Measure Theory
By addebook • Jun 28th, 2008 • Category: Mathematics •
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Measure Theory (Graduate Texts in Mathematics)
Publisher: Springer
Number Of Pages: 304
Publication Date: 1978-02-28
Sales Rank: 460993
ISBN / ASIN: 0387900888
EAN: 9780387900889
Binding: Hardcover
Manufacturer: Springer
Studio: Springer
Average Rating: 4
Useful both as a text for students and as a source of reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory which is most useful for its application in modern analysis. Topics studied include sets and classes, measures and outer measures, measurable functions, integration, general set functions, product spaces, transformations, probability, locally compact spaces, Haar measure and measure and topology in groups. The text is suitable for the beginning graduate student as well as the advanced undergraduate.
Review:
An excellent introduction to measure theory
There comes a time when the budding probabilist or statistician seeks a more comprehensive treatement of measure theory than is afforded in the first few sections of a graduate text in pstats. I chose Halmos’s “Measure Theory” for this purpose for two primary reasons: i) Paul Halmos in my opinion is one of the best expository mathematics writers in history, and ii) years ago I paid $1 for the above-mentioned text (original Van Nostrand print) at a local thriftstore. My only perceived drawback was that likely some of his approach to measure theory may be outdated.
After reading the text (up to the chapter on probability) my opinion of Halmos as a writer and mathematician not only has been elevated, but the book delivered the thorough study of measure theory that I had hoped for. Indeed, the author does an excellent job in presenting measure theory in its entire generality semi-rings, rings, hereditary rings, algebras, sigma algebras and their extensions are all considered in detail, as well as measures on these set systems: finitely additive , sigma additive, inner measures, outer measures, sigma-finite measures, the completion of measures, regular measures). I especially enjoyed his presentation of Fubini’s Theorem along with the concept of “section of a measurable set”, which helped the theorem fall out effortlessly. I also found his presentation of different types of convergence (e.g. pointwise, uniform, almost uniform, in measure, in mean) very good and helped give me the bigger picture on modes of convergence. Theorem 22A is essentially a generalization of the Borel-Cantelli Lemma.
The book does have a few downsides. In particular pi and lambda-systems are not used and in some sense replaced by the older notion of semiring. Also, the author’s definition of Lesbegue integrable seemed a bit more complicated than what is usually presented (e.g. for nonnegative measurable f to be integrable it requires a sequence fn of simple functions that is mean fundamental and converges in measure to f; compare this with the simpler definition of the integral of measurable f being the sup of Lesbegue integrals of simple functions g for which g <= f). But I consider these downsides minor and highly recommend this text to anyone who seeks a deeper understanding of measure theory.
My impression of measure theory has gone from seeing it as abstract mathematical machinery for simplifying analysis proofs, to a kind of mathematical philosophy that unifies the infinite with the discrete, and lays the proper foundations for inference, probabilistic reasoning, and learning; i.e. the foundations of cognitive intelligence.pass:
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