Definition:Total Variation/Measure Theory

Definition

Signed Measure

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

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

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

We define the total variation $\norm \mu$ of $\mu$ by:

$\norm \mu = \map {\size \mu} X$

Complex Measure

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

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

Let $\cmod \mu$ be the variation of $\mu$.

We define the total variation $\norm \mu$ of $\mu$ by:

$\norm \mu = \map {\cmod \mu} X$