absolutely continuous

Let $\mu$ and $\nu$ be signed measures or complex measures on the same measurable space$(\mathrm{\Omega },𝒮)$. We say that $\nu$ is absolutely continuouswith respect to $\mu$ if, for each $A\in 𝒮$ 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 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 $\epsilon >0$, there is a $\delta >0$ such that whenever .