Customer Reviews

Probability and Measure, 3rd Edition

5 star:		(6)
4 star:		(3)
3 star:		(2)
2 star:		(1)
1 star:		(1)

 

 

I've read portions of almost every measure theoretic probability theory book published. And I've come back to Billingsley. This is a hard book to read through and through if you are a novice; this is not Billingsley's fault - it is just that the subject is hard on first acquaintance.
Billinglsey develops everything from first principles, so if you have the intellectual gumption you ought to be able to read the main text with a knowledge of plain college algebra and a little epsilon-delta practice of the sort that comes from an undergraduate real analysis course. The small print asides are fascinating but they are often addressed to a card carrying mathematician. The November 2003 reviewer who complained that Billingsley uses expectation before defining the integral fails to notice - or at any rate, to point out - that he defines only the expectation of simple random variables in the first chapter, so what is involved is just a sum, not an integral. I could sing my praises on and on. But here is the kernel of this review in a line: this is one of the best books ever written on measure theoretic probability. Full stop.

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.

"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 (I cannot believe it is so overpriced though!!!!)

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.

This book is not for everybody. It is for the professional mathematician (or physicist, or alike). All concepts are very well explained, and Billigsley does go down to the core of everything. It is, as far as I'm concerned, among the best books in math ever written, with favorites such as Feynman's lectures and Herstein's algebra manual. If you are a mathematician and want to have the top reference in probability, this is it.

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.

This book was my first introduction into probability so I am somewhat sentimentally attached to it. The exposition is very, very careful and there is very little a reader has to do than to read carefully meaning that this text is a solid introduction to a serious probability for upper class undergraduates or beginning graduate students.
As a graduate student I very much appreciated the rigor and detailed approach in this textbook. Some of the topics like independence and martingales are developed with a rigor and details usually missing from undergraduate textbooks. If you do not like books missing any deductions this is your textbook.

But there are problems with the text, the major one being the outline and order of the exposition, sometimes it looks as if it was done at random though even that has been done with intention to facilitate understanding but has somehow gone a bit astray. For example it takes hundreds of pages to get into fairly elementary notions and then they are treated counterintuitive leaving reader without intuitive understanding of the topics. See for example exposition of expectation (general case) that is way too slow after hundreds of pages. Or discrete Markov Chains treatment early enough but completely technical and counterintuitive without ever leading a student to any concrete ideas. The integration topics are sprinkled all over the place as well.

Still I would think this is the text for introductory study of probability only with a instructor furnishing better choice and order of topics than the one presented in the textbook. It surely is one of the most detailed approaches geared toward a serious study. On the level of sofistication it does not come anywhere close to Kallenberg's "Foundations of Modern Probability" or Borovkov's "Probability Theory" but it does provide a treatise with important examples, without omissions (sometimes on the expense of elegance but every time for the benefit of reader), and with a rigor required while still being accessible to a beginning graduate student and thus it provides the important benefits of learning.

This is one of the plausible choices for two semester course in introductory graduate probability with perhaps Kai Lai Chung's classic Stanford textbook as the alternative.

The book is a great reference for someone who already knows probability theory but needs to refresh her/his memory. It covers all the basics in measure theory, integration and probability, although it hardly discusses non-probabilistic applications of measure theory (however, in the second edition there is a very nice section on Hausdorff Measure).

The book is not so great as a textbook. Despite covering most of the topics typically included in an introductory course on measure-theoretic probability, the order in which these are presented is far from optimal. There is a lot of repetition and this may confuse a reader who is learning the material for the first time.

Anyone attempting to use this book as a textbook must have some knowledge of elementary (non measure-theoretic) probability and a solid background in multivariable calculus (including epsilon-delta arguments).

First thing, this book isn't not appropriate for undergrads, therefore if you are an undergrad, please look for a suitable book for your level and don't post reviews that may mislead other customers. It is also not appropriate for business majors or fields that deal exclusively with applied statistics and probability.

If you are a graduate student in mathematics, statistics or economics (with concentration in econometrics), if you need theoretical probability, this is a must have book. The proofs are presented in a very solid way. Some are even beautiful masterpieces. The same can be said to some motivation for particular issues, for instance, using diatic expansion to motivate the whole probability theory in the first few sections. There aren't many typos, I personally haven't come across any yet. Shiryaev's Probability book is also very good; however, there are apparently more typos and in my personal opinion the explanations in Billingslley's book are a bit better.

If you think good things come in two's I would suggest you also take a look at a solid, concise and cheap book, Probability Essentials by Jacod and Protter.
Enjoy!!!

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.