When I read statistics papers, I often see $$\int g(x) dF_n(x) - \int g(x) dF(x) = TV([F_n(x)-F(x)]g(x)) + \int(F_n(x)-F(x))dg(x), $$ where $TV(\cdot)$ is total variation, $F_n(x)$ is an empirical distribution of $x$, $F(x)$ is a population distribution of $x$ and $g$ is a measurable function.
How does above equality holds? Also, how should I interpret $dg(x)$?