One possibility would be some combination of the book Probability With Martingales by David Williams, and the lecture notes of James Norris available freely online here and here, depending on your taste. As the names suggest, both of these develop measure theory with probability theory in mind. Section A of Williams’s book and the first set of Norris’s notes cover basic measure theory, and sections B and C of Williams’s book and the second set of Norris’s notes cover more advanced topics (conditional expectation, Martingales, Brownian motion). I’m only really familiar with the former, not the latter, and depending on what exactly you need/want to know you might only work through the former, and maybe not even all of that.
Norris’s notes in both cases are quite a bit terser and cover a little more material. Both the book and the notes have a large number of very helpful exercises compiled at the end. I remember finding some of Norris’s exercises in particular a lot of fun!
Other books I’m aware of: D. H. Fremlin has an encyclopaedic set of volumes on measure theory available for free on his website here. I’ve found this useful as a reference. Some people like Rudin’s Real and Complex Analysis for this kind of thing, but I wouldn’t recommend it for your goals. (It’s very difficult to jump into that book in the middle: you really need to follow the garden path from beginning to end, and I think that’d force you through a lot of functional analysis that you could survive without). Folland probably suffers from similar problems, but I’m not really familiar with it.
Disclaimer: I am not a probabilist.
One possibility would be some combination of the book Probability With Martingales by David Williams, and the lecture notes of James Norris available freely online here and here, depending on your taste. As the names suggest, both of these develop measure theory with probability theory in mind. Section A of Williams’s book and the first set of Norris’s notes cover basic measure theory, and sections B and C of Williams’s book and the second set of Norris’s notes cover more advanced topics (conditional expectation, Martingales, Brownian motion). I’m only really familiar with the former, not the latter, and depending on what exactly you need/want to know you might only work through the former, and maybe not even all of that.
Norris’s notes in both cases are quite a bit terser and cover a little more material. Both the book and the notes have a large number of very helpful exercises compiled at the end. I remember finding some of Norris’s exercises in particular a lot of fun!
Other books I’m aware of: D. H. Fremlin has an encyclopaedic set of volumes on measure theory available for free on his website here. I’ve found this useful as a reference. Some people like Rudin’s Real and Complex Analysis for this kind of thing, but I wouldn’t recommend it for your goals. (It’s very difficult to jump into that book in the middle: you really need to follow the garden path from beginning to end, and I think that’d force you through a lot of functional analysis that you could survive without). Folland probably suffers from similar problems, but I’m not really familiar with it.
Good luck!
Many thanks!