- Series: Cambridge Studies in Advanced Mathematics (Book 74)
- Paperback: 568 pages
- Publisher: Cambridge University Press; 2nd edition (October 14, 2002)
- Language: English
- ISBN-10: 0521007542
- ISBN-13: 978-0521007542
- Product Dimensions: 6 x 1.4 x 9 inches
- Shipping Weight: 2.1 pounds (View shipping rates and policies)
- Average Customer Review: 10 customer reviews
- Amazon Best Sellers Rank: #1,063,190 in Books (See Top 100 in Books)
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Real Analysis and Probability (Cambridge Studies in Advanced Mathematics) 2nd Edition
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"A marvelous work which will soon become a standard text in the field for both teaching and reference...a complete and pedagogically perfect presentation of both the necessary preparatory material of real analysis and the proofs througout the text. Some of the topics and proofs are rarely found in other textbooks." Proceedings of the Edinburgh Mathematical Society
This classic graduate textbook, now reissued in paperback, offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures.The new edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new excercises have been added, together with hints for solution.
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While I appreciate the wonderful integration of Real Analysis and Probability and short proofs, the brevity is often achieved by omitting details rather than choosing a simpler argument and so the book is a bit too hard on the students. Many proofs are too terse and have significant gaps which often take a lot of classroom time to get over, unless you are willing to leave them puzzled. The wording in the proofs is often counterintuitive, in particular it is usually not clear if the sentence continues the line of argument or starts a new one. This is an unnecessary hiccup for the reader and it would cost just few friendly words here and there to fix. Overall the book is harder to follow than Royden's Real Analysis. Many of the exercises are great and illuminative but many are just impossibly hard.
The second point to be emphasized is that this book fills in an important gap in probability literature as it reveals numerous links between this branch of mathematics and other areas of pure mathematics such as topology, functional analysis and, of course, measure and integration theory, while most books on advanced probability develop barely the latter connection, which is plainly insufficient for (future) researches on probability theory.
Finally, despite the complaint of some reviewers, the book is extremely well written and amazingly comprehensive. The sole prerequisite to reading it is a certain amount of "mathematical maturity" which perhaps these reviewers lack.