Best measure theoretic probability theory book?

I’m looking for a clear way to learn measure theoretic probability theory. Any suggestions?

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I would recommend Erhan Çinlar’s Probability and Stochastics (Amazon link).

Probability With Martingales by David Williams is a very enjoyable book.

You can try the lecture notes here: http://www.statslab.cam.ac.uk/~beresty/teach.html
They’re very good.

Noel Vaillant’s online Probability Tutorials are an excellent introduction to the real analysis, general topology and measure theory foundations of probability theory.

Kallenberg – Foundations of Modern Probability.

Probability and Measure, P. Billingsley.

A really comprehensive, easy to read book would be “An Introduction to measure and probability” by J.C Taylor (Amazon).
Lots of examples, exercises, and really nice geometric view of conditional expectation via Hilbert spaces.