Definition:Total Variation/Measure Theory/Complex Measure

Definition

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:

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

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.1$: Signed and Complex Measures