Let $\mu(\cdot)$ be a probability measure on the closed set $\mathbb{W} \subseteq \mathbb{R}^m$.
Consider $f: \mathbb{X} \times \mathbb{W} \rightarrow \mathbb{R}_{> 0}$ locally bounded, where $\mathbb{X} \subset \mathbb{R}^n$ is compact, such that for any $w \in \mathbb{W}$ the map $x \mapsto f(x,w)$ is continuous.
Define $\bar{f}:\mathbb{X} \rightarrow \mathbb{R}_{\geq 0}$ as $$\bar f(x):=\int_\mathbb{W} f(x,w) \mu(dw).$$
Prove (or find a counterexample) there exist $F_0 \in \mathbb{R}_{\geq 0}$ and $F:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ continuous, strictly increasing, with $F(0)=0$, such that:
$$ \bar f(x) \ \leq \ F_0 + F(|x|) \quad \forall x \in \mathbb{X}$$
(for some norm $|\cdot|$)
EDIT: $(x,w) \mapsto f(x,w)$ locally bounded.