Here I asked the question about approximating the function $g(x) := \mathbb{E}(f(x,Y))$, where $x \in R$ and $Y$ is a random variable. If you follow the link you will see that $g(x)$ can be approximated by finite sum $\frac{1}{n}\sum_{i=1}^nf(x,\mu_i)$ with any precision we desire.
My question: whats is the speed of the convergence? Can we deduce how large $n$ should be to make error $\leq \epsilon$
Feel free to impose any reasonable conditions on $f$ and distribution of $Y$