Equivalence of Definitions of Concentration of Complex Measure on Measurable Set
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Theorem
Let $\struct {X, \Sigma}$ be a measurable space.
The following definitions of the concept of Concentration of Complex Measure on Measurable Set are equivalent:
Definition 1
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Let $E \in \Sigma$.
We say that $\mu$ is concentrated on $E$ if and only if:
- $\map {\size \mu} {E^c} = 0$
Definition 2
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $E \in \Sigma$.
We say that $\mu$ is concentrated on $E$ if and only if:
- for every $\Sigma$-measurable set $A \subseteq E^c$, we have $\map \mu A = 0$.
Proof
From Characterization of Null Sets of Variation of Complex Measure, we have that:
- $\map {\size \mu} {E^c} = 0$ if and only if:
- for each $\Sigma$-measurable set $A \subseteq E^c$, we have $\map \mu A = 0$.
Hence the desired equivalence.
$\blacksquare$