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Let $f:[a,b] \rightarrow \mathbb R$ be of bounded variation. Let $V(f)_a ^x$ denote its total variation from $a$ to $x$. Define

$f_1(x) = \frac{V(f)_a ^x + f(x)}{2}$


$f_2(x) = \frac{V(f)_a ^x - f(x)}{2}$.

Then clearly $f = f_1 - f_2$. It is easily verified that $f_1$ and $f_2$ are increasing. I am trying to prove that the Lebesgue-Stieltjes measures induced by $f_1$ and $f_2$ are mutually singular.

So far, I have been able to show that $f_1$ is the supremum of the "positive variation" of $f$ over partitions and $f_2$ is the supremum of the negative variation. However, I am unsure how to turn this into a proof of mutual singularity. Any help would be appreciated.

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