Assume that the function $f:\mathbb{R}\rightarrow \mathbb{R}$ has bounded variation on finite intevals, and is right continuous. It can then be written as the difference of two increasing functions: $f = f_1-f_2$.
Assume that $g$ is a borel-measurable function. Let $\mu_1,\mu_2$ be the Lebesgue-Stieltjes integrals generated by $f_1,f_2$.
Do we then have that $|\int_0^\infty g d\mu_1-\int_0^\infty g d\mu_2|\le\int_0^\infty|g|dV$. Where V is the Lebesgue-stieltjes measure generated by $Var(f)$, the variation of f.