Probability theory with sets actually generalizes well to infinite and even uncountable cardinalities—rather than counting events you just have to switch to the more general notion of the measure of a (sub)set [1].
I think it's OK to skim the first few chapters about constructing Lebesgue measure from Jordan measure, as it makes the whole topic appear more difficult than it needs to be. From Chapter 4 (abstract measure spaces) onwards, the book is full of great insight.
A good one on probability & stochastic processes is "Measure Theory and Probability Theory" by Athreya & Lahiri.
[1] https://terrytao.files.wordpress.com/2011/01/measure-book1.p... [2] https://www.springer.com/gp/book/9780387329031
It's also been useful for understanding the "novelty" papers like "Neural Differential Equations" and so on.
I was lucky enough to take a good measure theory class in college, where it was fine to spend an entire day working through a proof from a textbook. Now that I'm working, though, it would be harder to justify that time commitment. So it really depends on your priorities.
I also can’t recommend this text enough for anyone looking to learn measure theory. Tao is not only one of the brightest research mathematicians but also is extremely good at communicating and framing concepts.