I'm reading an article and there is a passage that is not very clear to me. The situation is as follows:
$f$ is a continuous monotonically increasing function on $[a,b]$. Define:
$$ G := [x \in (a,b) : (M(f))'(x) = \infty] $$ Where $M(f)$ is the noncentered Hardy-Littlewood maximal operator. Suppose that $M(f)(x) > |f(x)|$ and that there exist $C > 0$ such that $M(f)(x) \leq C$ for any $x \in (a,b)$. Then, if the lebesgue measure of $M(f)(G)$ is zero, $M(f)$ is absolutely continuous.
In fact, the paper proves that $G$ = $\emptyset$. I didn't understand why the the zero measure implies absolute continuity.
