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My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics - mostly oriented toward the applications without emphasizing too much the rigorousness of mathematics.

Now I am very interested in learning what makes the core logic and mathematical framework of the probability theory, as a math branch. More specifically, I would like to learn answers to the following questions:

1) what are the necessary axioms, from which we can build the probability theory? 2) what are the core theorems and results in the mathematical theory of probability? 3) what are the derived rules for reasoning/inference, based on the theorems/results in the probability theory?

So I am seeking a book that covers the "heart" of the mathematical probability theory - no need much on applications, or discussion on extended topics.

I would like to appreciate your patience for reading my post and any informative responses.

Regards, finguy

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closed as off-topic by Andrés Caicedo, Mark Meckes, Chris Godsil, Willie Wong, Stefan Kohl Jan 28 '14 at 17:04

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This book is a nice small book on the subject:

David Williams, Probability with Martingales

For a more comprehensive treatment of the subject I suggest the following:

William Feller, An Introduction to Probability Theory and Its Applications

Patrick Billingsley, Probability and Measure

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