I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like point-free topology, where one basically replaces topological spaces by their locales of open sets, I figured there is a way to do something similar with $\sigma$-algebras and with probability spaces.
Any thoughts on that? Does somebody know, whether this has been studied before?
Here are some more thoughts: I suppose a problem is how to recover the sample space $\Omega$ from a point-free probability space, as there is a no guarantee that there is an injection $\Omega \to \sigma$ from the sample space to the $\sigma$-algebra of a probability space. I wonder, how important it is to have a sample space at all. I (think I) know, that probability theory is actually about random variables, but do we really need a sample space to talk about those? Also, considering that there is no obvious notion of a morphism between probability spaces, maybe there are other objects we should look at?
(I asked this question on MSE and a user suggested to ask this on MO. I studied mathematics for about a year in university, so my background is not actually that sophisticated. I only know a little bit category theory, analysis and linear algebra)