Let $\alpha:[a,b]\to\mathbb{R}$ be a function of bounded variation on $[a,b]$ and $f:[a,b]\to\mathbb{R}$ a bounded function.
It is well known that if $f$ is Riemann-Stieljes integrable respect to $\alpha$, then $f$ is Riemann-Stilejes integrable with respect to $V(x):=V_a^x(\alpha)$ (the total variation of $\alpha$ in $[a,x]$).
But, what about the reciprocal? That, is. If $f$ is Riemann-Stieljes integrable with respct to $V$, it is true that $f$ must be integrable with respect to $\alpha$?
I tried some standard calculations, but i cannot arrive to something useful so i wonder if there is some counterexample that i cannot figure out.
EDIT: I am quite sure that the reciprocal isn't true, so I reformulate my question to the next one.
Do there exist functions $f,\alpha:[a,b]\to\mathbb{R}$ enjoying the next threee properties?
- $\alpha$ is of bounded variation.
- $f$ is Riemann-Stieljes integrable with respect to $V_\alpha(x)$ (the total variation function of $\alpha$).
- $f$ is NOT Riemann-Stieljes integrable with respect to $\alpha$.