There's also the Axiom of Determinacy that rejects Choice and, when paired with the existence of a very strong measurable cardinal, gives a very broad class of measurable sets.
Could you give an example of a set whose measurability I might care about, other than subsets of R? something for random processes?
Could you give a reference for the combination?
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.