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what are some good books on probabilities and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of "real life" situations where they would be used

thanks

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    $\begingroup$ David Williams' "Probability with Martingales" is superb. $\endgroup$ – Sasha Jun 13 '12 at 0:15
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    $\begingroup$ I know this won't be a popular opinion among the more applied math types but if you learn the theory deeply enough, the "real world"/computational aspect will follow naturally. Robert Ash and Doleans-Dade's Probability and Measure Theory would be my suggestion as far as internalizing the theory but I can't speak to the applied aspect. $\endgroup$ – Wavelet Dec 18 '16 at 13:48

12 Answers 12

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I'd recommend Klenke's Probability Theory.

It gives a good overview of the basic ideas in probability theory. In the beginning it builds up the basics of measure theory and set functions.

There are also some examples of applications of probability theory.

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I think Chung's A Course in Probability Theory is a good one that is rigorous. Also Sid Resnick's A Probability Path is advanced but easy to read.

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    $\begingroup$ +1 for Resnick, probably the most readable of the graduate level probability textbooks without losing an ounce of rigor. $\endgroup$ – Mathemagician1234 Jun 13 '12 at 1:25
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    $\begingroup$ @Mathemagician1234 You comment +1 but I do not see that you actually gave me an upvote. $\endgroup$ – Michael Chernick Jun 13 '12 at 2:46
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    $\begingroup$ My bad-sorry.I've fixed the error. $\endgroup$ – Mathemagician1234 Jun 13 '12 at 2:48
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I like Olav Kallenberg's Foundations of Modern Probability - about as complete and up-to-date a textbook as you can find on the subject.It's not easy reading,despite its well written nature, because Kallenberg really packs a LOT into it. But it's certainly worth the effort. I personally wouldn't try and learn measure theory from it,though-it'll definitely be much easier going if you've already had a graduate real analysis course.

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    $\begingroup$ And this deserved a downvote,why,exactly? Ah,my fan club is at work again,I see. $\endgroup$ – Mathemagician1234 Aug 8 '12 at 17:56
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Feller's books are the standard reference. Personally I used Measure theory and probability theory by Athreya and Lahiri, which gives basic informations about some of the topics mentioned above, to begin with.

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Probability by Albert Shiryaev

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  • $\begingroup$ Note that the third edition of the book was recently published. $\endgroup$ – Gabriel Romon Mar 25 '17 at 9:22
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Jeffrey Rosenthal's A first look at rigorous probability theory will probably lack in real life examples but it is quite compact and very clearly written. Beautiful piece of work in my opinion.

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I learned probability from Grimmett & Stirzaker, Probability and Random Processes. It has a lot of exercises with a good mix of difficulties. It was a standard fixture on the desks of quants at the bank where I used to work. It's pleasant to read, includes interesting applications, does its best to build intuition and the occasional joke is pretty funny (YMMV).

Caveats: I stopped just short of the material on the Itô calculus, and if/when I come to study that subject I'll probably seek out a more leisurely treatment. Also, although it does things in terms of sigma-algebras etc the book aims to teach probability, not rigorous measure theory.

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Probability And Measure by Patrick Billingsley

Foundations of Modern Probability by Olav Kallenberg

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Jacod--Protter's Probability Essentials. https://www.springer.com/gp/book/9783540438717

I am surprised no one has mentioned Jacod--Protter. If you are stuck with conditional expectation, you might want to read Jacod--Protter. (also contains "real life" motivations, too).

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I really like Probability with Martingales by D. Williams and Probability: Theory and Examples by Durrett.

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I first learned measure theory from Real Analysis by McDonald and Weiss. It has a chapter on probability from a measure theoretic perspective.

Currently I'm reading through Probability Theory: A Comprehensive Course by Klenke.

I recommend both of these books.

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