Let $m$ be a probability measure on the compact set $W \subset \mathbb{R}^m$, so that $m(W)=1$.
Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that
$\forall w \in W$ $\ x \mapsto f(x,w)$ is continuous;
$\forall x \in X$ $\ w \mapsto f(x,w)$ is measurable.
Assume that for any $x \in X$, we have $\int_W f(x,w) m(dw) < \infty$.
Say if the following property is true.
There exists $\delta>0$ such that the family of functions $\{ w \mapsto f(\xi,w) \mid \xi \in \{x\} + \delta \overline{\mathbb{B}} \} $ is Uniform Integrable.