Advanced probability theory books?

[bpet]

I'm interested in learning the calculus of general random variables, i.e. those that do not necessarily have a density or mass function - such as mixtures of continuous / discrete / Cantor-type variables.

There seem to be several different approaches:

1. Via densities, using delta functions etc, e.g. $$E[X]=\int_{-\infty}^{\infty}x f(x)dx$$

2. Via cumulative distributions, using Stieltjes-type integrals, e.g. $$E[X]=\int_{-\infty}^{\infty}xdF(x)$$

3. Via probability measures, e.g. $$E[X]=\int x d\mu(x)$$

Each seems to have a well developed rigorous theory. What would be the best approach to focus on, and what's a good accessible book on the subject?

 

[jbunniii]

If you learn (3), then you will develop (1) and (2) as part of the process. $F(x) = \mu((-\infty,x])$, and the probability density function $f(x)$ exists if $F$ is an absolutely continuous function.

I like Billingsley's https://www.amazon.com/dp/0471007102/?tag=pfamazon01-20 because it's a very readable yet rigorous treatment that doesn't assume that you already know measure theory and Lebesgue(-Stieltjes) integration.

 

[bpet]

Thanks - sounds like measure theory is the way to go and will be useful for more advanced topics.

Having a very basic and incomplete knowledge of Lebesgue integration, I'm tossing up between Billingsley and Shiryaev's https://www.amazon.com/dp/0387945490/?tag=pfamazon01-20 - the gist of the reviews seems to be that B is more of a gentle essay-style introduction whereas S is more concise and efficiently organized. Any thoughts on this?

 

[jbunniii]

Thanks - sounds like measure theory is the way to go and will be useful for more advanced topics.

Having a very basic and incomplete knowledge of Lebesgue integration, I'm tossing up between Billingsley and Shiryaev's https://www.amazon.com/dp/0387945490/?tag=pfamazon01-20 - the gist of the reviews seems to be that B is more of a gentle essay-style introduction whereas S is more concise and efficiently organized. Any thoughts on this?

I haven't read Shiryaev's book, so I can't compare the two. Billingsley isn't organized as a reference; he deliberately interleaves the probability material with measure theory on an "as-needed" basis, which is nice because everything seems properly motivated as you read through it. I would not say that his book is gentle per se (parts of it are quite tough), but it flows pretty well and he does a good job letting you know what he's doing and why.

Besides Billingsley and Shiryaev, another commonly used probability book at this level is Chung's https://www.amazon.com/dp/0121741516/?tag=pfamazon01-20. I've only skimmed it, and it looks fine, but a lot more dry than Billingsley. For example, Billingsley has a cool chapter about gambling theory, and often sprinkles interesting side topics such as "Strange Euclidean Sets" and the Banach-Tarski paradox, but Chung takes more of a no-nonsense approach. Chung is probably more appropriate for a graduate course, whereas Billingsley seems better for self-study. Just my opinion.