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I'm looking for a clear way to learn measure theoretic probability theory. Any suggestions?

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9 Answers 9

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I would recommend Erhan Çinlar's Probability and Stochastics (Amazon link).

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    $\begingroup$ This is not a good suggestion. This book is a good reference, but a very poor resource for learning. It has very few examples, the ideas are not well motivated, and the notation prioritizes brevity over clarity. In my view, Billingsley's book is better in every way. $\endgroup$
    – tooAnnoying
    Commented Mar 19, 2021 at 16:49
  • $\begingroup$ Agree with @tooAnnoying, although I found the book A User's Guide to Measure Theoretic Probability by David Pollard (2002) to be much more accessible than Probability and Stochastics by Çinlar and Billinglsey's book. $\endgroup$
    – mhdadk
    Commented Nov 5, 2023 at 18:33
  • $\begingroup$ @mhdadk: I would suggest posting that suggestion as a new answer instead of as a comment. $\endgroup$
    – Willie Wong
    Commented Nov 6, 2023 at 4:44
  • $\begingroup$ @WillieWong done. $\endgroup$
    – mhdadk
    Commented Nov 6, 2023 at 13:14
  • $\begingroup$ I strongly disagree with @tooAnnoying. Cinlar's book is great for someone who is confused about measure theory in the context of probability and statistics, but has to be read linearly, in my opinion. I have tried countless books and all my coursemates know how much I have struggled with those. Probability and Stochastics is great because it is clear about notation, the examples are few but usually are critical in statistics and often you won't find them elsewhere. Personally, it has been the best book to learn measure theory in the context of MCMC and statistical machine learning $\endgroup$
    – Euler_Salter
    Commented Jan 19 at 14:51
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You can try the lecture notes here: https://web.archive.org/web/20190310150708/http://www.statslab.cam.ac.uk/~beresty/teach/pmnotes.pdf They're very good.

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Probability With Martingales by David Williams is a very enjoyable book.

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Probability and Measure, P. Billingsley.

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  • $\begingroup$ This is clearly the best suggestion here. $\endgroup$
    – tooAnnoying
    Commented Mar 19, 2021 at 16:50
  • $\begingroup$ I have found Billingsley eminently readable, certainly compared to the graduate texts by Rick Durrett and even Kai Lai Chung. He begins with an excellent introductory chapter about normal numbers which gives a lot of the motivation for the techniques needed for analyzing infinite probability spaces in an understandable context (for those who know basic analysis, that is). He has lots of examples throughout and he is even in the development, not accelerating dramatically in later chapters as many authors tend to do. $\endgroup$
    – fromscratch
    Commented Dec 14, 2023 at 9:24
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Noel Vaillant's online Probability Tutorials are an excellent introduction to the real analysis, general topology and measure theory foundations of probability theory.

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Kallenberg - Foundations of Modern Probability.

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    $\begingroup$ It is a great book, but certainly not for learning about measure theoretic probability for the first time. $\endgroup$
    – Michael Greinecker
    Commented Dec 24, 2011 at 16:48
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Also try A First Look at Rigorous Probability Theory by J. S. Rosenthal. It shows the reader why measure theory is important for probability theory. The author, however, presupposes a knowledge of analysis from the reader.

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A really comprehensive, easy to read book would be "An Introduction to measure and probability" by J.C Taylor (Amazon). Lots of examples, exercises, and really nice geometric view of conditional expectation via Hilbert spaces.

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I highly recommend A User's Guide to Measure Theoretic Probability by David Pollard (2002). Here are some reasons why you should consider it over the other recommendations:

  1. Unlike Probability and Stochastics by Çinlar, Pollard's book is much more conversational and motivates the need for definitions, rather than presenting them out of thin air (at least that was my experience with Çinlar's book).
  2. I felt that Billingsley's book is too dense for a first course on measure-theoretic probability. It feels more like a reference textbook rather than a book that aims to teach. Pollard's book, on the other hand, gently guides you through the different topics, with each topic building infinitesimally over the previous one. I don't feel that there are big jumps between topics.
  3. Rosenthal's A First Look at Rigorous Probability Theory felt like a condensed version of Billingsley's book, so I didn't like it.
  4. The emphasis in Pollard's book on indicator functions at the very start makes several concepts easier to understand later on.

This being said, Pollard's book assumes that you have taken at least one course on axiomatic probability theory.

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