Pretty simple question. I know that when it comes to Riemann-integrable functions, we know that a function $f$ on a compact interval $[a, b]$ is Riemann-integrable iff $f$ is bounded and the set of discontinuities has Lebesgue measure $0$. Does an analogous characterization exist for the functions which are Riemann-Stieltjes-integrable with respect to some function $g$? Perhaps with some restrictions on $g$ like continuity? Or is it actually significantly hairier? I don't know much about Riemann-Stieltjes integrals, and Wikipedia wasn't very helpful in this regard.
Thanks!