Suppose that $g$ is a monotone function on $[a,b]$ and $f$ is a bounded function defined on $[a,b]$. Show that $f$ is Riemann-Stieltjes integrable with respect to $g$ if and only if $f$ is almost everywhere continuous with respect to the measure induced by $g$
I'm trying to imagine what could possibly fail if we found a set of non-zero measure w.r.t to $m_g$ (the measure induced by $g$). I'm picturing that if such a set were to be found, then it would be impossible to find a partition that makes $\mathrm{U}(f,g,P) - \mathrm{L}(f,g,P) < \epsilon$ for a given $\epsilon>0$ because there will be a gap inside the interval $[a,b]$ that's too big and our partitioning will fail to cover that in an appropriate way. But I don't know how to write this down rigorously. I also don't know how the measure induced by $g$ looks like and I have asked a related question here.
Any help is appreciated.
For future reference, the proof of this fact has been published as an article titled Riemann-Stieltjes and Lebesgue-Stieltjes Integrability by H. J. Ter Horst in The American Mathematical Monthly. Please check Theorem C of the aforementioned article.
The DOI of the article is 10.2307/2323739 and it can be accessed online for free on JSTOR.