Definition:Absolute Continuity/Measure

Definition

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

Let $\mu$ and $\nu$ be measures on $\struct {X, \Sigma}$.

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

$\nu \ll \mu$

if and only if:

$\ds \forall A \in \Sigma : \map \mu A = 0 \implies \map \nu A = 0$

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absolutely continuous: 2.
• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.2$