Take an unbounded continuous function g which maps a random variable X and a constant $\theta$ to $\mathbb{R}^n$.
Consider now the correspondence defined by:
$B(\theta) = \{ M\, mbl\, wrt\, X: \mathbb{E}[M]=1, \mathbb{E}[M g(X,\theta)] = 0, M\geq 0\} $
Where the expectations are taken wrt $X$. Now, pick a sequence $\{\theta_j\}$ with $lim_j \theta_j = \theta$ and consider an $M$ s.t. $M\in B(\theta_j)$ $\forall j$. I wish to find assumptions on $g$ (other than boundedness) so that I can prove that $M\in B(\theta)$.
This boils down to showing that $\mathbb{E}[M g(X,\theta)] =0$, which would be trivial if I can simply bring the limit inside the expectation, i.e. if $lim_j \mathbb{E}[M g(X,\theta_j)] = \mathbb{E}[lim_j M g(X,\theta_j)]$, since $M\in B(\theta_j) \forall j$ $\implies$ $lim_j \mathbb{E}[M g(X,\theta_j)] = 0$
I cannot seem to find any way to apply standard MCT/DCT assumptions since g is unbounded. Since M is only in $L^1$ I cannot seem to find a way to apply uniform integrability either. However, since the only variation comes from a constant changing in a continuous function I cannot help but to think that there must be some assumptions which can guarantee this to work.
Any help would be appreciated!