Could you give an example of a set whose measurability I might care about, other than subsets of R? something for random processes?

Well, I guess this pretty much depends on the area you're working on. I'm interested in the foundation of mathematics, for which measurable sets are of big importance (for example, they are the smallest critical point of embedding of transitive models, or they are the smallest large cardinal property that cannot be shown to exists inside the smallest inner model). Outside of that area, I guess the interest is all about R and descriptive set theory.

Edit: It's not true that measurable cardinals are the smallest large cardinals that do not exists in L. Technically, the consistency strength is called 0#, and between that and measurables there are Ramsey cardinals.

Could you give a reference for the combination?

Well, the definitive source is Kanamori's book "The higher infinite", but it's advanced. Some interesting things can be scooped up from Wikipedia's entry about the axiom of determinacy.