Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $\int_W \mu(dw) = 1$.
Consider a function $f: X \times Y \times W \rightarrow \mathbb{R}_{\geq 0}$, with $X \subset \mathbb{R}^n$ compact, $Y \subset \mathbb{R}^m$ compact, such that: $\forall w$ $f(\cdot,\cdot,w)$ is continuous, $\forall (x,y)$ $f(x,y,\cdot)$ is measurable.
Assume that for any compact $\underline{W} \subset W$ we have
$ \max_{y \in Y} \int_{\underline{W}} f(x,y,w) \mu(dw) \leq F(x) $
On the whole $W$, we assume that
$ \max_{y \in Y} \int_W \sup_{x \in X} f(x,y,w) \mu(dw) < \infty $
This implies that for any $y$ the family $\{w \mapsto f(x,y,w)\}_{x \in X}$ is Uniformly Integrable.
Are the assumptions sufficient to say the following?
$ \max_{y \in Y} \int_W f(x,y,w) \mu(dw) \leq F(x) $