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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.

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    $x$ is fixed. I'll try to be more clear in the question.2012-06-18

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