Definition:Absolute Continuity/Signed Measure

Definition

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a measure on $\struct {X, \Sigma}$.

Let $\nu$ be a signed measure on $\struct {X, \Sigma}$.

Let $\size \nu$ be the variation of $\nu$.

We say that $\nu$ is absolutely continuous with respect to $\mu$ if and only if:

$\size \nu$ is absolutely continuous with respect to $\mu$.

We write:

$\nu \ll \mu$

Also see

• Results about absolutely continuous signed measures can be found here.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.2$: Absolute Continuity