Pairs $(f,g)$ for which the Riemann-Stiltjes integral $\int_a^bf(x)\,dg(x)$ exists

Question

Let us deal with $f,g:\Bbb R\to\Bbb R$.

I know that $\int_a^bf(x)\,dg(x)$ exists for $f$ continuous and $g$ of bounded variation.

But is there a way to get exactly all the pairs $(f,g)$ for which the Riemann-Stieltjes exists?

Even a good reference, a link, some pdf would be a great answer.

Answer 1

I haven't read it in a few semesters, but Rudin's Principles of Mathematical Analysis is a standard source which covers the Riemann-Stieltjes integral. Not sure if if contains the exact answer you're looking for, but it might be a good place to start.