Let $m(\cdot)$ be a probability measure on $Z$, so that $\int_Z m(dz) = 1$.
Consider a continuous function $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$, $Y \subset \mathbb{R}^m$ is compact, $Z \subseteq \mathbb{R}^p$ is closed.
Define
$ \hat{f}(x) \ := \ \inf_{y \in Y} \ \int_Z f(x,y,z) m(dz) $
Under which conditions is function $\hat{f}(\cdot)$ continuous?
Instead of assuming continuity of $f(\cdot)$, let us assume that for each $z \in Z$ the map $(x,y) \mapsto f(x,y,z)$ is continuous. Now under which conditions we have continuity of $\hat{f}(\cdot)$?
Note: in the non-probabilistic case, $Y$ compact and $(x,y) \mapsto f(x,y)$ continuous are sufficient to establish continuity of $x \mapsto \inf_{y \in Y} f(x,y) $.