Suppose we have a conditional expectation, that is, an integral of this form $\int g(x,y)f(y|x)dy$, where $g(x,y)$ and $f(y|x)$ are continuos functions in their arguments, and $f(y|x)$ is a conditional density.
I have often seen stated in papers that if $|g(x,y)| < M$ then we can switch the limit and integration operators, that is
$$\lim_{x \to x_0} \int g(x,y)f(y|x)dy = \int \lim_{x \to x_0}g(x,y)f(y|x)dy$$.
It is probably trivial, but I don't see how this is a sufficient condition. I know that the dominated convergence theorem requires the existence of a dominating function of the integrand. However, here the condition is only on $g(x,y)$, while I think there should be another on the density $f(x|y)$.
I thought it could be an application of the Generalized Dominated Convergence Theorem, as $|g(x,y)f(y|x)| < Mf(y|x)$, where $\lim_{x \to x_0}\int Mf(y|x)dy = \int\lim_{x \to x_0}Mf(y|x)dy$ by Scheffè's Lemma, but I'm not sure.
Thank you in advance for any help!