Let $\mu$ and $\nu$ be two measure on a sigma algebra $\mathfrak {B}$ of subsets of $X$.Recall that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu(A)=0$ for any $A \in \mathfrak {B}$ such that $\mu(A)=0$
what is the motivation for this definition?why do we call it 'absolutely continuous'? I don't see any connection of above definition with the actual defition of absolute continuity of real valued function.
A necessary and sufficient condition for a finite Borel measure $\mu$ on $\mathbb{R}$ to be absolutely continuous with respect to the Lebesgue measure is that the function $$ x \mapsto \mu((-\infty, x]) $$ be absolutely continuous in the real analysis sense.