The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that:
A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact.
Now $X_n \in L^1(\mu)$ means that $\sup_n \int_\mathbb{X} |X_n(x)| \mu(dx) < \infty$.
A family $\mathcal{X}$ is relatively weakly compact if for every sequence $\{X_n\}$ in $\mathcal{X}$ there exist a subsequence $\{X_{\tilde{n}}\}$ and $X_* \in \mathcal{L}^1(\mu)$ such that $$ \lim_{\tilde n} \int_A X_{\tilde n}(x) \mu(d x) = \int_A X_*(x) \mu(dx) \quad \forall A \subseteq \mathbb{X} \ s.t. \int_A \mu(dx) > 0 $$
Given $Y: \mathbb{R}^m \times \mathbb{X} \rightarrow \mathbb{R}_{\geq 0} $ continuous in the first argument, locally bounded in the second, is the family $\{ Y(n,\cdot) \}_{n \in \mathcal{N}}$, $Y(n,\cdot) \in L^1(\mu)$, being $\mathcal{N}$ is compact, "relatively weakly compact"?