Law of large numbers: average value of a function approaching the expected value when the domain is sampled according to a different distribution

Question

Let $x_1$, $x_2$, ... , $x_n$ be sampled according to some distribution $f$.

Will the law of large numbers hold for some function $g(x)$ where

$$\frac{g(x_1) + g(x_2) + \cdots + g(x_n)}{n} \longrightarrow E[g(x)]\ ?$$

If so how? Will this work for all sampling distributions $f$?