Definition (Absolute continuity of measures)

Let $\mu,\nu$ be measures on $(X,\mathcal{F})$, $\mu$ is said to be absolutely continuous, ($\mu\ll\nu$), with respect to $\nu$ if $$\forall A\in\mathcal{F}:\quad\nu(A)=0\implies\mu(A)=0$$

Definition (Absolute continuity of functions)

$F:\mathbb{R}\to \mathbb{R}$ is said to be absolutely continuous if and only if $\forall\epsilon>0$, $\exists\delta$ such that whenever $\{ (a_{j},b_{j}) \}_{j=1}^{n}$ are pairwise disjoint intervals with $$\sum_{j=1}^{n}(b_{j}-a_{j})<\delta$$then we have $$\sum_{j=1}^{n}|F(b_{j})-F(a_{j})|<\epsilon$$