TITLE: A Point-free probability theory?
QUESTION [10 upvotes]: 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)

REPLY [5 votes]: Possibly related... Caratheodory's book on measure and integration without points ...
Carathéodory, C. Mass und Integral und ihre Algebraisierung. (1956)
translated
Carathéodory, C. Algebraic theory of measure and integration. (1963) 
Translated from the German by F. E. J. Linton