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: 
I suggest David Pollard's [A User's Guide to Measure Theoretic Probability](http://books.google.fr/books/about/A_User_s_Guide_to_Measure_Theoretic_Prob.html?id=B7Ch-c2G21MC&redir_esc=y), published in 2002.
I'd recommend Doob's *Measure Theory*. It does a nice job at blending probability with measure theory, perhaps close to the style you're looking for.