$\Theta \in \mathbb{R}^d$ compact, $\rho(x,\theta): \mathbb{R}^p\times\Theta\in\mathbb{R}^+$ continuous in $\theta \in \Theta$ for all $x$.
If $u(x,\theta, \eta)=\sup_{v\in B(\theta, \eta)}\rho(x,v)$ and $l(x,\theta, \eta)=\inf_{v\in B(\theta, \eta)}\rho(x,v)$, then $E(|u(x,\theta, \eta|)<\infty$ and $E(|u(x,\theta, \eta|)<\infty$
Because of the very definition of $u$ and $l$ follows naturally that $l(x,\theta,v)\leq \rho (x,v) \leq u(x,\theta,v)$, then $E(|l(x,\theta, \eta)|)<E(|u(x,\theta, \eta|))<\infty$, so it suffices to show that $E(|u(x,\theta, \eta|)<\infty$. But I don't think this is true in general.
Could it be true applying some additional conditions? I am thinking of $E(u)<\infty$.