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

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

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

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

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Kallenberg - Foundations of Modern Probability.

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It is a great book, but certainly not for learning about measure theoretic probability for the first time. – Michael Greinecker♦ Dec 24 '11 at 16:48
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Noel Vaillant's online Probability Tutorials are an excellent introduction to the real analysis, general topology and measure theory foundations of probability theory.

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