Definition:Total Variation/Measure Theory
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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:
- $\norm \mu = \map {\cmod \mu} X$