Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $\int_W \mu(dw) = 1$.
Consider a locally bounded function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, with compact $X \subset \mathbb{R}^n$, such that $\forall w$ $f(\cdot,w)$ is continuous, $\forall x$ $f(x,\cdot)$ is integrable.
Find $f(\cdot)$ such that $ \int_W \sup_{x \in X} f(x,w) \mu(dw) = \infty $