PlanetMath: absolutely continuous

(more info)

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absolutely continuous

(Definition)

Let $\mu$ and $\nu$ be signed 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 $\vert\mu\vert(A)=0$ , it holds that $\nu(A)=0$ . This is usually denoted by $\nu \ll \mu$ .

Remarks.

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

$\nu\ll\mu$ ;
$\nu^+\ll\mu$ and $\nu^-\ll\mu$ ;
$\vert\nu\vert\ll\vert\mu\vert$ .

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