Real Analysis and Probability (Cambridge Studies in Advanced Mathematics Book 74) 2nd Edition by R. M. Dudley (PDF)

Description

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale’s convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The 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 exercises have been added, together with hints for solution.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I have been teaching a one semester course of Real Analysis (measure and integration) from this book. The students have already been through a course based on Rudin’s Principles of Mathematical Analysis though not the Lebesgue integral there, and pretty comfortable with metric spaces and such and the standards of mathematical proof. So as the next step in analysis this book seems to be in the right place esp. because the book advertises itself as self-contained.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.

⭐First of all I should say that this book was written for those interested in the foudations of probability theory (the same is also true for Prof. Kallenberg’s book). Therefore beginners learning real analysis and probability for the first time and those looking for applications should look elsewhere to find out appropriate books (instead of underrating such an important text like Prof. Dudley’s book). 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.

⭐Very thorough, includes an appendix on axiomatic set theory.

⭐Simply superb, you will fall in love with this book. Probably not the best for a first or ‘quick’ reading, but if you persevere, you will reap the rewards. The more you read, the more you appreciate what this book has. Specially, the historical notes at the end of each chapter are priceless. Great stuff from Professor Dudley.

⭐I had the older copy many years ago. The second edition is even better and with paperback. This book builds better connection between real analysis and probability than the earlier Robert Ash’s text. It’s a good reference for anyone who is interested in having some foundations for theoretical probability.

⭐This is a text book for math major students. I believe nothing is more terrible than a book full of theorems without adequat samples. And this happen to be one. The “A Probability Path” is much better than this one.

⭐The book is almost identical in the content to the later published Cambridge University Press copy, except for omission of the Stone-Weierstrass theorem. It has wider pages, not as bulky, and as a result somewhat easier to grasp.

⭐This is absolutely a classic book on real analysis and probability, although it is a little hard to read. Highly recommend to people working in machine learning and/or pattern recognition, since it provides almost all mathematical foundations needed to do deep research in these two fields, for example, on statistical learning theory.

⭐I like the way subject is developed, interesting book, good level of details

⭐Really impressive~Guide book for my review of probability theory

⭐the book is well written, even if it’s a bit expensive in my opinion. But finding a book that’s both well written and theoretically consistent and thorough is difficult with all the trash books these days.The packaging choice was very poor however. My soft cover copy arrived bent because the parcel was too small and i am flattening it under a huge pile of other books. Very bad packaging indeed!

⭐This book is just great! It contains almost every essential ingredient in real analysis, especially measure theory, and probability theory. It presents clear proofs and offers more than enough exercises. No more needed to say. Just the best!

⭐Moderne et très avancé sur le sujet. Rigoureux, complet et très agréable à lire. Contient des ressources nombreuses et précieuses en matière de problèmes. Livré dans les délais. Neuf. Emballage propre. Surpris par l’état de l’article. Grosse tâche d’encre sur la tranche. Impossible de ne pas la voir avant la livraison.

⭐

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