Problem. If $f$ is absolutely continuous, there is a decomposition $f = g - h$ with $g$ and $h$ absolutely continuous inscreasing funcions and such that $g' = f'_{+}$ and $h' = f'_{-}$ where $f'_{+} = \max\{0,f'\}$ and $f'_{-} = \max\{0,-f'\}$.
My problem is show that $g$ and $h$ are absolutely continuous. The decomposition follows from: $f$ is, in particular, a bounded variation function and so, $f$ is a sum of two increasing functions. But I dont know how to show that $g$ and $h$ are absolutely continuous. If I show that, at least, $g$ or $h$ are absolutely continuous so it is enough to write $f - g = h$ or $g = f + h$. Can someone help me?
If $f$ is ac. then $f(x) = \int_0^x f'(t)dt$ for some $L^1$ function $f'$.
Then $\max(0,f')$ and $\min(f',0)$ are $L^1$ functions and $g(x) = \int_0^x \max(0,f'(t))dt$ is non decreasing, $h(x) = - \int_0^x \min(0,f'(t))dt$ is non decreasing and $f = g-h$.
$g,h$ are ac. since $\max(0,f')$ and $\min(f',0)$ are $L^1$.