Vtyjkyuk

Say I have a sample space $Omega$. I want for every $omega in Omega$, a function $f: Ato B$, 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 $ain A$?

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 $a$, $omegato f_omega(a)$ 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.