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I'm looking for a probability textbook that would first build intuition using finite sample spaces and uniform probability mass over the sample space in a first part (coin toss, die, Bernoulli Urn, etc.)

Then introduce Kolmogorov axioms and tackle infinite sample spaces in a second part. It's best if there are interesting applications in this second part.

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Probability by A.N. Shiryaev

It goes exactly by the scheme you've drawn: first elementary things, then axiomatic theory. This way it is adapted to the sctructure of Probability Theory courses in Moscow University: elementary Probability Theory on the first year of study, followed later by the axiomatic theory, which goes side-by-side with Measure Theory.

The book is quite advanced and not easiest too read (although not so hardcore as Kallenberg; I'd put it on approximately the same level as Kai Lai Chung). This is not bad by itself - I like books which challenge a reader, and Shiryaev, in my opinion, gives a challenge exactly a bit below the level of scaring off an unprepared reader. And it is still very dydactic, a pleasure to read.

All in all, one of my favorite textbooks; my own lecture course is heavily based on it.

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  • $\begingroup$ Thank your for this detailed review and comparisons. This is very useful. I will check the book ASAP $\endgroup$ – Julien__ May 2 '18 at 10:21
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I think William Feller's An introduction to Probability theory and applications might be what you are looking for.

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  • $\begingroup$ Feller's book is very interesting (+1) and I highly recommend anyone reads it. However, it starts with discrete sample space, not finite sample space. $\endgroup$ – Julien__ May 3 '18 at 18:46

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