I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that
$$\int_0^t g(x) df(x)$$ is of bounded variation?
Clearly if $f \in C^1$ this holds, because then the integral is absolutely continuos and therefore of bounded variation.
Are there other easy-to-check sufficient conditions?