Context: The simplest random variables are real-valued, and obviously real numbers have commutative multiplication. But random variables can take values in any measurable space (at least this is my understanding and also that of Professor Terry Tao), i.e. random variables can also be random vectors, random matrices, random functions, random measures, random sets, etc. The whole theory of stochastic processes is based on the study of random variables taking values in a space of functions. If the range of the functions of that space is the real numbers, then yes we have commutative multiplication, but I don't see how that's the case if we are e.g. talking about functions into a Riemannian manifold.
EDIT: To clarify what I mean by "classical probability theory", here is Professor Tao's definition of random variable, which is also my understanding of the term (in the most general sense):
Let $R=(R, \mathcal{R})$ be a measurable space (i.e. a set $R$, equipped with a $\sigma$-algebra $\mathcal{R}$ of subsets of $R$). A random variable taking values in $R$ (or an $R$-valued random variable) is a measurable map $X$ from the sample space to $R$, i.e. a function $X: \Omega \to R$ such that $X^{-1}(S)$ is an event for every $S \in \mathcal{R}$.
Then, barring that I am forgetting something obvious, classical probability theory is just the study of random variables (in the above sense).
/EDIT
