Consider a real-valued random variable $X_i$ defined on the probability space $(\Omega, \mathcal{F}, P)$.
Let $m(X_i;\theta)$ be a random function which depends on the parameter $\theta \in \Theta \subseteq \mathbb{R}$.
(*) Suppose that for every sufficiently small open interval $(a,b) \in \Theta$ $E_P(\sup_{\theta \in (a,b)}m(X_i;\theta))<\infty$.
Define $\Theta_0:=\{\theta_0 \in \Theta \text{ s.t. } E_P(m(X_i; \theta_0))=\sup_{\theta \in \Theta}E_P(m(X_i; \theta))\}$
I want to show that (*) implies that $E_P(m(X_i; \theta_0))<\infty$ $\forall \theta_0 \in \Theta_0$
I think that one way to proceed is by exchanging expectation and supremum but I'm not quite sure.