I recently learned the basics of Riemann-Stieltjes integral, and based on the sources I found, we can define the expectation of random variables quite naturally with the R-S integrals: if $X$ is a random variable and $F_X$ is its distribution function defined as $P(X \leq x)$, then its expectation is $ E[X] = \int_{-\infty}^{\infty} x dF_X(x), $ or if $Y = g(X)$ such that $g : \mathbb{R} \rightarrow \mathbb{R}$ is measurable, then $ E[Y] = \int_{-\infty}^{\infty} g(x) dF_X(x), $ if the integral exists.
Now, consider the random variables $X_n(y)$ and $X(0)$ where $X_n(0) \rightarrow X(0)$ in the sense of weak convergence, $y \in [0,1]$, and their respective distribution functions $F_{X_n(y)}$ and $F_0$. My question is, for a fixed $x \in [0,1]$, how do I interpret the following R-S integral? Does the expectation interpretation above apply here? $$ \int_0^1 F_{X_n(y)}(x) dF_0 (y) $$
I made another post on MSE with more details on the context of the problem. The source paper claims that this integral is equal to $F_0(x)$, but I don't understand why.
This is my first post here, hopefully it's not too trivial...