March, 1999 List Price(USD): $12.95 |
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Title: Real Analysis (3rd Edition) by H.L. Royden ISBN: 0024041513 Publisher: Prentice Hall Pub. Date: 01 May, 1988 List Price(USD): $102.67 |
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