I basically want to prove that under some conditions $E[f(X)\mid X \in S]$ is uniformly continuous in $S$
To be more specific, suppose $f:K \subset \mathbb{R}^n \rightarrow \mathbb{R}$ is a bounded uniformly continuous function where $K$ is compact. Also suppose $X: \Omega \subset \mathbb{R}^M \rightarrow K$ is a random variable with a uniformly continuous bounded PDF and $\Omega$ is compact (You can assume the mapping for $X$ is one-to-one. Also you can assume $n=2$ if you want)
Suppose $h(v)=g(S)=E[f(X)\mid X \in S]$ where $v=(x_1,y_1,\ldots,x_n,y_n)$ and $S=[x_1,y_1]\times[x_2,y_2]\times\cdots\times [x_n,y_n]$. I want to prove that $h$ is uniformly continuous in its arguments (or $g$ is uniformly continuous wrt Hausdorff distance or Symmetric difference pseudometric or Fréchet-Nikodym metric)
Is there a general theorem which can encompasses this?