Let $g$ be a function of finite $q$ Variation and $f$ be a function of finite $p$ Variation, and $\frac{1}{q}+\frac{1}{p}>1$. What can be said about the variation of $H$ with $$H(t):=\int_{a}^{t}g(s)df(s)\text{ ? }$$
The $p$ variation $Var_{p}(f;[a,b])$ of a Function $f$ over $[a,b]$ is defined as: $$Var_{p}(f;[a,b])=\sup \left(\sum_{i=1}^{n}\left|f(t_{i})-f(t_{i-1})\right|^{p}\right)^{\frac{1}{p}}$$
where the supremum runs over all partitions of $[a,b]$
If $f$ would be monotone on $[a.b]$ then I could use a mean value theorem for RS Integrals. Is there a similar mean Value theorem for a function $f$ which is not monotone? I think, that then I could conclude: $H$ has finite $p$ variation.
I have alread asked this on mathstackexchange