I'm taking stochastic probability class but I'm now only taking analysis (with Rudin's PMA) class. The stochastic probability class doesn't depend heavily on the theoretic structures: rather, the professor wants to give the intution and that's fine with me because I've taken set theory class and basic probability class before, so I can understand almost every theorems.

The problem is that I haven't studied rigorously in the previous basic probability class. I think it's because the professor just wanted to nurture us the intuition and become prepared to be rigorous after having learned the Measure Theory (I'm just guessing here). And the current stochastic probability class is mainly dealing with a general probability space, Martingale and Marcov chains, so I somewhat feel there is a gap between the two classes (our textbook is Basic Stochastic Processes by Brzezniak and Zastawniak, a small book indeed). So I want a book that covers the whole probability theory in a rigorous framework. For example, we haven't proved the central limit theorem or had gone deep into the probability generating function. I know what a Poisson point process is, but we used it only to solve some questions. But, as far as I know, without measure theory I cannot understand those concepts any deeper, am I right?

In conclusion, should I bother with basic probability one more time to review stuff? Or just wait patiently until I study measure theory and then grab a decent book about probability? I've looked into the probability textbook from Cambridge Press, and I still couldn't conclude myself which way is the better. I need a piece of advice from other mathematics majors.

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    $\begingroup$ If you are not solid on your probabilistic intuition, then measure theory will just make that worse. Measure theory extends your intuition and makes it rigorous enough to handle limiting processes (where it shores up and generalizes non-measure theoretic probability). A good pre-measure theoretic book is "Basic Probability Theory" by Ash (and its free from his website). Its "elementary" only in the sense that its not measure theoretic. $\endgroup$ – user76844 Oct 7 '14 at 1:05
  • $\begingroup$ So you mean that I should review the stuff with a more concrete framework yet without measure theory, right? I've seen the book and it looked quite promising. Thanks for the recommendation. $\endgroup$ – Taxxi Oct 7 '14 at 1:10
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    $\begingroup$ Yes, always start with concrete. For example, Feynmann said that whenever he was confronted with something abstract, he would first construct a concrete example. So, if it was good enough for Feynmann, then I think we can all benefit from concreteness. Even now, although I write as if I think with just abstract symbols, I actually have a paradigmatic concrete example in my head. $\endgroup$ – user76844 Oct 7 '14 at 1:28
  • $\begingroup$ @TaxxiDriver, physicist ET Jaynes in his book Probability Theory more or less states that measure theory has no impact on statistics. Also, Renyi's book is good at least the back on information theory re generalized Shannon entropy. $\endgroup$ – alancalvitti Nov 22 '14 at 18:35
  • $\begingroup$ @alancalvitti There's no need, because plenty of people have already written good books on probability (Ross, Chung, Durrett, Fristedt, Gray, Shiryaev, Kallenberg, Cinlar, Loeve, Neveu, ...), books which reflect and convey a proper understanding of the subject. The whole point of a sigma algebra is that it is a Boolean algebra with closure under countable operations to be able to describe limiting behavior. Boolean algebra can describe classical logic as well as basic set operations -- this is no secret, it's the entire basis for set-builder notation, for example. $\endgroup$ – Chill2Macht Apr 12 '17 at 6:57

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