30

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

Willie Wong
  • 73,139
John Salvatier
  • 445

9 Answers9

17

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

Willie Wong
  • 73,139
  • 4
    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. – tooAnnoying Mar 19 '21 at 16:49
  • 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. – mhdadk Nov 05 '23 at 18:33
  • @mhdadk: I would suggest posting that suggestion as a new answer instead of as a comment. – Willie Wong Nov 06 '23 at 04:44
  • @WillieWong done. – mhdadk Nov 06 '23 at 13:14
  • 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 – Euler_Salter Jan 19 '24 at 14:51
14

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.

perepelart
  • 444
A.A
  • 95
  • They are indeed quite nice. I recommend them. – Amey Joshi Aug 12 '20 at 03:51
  • 4
    The link is not working. – Silvinha Mar 16 '21 at 23:07
  • 2
    Updated link: https://homepage.univie.ac.at/nathanael.berestycki/teach.html – ilija139 Sep 27 '21 at 10:47
  • Just found these, and the Probability and Measure notes at https://homepage.univie.ac.at/nathanael.berestycki/teach/pmnotes.pdf have a very nice short but not too short intro to a lot of things. I find it a great complement to Billingsley and Rosenthal. – postylem Oct 26 '21 at 01:54
  • 1
    They're by Norris and can be found from his site: http://www.statslab.cam.ac.uk/~james/Lectures/ – Okoyos Jan 02 '23 at 04:57
  • @postylem I cannot find any lecture notes for probability theory on that website. Could you please clarify? – Ypbor Feb 10 '23 at 08:14
  • 1
    @Ypbor they’re not there anymore it’s true. the link that Okoyos posted looks to be the same that i had linked to before http://www.statslab.cam.ac.uk/~james/Lectures/pm.pdf – postylem Feb 11 '23 at 13:02
11

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

ShreevatsaR
  • 41,374
9

Probability and Measure, P. Billingsley.

Aschkan Mk
  • 1
  • This is clearly the best suggestion here. – tooAnnoying Mar 19 '21 at 16:50
  • 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. – fromscratch Dec 14 '23 at 09:24
6

Noel Vaillant's online Probability Tutorials are an excellent introduction to the real analysis, general topology and measure theory foundations of probability theory.

Henry
  • 157,058
6

Kallenberg - Foundations of Modern Probability.

JT_NL
  • 14,514
5

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.

Harikrishnan R
  • 56
4

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.

Don Shanil
  • 363
2

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.

mhdadk
  • 1,403