This is partially inspired by this question and this blog post.
When trying to express classical probability in the "free probability" setting one takes an algebra of random variables equipped with the expectation as a replacement for a (localizable / pointfree) measure space (more precise statements about this relationship are stated here (ignoring the measure) and here e.g.).
While even in the case including the measure/expectation we may get a nice equivalence / adjunction, it seems to me that doing probability theory in this "free" setting is not without its problems: replacing the measure space $X$ with $L_\infty(X)$, in practice, has the glaring issue that we can't "directly" talk about variables which are not essentially bounded without using the equivalence to pass back to the measure spaces; same with the algebra of random variables with finite moments (I'm thinking of Cauchy-distributed variables as a counterexample).
One naive way of fixing this is looking at the space of all random variables instead. I have two questions (and hope that is not to much for just this post):
- Are there obvious reasons why this idea is fundamentally flawed in some way?
- If not, has anyone explicitely worked something out in this direction? By this I mean: defining appropriate categories (the space of all random variables is not even normed anymore), functors and proving that they are adjoint / an equivalence (up to duality)