I'm trying to complete exercise 35 b) on from chapter 3.5 from Folland's Real Analysis that goes like if $F,G$ are NBV and $- \infty < a< b< \infty $ and there are not points in $[a,b]$ where both $F$ and $G$ are discontinuous then $$\int_{[a,b]} FdG + \int_{[a,b]} GdF = F(b)G(b) - F(a-)G(a-).$$ In part a of the problem we proved that
$$ \frac12\int _{[a,b]} (F(x) +F(x-)) dG(x) + \frac12 \int_{[a,b]} (G(x) +G(x-)) dF(x) = F(b)G(b) - F(a-)G(a-) $$
I proved this part and I am thinking to use to solve part b, perhaps by showing that $dF(x)$ and $dG(x)$ are $0$ on the points of discontinuity of the other function but I am not sure how to prove that because here we are working with the Lebesgue-Stieltjes measure and it is a signed measure. Perhaps there is something I am missing or an easier way to go about it?
Indeed the discontinuities of $F$ forms a $\mu_G$-null set. To prove this, recall that $G$ does not have discontinuities in common with $F$. So if $F$ is discontinuous at $x$, then $G$ is continuous at $x$ and hence $\mu_G(\{x\})=G(x+)-G(x-)=0$. Finally, since $F$ is NBV, there are only countably many points of discontinuity and $\sigma$-additivity of $\mu_G$ takes care of it.
Thus $F=(x\mapsto F(x-))$ $\mu_G$-a.e., and similarly with $F,G$ interchanged. This gives the desired result.