Has there ever been an analysis of random variables with multiple separate probability spaces? Whenever we analyze random variables, we analyze them as functions from the same probability space to e.g. $\mathbb R$. 

I am wondering if anyone has ever analyzed what kind of possible properties might arise from having multiple separate probability spaces, with separate probability measures. 

Intuitively, I assume we cannot analyze such a thing except by treating them completely separately. Nevertheless, I'm wondering whether someone has come up with some new concepts that allows one to get interesting results from analyzing multiple separate probability spaces, and their interaction, in some way. 

I know this is a vague question, and very exploratory, as I don't have any specific reason to believe that it would even make sense.
If we have a function $f:\Omega\to X$ then the measurable spaces $(\Omega,\mathscr A)$ and $(X,\mathscr B)$ are given. The pair of $\sigma$-algebras with respect to which $f$ is measurable may be different. More importantly, the measure $P_{\theta}$ over $(\Omega,\mathscr A)$ may vary too. Then, $f$ may have different distributions over $X$ as the parameter $\theta$ changes.

This is quite common in quantum mechanics. Consider, for example, the famous Stern-Gerlach experiment demonstrating the quatized nature of the magnetic momentum of certain objects, say, electrons. If an electron beam goes through a cascade of Stern-Gerlach apparatuses then the probability of observing "up" spin or "down" spin depends on the relative situation of the Stern-Gerlach magnets. So, the probability space modelling the behavior of the electrons is equipped with a probability which depends on the parameter related to the said relative situation (usually the angle between the vector of inhomogeneities). At the same time it is meaningless to talk about a common probability space unless one changes the angle randomly.
If we have a function $f:\Omega\to X$ then the measurable spaces $(\Omega,\mathscr A)$ and $(X,\mathscr B)$ are given. The pair of $\sigma$-algebras with respect to which $f$ is measurable may be different. More importantly, the measure $P_{\theta}$ over $(\Omega,\mathscr A)$ may vary too. Then, $f$ may have different distributions over $X$ as the parameter $\theta$ changes.

This is quite common in quantum mechanics. Consider, for example, the famous Stern-Gerlach experiment demonstrating the quatized nature of the magnetic momentum of certain objects, say, electrons. If an electron beam goes through a cascade of Stern-Gerlach apparatuses then the probability of observing "up" spin or "down" spin depends on the relative situation of the Stern-Gerlach magnets. So, the probability space modelling the behavior of the electrons is equipped with a probability which depends on the parameter related to the said relative situation (usually the angle between the vector of inhomogeneities). At the same time it is meaningless to talk about a common probability space unless one changes the angle randomly.