Definition:Absolute Continuity/Complex Measure

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

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

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

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

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


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

We write:


 * $\nu \ll \mu$