Good books on "advanced" probabilities 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
 A: 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.
A: 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.
A: Probability by Albert Shiryaev
A: 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. 
A: Probability And Measure by Patrick Billingsley
Foundations of Modern Probability by Olav Kallenberg
A: I really like
Probability with Martingales by D. Williams  and Probability: Theory and Examples by Durrett.
A: Here is a list of great books in probability, found in this blog:


*

*The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)

*An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

*Discovering Statistics Using R

*Fifty Challenging Problems in Probability with Solutions

*Introduction to Probability Theory

*Probability Theory: A Concise Course

*Introduction to Probability, 2nd Edition

*Ramanujan's Lost Notebook

*Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)

*Understanding Probability: Chance Rules in Everyday Life

*Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)

*A Course in Probability Theory, Third Edition

*Probability and Statistics (4th Edition)
A: 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.
A: 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.
A: 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.  
A: 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).
A: 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.
