For a metric space $S$(without any further assumption like separable or anything), a random variable $\xi:\Omega \rightarrow S$, then given an open set $A\subset S$, is it always possible to approximate $1_{A}$ by bounded continuous functions in expectation? That is, do there exist $f_n$ bounded and continuous on $S$, so that $\lim\limits_{n\rightarrow \infty}E(f_n(\xi))=E1_A(\xi)$?
If $1_A$ itself can be approximate point-wisely by $f_n$, then it's certainly also true in expectation, but with a mere metric space $S$, I'm not sure this is always doable.
We can prescribe $f_n:S\to\mathbb R$ by $s\mapsto\min\left(1,nd\left(s,A^{\complement}\right)\right)$.
Then $f_n$ is continuous.
This with $s\in A\iff d(s,A^{\complement})>0$ because $A$ is open.
Consequently $\lim_{n\to\infty}f_n(s)=1_A(s)$ for every $s\in S$.