Say I have a sample space $\Omega$. I want for every $\omega \in \Omega$, a function $f: A\to B$, for some arbitrary sets $A,B$. (note this is not in the context of random walks, or sample statistcs or anything. The function and $A,B$ are arbitrary.)
What is the generally accepted most straightforward notation for this? e.g. how should I write the expected value of $f$ given some $a\in A$?
I was thinking of $E(f_\omega (a)|...)$, but this breaks with the convention that the $\omega$ is not written in the expectation operator since it's not a free variable. However, $E(f(a)|...)$ would instead suggest that for any $a$, $f(a) : \Omega \to B$ is a random variable. So should the notation be $f: A \to (\Omega \to B)$ ? This feels a bit clunky, and we can no longer speak of "a function $f_\omega$ for a given $\omega$".
For fixed $a$, $\omega\to f_\omega(a)$ is a random variable on $\Omega$ (under appropriate measurability assumptions on $f$). Denote this variable by $f(a)$ and write $\mathbb E(f(a))$. Mention in the text that this is what you mean by that notation.
Note that taking the expected value of $f$ doesn't make sense unless $B$ is $\mathbb R^n$ or possibly a Banach space or something.