Question: 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.)

Answer: 
My favorite introduction to measure theoretic probability is [Probability with Martingales](http://rads.stackoverflow.com/amzn/click/0521406056) 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](http://rads.stackoverflow.com/amzn/click/081764055X) 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](http://rads.stackoverflow.com/amzn/click/0821828525) by S. R. S. Varadhan
Although it is little terse, I like Durrett's [Probability: Theory and Examples](http://www.math.duke.edu/~rtd/PTE/pte.html).