Probability and Measure, 3rd Edition

Rating:
Summary: Probability Theory Bible
Review: "Probability and Measure" by P. Billingsley covers a lot of topics in probability theory, and in this sense it is a standard reference, but what I did not like much is that the concepts are somewhat scattered around the book, so one has to jump back and forth all the time. (May be this is an artefact of my graduate course in probability theory that had topics in the syllabus ordered in a way different from Billingsley.) I found another book more useful and more clearly written -- see Borovkov et. al., "Probability theory".

Rating:
Summary: Great text
Review: I have found Billingsley's text to be the most understandable probability/measure theory text that I have encountered. It is not necessary, but a basic background in measure theory would be very helpful.

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Summary: Standard text but ...
Review: the main problem with Billingsley's book lies in its organization of topics and results.
Yes it has all the standard results that need to be covered in a first (rigorous) course on probability theory and the proofs and exercises are good (thats why the three stars) but it is incredibly hard to study them from this book because of poor organisation which makes for lack of continuity (thats why no more than three).
Stick to Chung (and move to something more specialized thereafter). Unfortunately, Parthasarathy's 1977 Macmillan book is now out of print and only available in libraries ... I find that to be the best book at this level.

Rating:
Summary: Readable
Review: The only book on probability in graduate or senior undergraduate level that is readable. Sounds strange? Not really, probabilists in general do not know how to write readable books and examples are abundant. This book is exception. However this is not a definitive book, once you are done with it, you have to go further to learn more.

Rating:
Summary: Very rich textbook
Review: This book (or a previous version) has been a classic for decades. The book employs an unusual sequence in its exposition of the concepts of measure theory and probability. Therefore, it probably does not coincide with the order in which things are taught in most university courses. The book also has an astonishing number of little facts and factoids that are not central to the theory. That is, the main results could have been developed without all the extra information.

If a reader is looking for a linear, straightforward development of the subject, this book is not the right one. In particular, this book will not coincide with the format of most university courses on measure or probability. It does not focus on quickly getting to the main results. It is useful as a reference, but some of the concepts are illustrated with detailed examples developed earlier in the text. At times, this makes it difficult to use it as a quick reference.

If a reader is looking for an in-depth introduction to the subject, aims to develop an understanding of the intricacies, and is willing to spend a significant amount of effort learning about them, this book is suitable. It has a very rich set of exercises. These wonderfully enhance the reader's understanding of the subject material. The material is interspersed with a large number of digressions and general mathematical truths (e.g. ''Since both [statements] are theorems ... it makes no sense to say that [one] implies [the other]'' etc.). Thus, for someone who wants to take the time to enjoy the subject of probability and measure theory, this is a great book.

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Summary: a very good text book
Review: This book gives abundant examples and statements that help to deepen your understanding. It does not require much statistic background to follow the book, although sometimes the reasoning is not that obvious to me. But maybe because I am not a math student. I also feel that topics are a bit scattered in the book.

Rating:
Summary: Excellent introduction and reference.
Review: This book is a thorough introduction and excellent reference book for the ideas involving probability as a measure. I would recommend this book to anyone who needs a deep understanding of probability, expectation, integration, random variables, and so forth. Good also as a graduate level or other measure-theoretic probability course. Two years after I learned these ideas, I still refer often to the text.

Rating:
Summary: Some nice examples, poorly organized
Review: This is probably a very nice text book if you already know probability. There are undeniably some insightful examples. However, it is often hard to follow the sequence of topics in the book.

It is at least amusing that the integral is only developed a couple of chapters after expectation has been in use...