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.