Question: Are there any good references for facts about function-valued random variables? In particular for facts like the following:
Let $X$ be a topological space, $Y$ be a random variable with values in $X$, and $F$ a random variable with values in $C(X, \mathbb R)$. Define the function $\overline F: X \to \mathbb R$ by $\overline F(x) = \mathbb E[F(x)]$. If $Y$ and $F$ are independent then $\mathbb E [F(Y)] = \mathbb E [\overline F(Y)] $.
Background:
Consider the online optimisation setup where we have a fixed set $X$ and each turn select an action $x_n \in X$. We get some real-valued reward $r_n(x_n)$ and some feedback and use this to decide the action on the next turn.
I am interested in when the rewards are noisy realisations of some fixed function. That means $r_n(x_n) = r(x_n) + \epsilon_n(x_n)$ for some fixed (but unknown) function $r: X \to \mathbb R$ and i.i.d functions $\epsilon_1, \epsilon_2,\ldots : X \to \mathbb R$ such that each $\mathbb E[\epsilon_n (x)] =0$.
The only difficulty defining function-valued random variables seems to be deciding what a measurable set of functions is. In order to prove anything useful we will need the evaluation function $(f,x) \mapsto f(x)$ to be measurable. The first roadblock comes from Auman's Borel Structures for Function Spaces which says there is no good notion of measurable sets in the set of all measurable functions.
Fortunately we can do better if we assume more structure. If $X$ has a topology and we restrict attention to continuous functions, then we can give the function space our favourite topology and the induced Borel Structure. It turns out if $X$ is sufficiently nice then the Borel structures induced by the uniform and pointwise topologies are the same. From this point we just assume as usual there is a measure space $(\Omega,P)$ and measurable functions $\epsilon_n: \Omega \to C(X, \mathbb R)$ that satisfy the usual independence relations.
In this setting the goal is to bound $\sum_{i=1}^N r(x_i) $ on expectation. Naturally this leads to considering $\mathbb E[\epsilon_n(x_n)]$. By construction $x_n$ is a function of $\epsilon_1,\ldots, \epsilon_{n-1}$ so is independent of $\epsilon_n$. From this I can for example prove using standard (but lengthy) measure theory arguments that $\mathbb E [\epsilon_n(x_n)] =\mathbb E [\overline {\epsilon_n}(x_n)] $ where we define $\overline {\epsilon_n}(x) = \mathbb E[\epsilon_n(x)]$. For the proof observe when $\epsilon_n, x_n$ are simple the equality is obvious. Use Arzelà–Ascoli to see $\epsilon_n$ is a limit of simple random variables. The same is true for $x_n$. Then use dominated convergence.
However this is an awful lot of work and doesn't have any fundamentally new ideas from the standard undergraduate measure theory course. I wonder are there any good references for facts like the above. So far I can find very little about function-valued random variable as all!
