It isn't clear to me how to tweak Monte Carlo integration to evaluate Riemann-Stieltjes integrals.
So I have the following integral over $S$ which is some compact subset of $\mathbb{R}^n$ where $n$ is quite large (like 20 or so).
$$I = \int_{S} g(x) dF(x)$$
In particular I'm assuming that $F$ is some distribution function for a random variable $X$. $F$ is not well behaved. Namely it might not be continuous. I can sample from $X$ but computing specific values of $F$ is hard/intractable to do directly. If the integral was just $\int_{\mathcal{S}} g(x) dx$ I could approximate the integral with $N$ samples as from some distribution (e.g. not the same as $F$) with PMF $p$ (ideally $p$ proportional to $g$) with samples $x_1,\ldots,x_N$:
$$\frac{1}{N}\sum_{i=1}^N \frac{g(x_i)}{p(x_i)}$$
But that doesn't account for $F$. It's not really clear to me how to tweak standard Monte Carlo integration to account for $F$ however. If we sample according to $F$ we can just use the following simple formula
$$I \approx \frac{1}{N}\sum_{i=1}^N g(x_i)$$
But if $g$ is only large on very specific values this is going to take a ton of samples to converge.
Is there any source that explains how to do this?