Consider a probability measure $m$ over $W \subseteq{R^m}$, so that $m(W) = 1$.
Consider a function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, with compact $X \subset \mathbb{R}^n$, such that the following proposition holds true.
For any $\epsilon > 0$ there exists $c > 0$ such that $m(\{w \in W \mid f(x,w) \geq c \}) < \epsilon \ $ for all $x \in X$.
In other words, the measure of $\{f\geq c\}$ can be made arbitrarily small, uniformly on $X$.
What are the (weakest) conditions to have the family $\{f(x,\cdot)\}_{x \in X}$ Uniformly Integrable?