Let $\mu(\cdot)$ be a probability measure in $X$.
Consider a function $f: Z \times X \rightarrow \mathbb{R}_{\geq 0}$, with $Z \subset \mathbb{R}^n$ compact, such that:
$\forall x \in X, \quad z \mapsto f(z,x)$ is continuous;
$\forall z \in Z, \quad x \mapsto f(z,x)$ is measurable;
The family $\mathcal{F} = \{ f(z,\cdot) \}_{z \in Z}$ is a subset of $L^1(\mu)$:
\max_{z \in Z} \int_X f(z,x) \mu(dx) < \infty
Is the closure of $\mathcal{F}$ in the weak topology of $L^1(\mu)$ compact?