I want to post the following as a comment on many of the answers and comments already given.
Several people have said, "Well, watch out -- probability theory is not really the study of probability measures, but rather the study of certain quantities preserved under certain equivalence relations on probability measures, like distribution functions."
I certainly accept this point. In fact, I had more or less accepted it before I asked the question, although I admittedly didn't give much indication of this in the question itself. To be clear, I am aware "rewriting" impulses I have when reading about basic measure-theoretic probability are taking me in a direction away from the material of mainstream probability theory. I have two responses to this:
1) Okay, let's agree that the definition and study of a category of probability spaces is not the domain of probability theory per se. But this does not mean that it's not useful or worth studying.
1a) If this branch of mathematics is not probability theory, what is it? [User "coudy" gave an answer saying that this is ergodic theory. I was unduly dismissive of this answer at first, and I apologize for that. I still don't think that "ergodic theory" is exactly the answer to my question, for instance because so far as I understand the subject it focuses almost exclusively on the dynamical aspects of iterating a measure-preserving transformation of a probability space. (By way of analogy, the branch of mathematics that studies the category of finite type schemes over a field $K$ is arithmetic geometry, not arithmetic dynamics.)
1b) While I agree that probability theory is at present not concerned with such structuralist questions, is it clear that it shouldn't be? Or, in less polemical terms, is there no advantage or insight to be gained by studying the structural aspects of probability spaces?
2) I think an outsider to probability theory has a right to ask: "Okay, if probability spaces are really not the point of probability theory, why do they appear so prominently in all (so far as I know) modern foundations of the subject? Wouldn't -- or couldn't -- it be better to isolate exactly the structure that probability theory actually does care about and study this structure explicitly from the outset?"
By way of analogy, consider the notion of a "differentiable atlas" in the study of smooth manifold theory. Gian-Carlo Rota referrred to atlases as a polite fiction, meaning (I think) that they are present in the foundations of the subject but do not really exist in the sense that the practitioners of the subject do not think about them and ask questions about them. They don't do any harm so long as you don't take them very seriously, but I have seen students get caught up on this point and "ask too many questions". The more modern approach of a structure sheaf seems like an improvement here -- it does the same work as an atlas but is something that the practitioners of the subject actually care about, so it is not at all a waste of time to "think deeply about structure sheaves". Indeed, the concept of "structure sheaf" is incredibly prevalent in other areas of mathematics, to the extent that if you are founding a new branch of geometry, knowing about structure sheaves will ease the birthing process.
So the dual question to 1) here is "What is the kind of mathematical structure that probability theorists are intersted interested in studying?" Note that I think that (Happily, many of the very nice answers above do in fact address this question.question.)