Absolute Continuity of Measures

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space and let $\mu$ be a measure on $(X,\mathcal{F})$.

(i) Absolute continuity of a measure. A measure $\nu$ on $(X,\mathcal{F})$ is absolutely continuous with respect to $\mu$, written $\nu\ll\mu$, if $\nu(A)=0$ for every $A\in\mathcal{F}$ with $\mu(A)=0$.

(ii) Absolute continuity of a finite signed measure. A finite signed measure $\nu$ on $(X,\mathcal{F})$ is absolutely continuous with respect to $\mu$, written $\nu\ll\mu$, if $\nu(A)=0$ for every $A\in\mathcal{F}$ with $\mu(A)=0$.