# Are there any measurable spaces of functions

I am approaching this question from a probability perspective, and am hoping for some kind of framework to help understand all of this. I believe I may have even asked a similar question on here in the past (so, if I did, sorry).

In what I'll call "naive probability" (in analogy to naive set theory), we can define and construct all kinds of "random objects". Specifically, random real numbers are defined in terms of "random variables", which are a measurable maps from the sample space under consideration to the real numbers. Similarly, we can define "random pairs" $$(X,Y)$$. The specifics of how we might define a random pair might vary, but at the end of the day, a random pair is a measurable function from the sample space into $$\mathbf{R}^2$$ or similar.

And this is where my questions sort of begin. In the back of my mind, I have a goal of defining a "random random variable". The intuition seems clear enough: a random random variable $$X_Y$$ would be a mapping $$X_Y\colon \Omega_Y \rightarrow (\Omega_X \rightarrow \mathbf{R})$$, so that, if we were to condition $$X_Y$$ on $$Y \in A$$, "it" is a random variable (on $$\Omega_X$$) in its own right.

And this does somewhat seem a promising approach, but then we end up dealing with somewhat tricky issues like defining a notion of measurability for collections of functions (presumably by restricting the class of functions we quantify over, in some way, and then defining a topology or $$\sigma$$-algebra on that space of functions).

Certainly, if the space of random variables we are considering "is" a continuum or perfect Polish space (say, it has a single real parameter we care about), or discrete, then the standard constructions just work, via the Borel isomorphism theorem, etc. That covers a whole lot of cases, of course.

Following similar motivation, it seems that we ought to be able to define random probability measures $$\mu_Y(X)$$, which would have the signature $$\Omega_Y \rightarrow (\Omega_X \rightarrow [0,1])$$, which would appear to "induce" a potentially distinct definition of a random random variable, as the random variable related to the probability measure $$\mu_Y$$. Again, we have the technical difficulties of trying to limit or characterize the space of random functions we are quantifying over. That said, it appears that "regular conditional probability measures" and the "disintegration theorem" sort of implement this idea. (So I'll be reading Faden's 1985 paper as soon as I finish my question)

The big question: are these potentially distinct definitions equivalent?

Smaller questions: OK, so does this seem insane? Am I on the right track? Is there a field which asks or even answers these or similar questions? Can we come up with rich spaces of functions to quantify over?

Thank you

• The idea of "random random variable" is somewhat similar to multiple steps in a random process. So the distinction between the two approaches in your question is analogous to the difference, in the theory of random processes, between (1) sample paths and (2) the joint distribution. Sep 9, 2021 at 17:21
• See Kallenberg, Olav. Random measures, theory and applications. Vol. 1. Cham: Springer International Publishing, 2017. Sep 10, 2021 at 2:41