Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, 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:
(1) $\forall x \in X$ $\ \int_W f(x,w) m(dw) < \infty $
(2) For all $x \in X$ we have the following property.
$\forall \epsilon, \delta > 0$ $\ \exists c >0$ such that
$ \sup_{\xi \in \{x\}+\delta \overline{\mathbb{B}} } m\left( \{w \in W \mid f(\xi,w) \geq c \} \right) \leq \epsilon $
(1) Prove that (Uniform Integrability) for any fixed $x \in X$ we have:
$\forall \epsilon >0$ $ \ \exists \delta, c>0$ such that
$ \sup_{\xi \in \{x\}+\delta \overline{\mathbb{B}} } \int_{\{ f(\xi,w) \geq c \}} f(\xi,w) m(dw) \leq \epsilon $
(2) Can assuming $(x,w) \mapsto f(x,w)$ continuous help in getting the proof?
Note: this question is a variant of Question on UI.