Definition:Variation/Complex Measure

Definition

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

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

Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.

We define the variation $\cmod \mu : \Sigma \to \R$ of $\mu$ by:

$\ds \map {\cmod \mu} A = \sup \set {\sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } : \set {A_1, A_2, \ldots, A_n} \in \map P A}$

for each $A \in \Sigma$, where the supremum is taken in the set of extended real numbers $\overline \R$.

Also see

• Variation of Finite Signed Measure as Signed Measure coincides with Variation as Complex Measure shows that if a complex measure $\mu$ is also a signed measure, its variation coincides with the definition of the variation of a signed measure.
• Variation of Complex Measure is Finite Measure shows that the variation of a complex measure $\mu$ is a finite measure.

• Results about the variation of a complex measure can be found here.

Sources

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