Let $\mu$ and $\nu$ be signed measures or complex measures on the same measurable space $(\Omega, \mathscr{S})$ We say that $\nu$ is absolutely continuous with respect to $\mu$ if, for each $A\in \mathscr{S}$ such that $|\mu|(A)=0$ it holds that $\nu(A)=0$ This is usually denoted by $\nu \ll \mu$

Remarks.

If $\mu$ and $\nu$ are signed measures and $(\nu^+, \nu^-)$ is the Jordan decomposition of $\nu$ the following propositions are equivalent:

1. $\nu\ll\mu$
2. $\nu^+\ll\mu$ and $\nu^-\ll\mu$
3. $|\nu|\ll|\mu|$

If $\nu$ is a finite signed or complex measure and $\nu\ll\mu$ the following useful property holds: for each $\varepsilon>0$ there is a $\delta>0$ such that $|\nu|(E)<\varepsilon$ whenever $|\mu|(E)<\delta$