Assume that I have a random variable $Z \sim H$, and there is a second random variable $Y$ with distribution $G$ and $Y|Z \sim F$, then $$ G(y) = \int F(y|z)dH(z) $$ Now, maybe $G$ is not known analytically, and I want to estimate $G$ based on knowing $F$ and $H$, and having access to a stationary and ergodic sample $Z_1,...,Z_n$ from $H$.
A natural estimator is $$ \hat G_n(y) = \frac{1}{n} \sum_{i=1}^n F(y|z_i). $$
Assuming that the first moments of all involved distributions are finite, does it hold that the expected value of $\hat G_n$ converges to the expected value of $G$, and how can I show this?