Rigorous probability text for math major

Most probability texts that do not use measure theory seemed to be geared toward engineers and the like, while more advanced texts already assume a strong background in measure theory and Lebesgue integration.

I'm not too familiar with measure theory, (my analysis background is limited to Rudin's PMA Ch. 1 - 10), so I'd appreciate some recommendations that do not assume an analysis background beyond Rudin's PMA. I'd be okay with a text that uses measure and Lebesgue theory as long as it sufficiently presents the material on its own.

Thank you very much.

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I like Robert Ash's Real Analysis and Probability.

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I like this one too. –  Dilip Sarwate Mar 13 '13 at 1:46

Williams' Probability with Martingales.

He assumes little background in measure theory but walks you through the development of the necessary machinery, and more importantly, helps you understand why it's necessary. I strongly recommend.

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I like a recent book by John Walsh Knowing the Odds: An Introduction to Probability.

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"Foundations of Modern Probability" by Olav Kallenberg is imho very good.

It contains a HUGE amount of material and is self-contained.

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