Consider a locally-bounded function $f: X \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact and $Z \subseteq \mathbb{R}^m$ is closed.
Let $m(\cdot)$ be a measure on $X$; let $\mu(\cdot)$ be a probability measure on $Z$. Define $F(x) = \int_Z f(x,z)\mu(d z)$.
Given $(\epsilon, \alpha, \delta)$, it is known that we can find a "probable minimum" of $F(\cdot)$, i.e. $x^*$ such that, with probability $(1-\delta)$, we have
$ F(x^*) \ \leq \ \min_{x \in X\setminus S} F(x) + \epsilon $
where $m(S)/m(X) \leq \alpha$.
Question: given $(\epsilon, \delta)$, under which additional conditions is it possible to find $y^*$ such that
$ F(y^*) \ \leq \ \min_{y \in X} F(y) + \epsilon $
holds with probability $(1-\delta)$?