Definition:Variation/Signed Measure

Definition

Definition 1

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

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

Let $\struct {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.

We define the variation $\size \mu$ of $\mu$ by:

$\size \mu = \mu^+ + \mu^-$

Definition 2

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

Let $\mu$ be a signed 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 $\cmod \mu : \Sigma \to \overline \R$ 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 Signed Measure is Measure

• Equivalence of Definitions of Variation of Signed Measure

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