Real Analysis and Probability

Rating:
Summary: A classic text in theoretical probability
Review: 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.

Rating:
Summary: Yakkkkkkkkkkkk............
Review: 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.

Rating:
Summary: A classic
Review: 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.

Rating:
Summary: Not a good book for learning new material!
Review: This is definitely not a good book for learning the material for the first (or even second!) time. It's difficult to read, has too few examples, and the material is way too condensed for studying analysis. Based on what the author presents, the problems are generally difficult to solve.

Rating:
Summary: one of the best
Review: This is one of the best textbooks on real analysis and probability (at the graduate level). You will need a solid undergraduate course in analysis before being able to read this one. In any case, the exposition is quite elegant and clear. All the major theorems are proved. Also provides good exercises ranging from routine to quite challanging. The first half of the book presents the elements of advanced real analysis and topology (including the essentials of functional analysis); the second half presents probability theory (including martingales and stochastic processes). Very comprehensive.

Rating:
Summary: Fun for those who like abstract math
Review: You will find this an excellent book, as long as you belong to its target public. The book is targeted towards real mathematicians interested in a very theoretical approach of probability and the underlying real analysis framework. Although this book is very much self contained (in principle you do not need any pre-knowledge of real analysis since everything is explained from the beginning), the reader should have a rather high level of maturity in abstract math. It is definitely not a book for beginners, since it has a high level of abstraction. If you only want to learn the more practical 'calculus alike' aspects based on intuition, you should buy another book. On the other hand, if you like highly theoretical and abstract math, if you want rigor,if you are a mathematical researcher,... this book deserves a closer look.
Readers of books at this level will definitely need to invest more time than with the average math books, but will be rewarded with the indescribable feeling of understanding the creative thoughts of some great mathematicians.

Key points are :
-explains everything you need from zero. The first chapter for instance starts with basic set theory, subsequent chapters
describe basic topology, Hilbert an Banach spaces and functional analysis. Further chapters then move to probability based on the theoretical underpinning of the first half of the book.
-contains not always the most intuitive proofs, but definitely the most beautiful, creative and elegant ones.
-contains interesting notes and historical aspects at the end of each chapter.
-does not cheat on the proofs : there are no gaps in the proofs that are left as an exercise to the reader. Everything is explained in full detail.
-is up to date with the most recent theoretical developments.

If you like abstract math, give it a try and enjoy