I'm looking for good lecture notes (or concise books) that develop probability theory from a measure theoretic point of view. In particular, I'm looking for a text where the measure theoretic part is developed only as far as needed for probability theory. (I'm not really interested in measure theory on its own.)
My favorite introduction to measure theoretic probability is Probability with Martingales by David Williams. The book is very well written and fun to read.
A text that is easier, but IMO less fun is A Probability Path by Sidney Resnick. That book provides you with all the details and does everything in small steps.
A very concise book that contains the essentials is Probability Theory by S. R. S. Varadhan
I suggest A first look at rigorous probability by Jeffrey Rosenthal.
Although it is a rather thick book I recommend Billingsley's Probability and Measure.
Although it is little terse, I like Durrett's Probability: Theory and Examples.
I suggest David Pollard's A User's Guide to Measure Theoretic Probability, published in 2002.
How about www.probability.net for a nice introduction...