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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

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  • $\begingroup$ Since $g$ is bounded, if you view it as a limit of step functions you should get a bound for the $q$-variation of $H$, regardless of $p$, as long as $q\geq 1$. Is this the sort of thing you want? $\endgroup$
    – Will Sawin
    Jul 3, 2012 at 22:43
  • $\begingroup$ Can you give a more detailed explanation? I have got the following aim: I want to integrate with respect to $H$. So I want to consider functions $\Psi(t):=\int_{t}^{a}\alpha(s)dH(s)$. So I need to know the smallest number $r$ such that $Var_{r}(H,[a,b])$ is finite.After that I could answer the question for which $\alpha$ the function $\Psi$ is well defined. This would follow from the condition $\frac{1}{w}+\frac{1}{r}>1$ $\endgroup$
    – Peter Moor
    Jul 3, 2012 at 23:05

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