Equivalence of Definitions of Concentration of Signed Measure on Measurable Set

Theorem

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

The following definitions of the concept of Concentration of Signed Measure on Measurable Set are equivalent:

Definition 1

Let $\mu$ be a signed 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 signed 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 Signed 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.

$\blacksquare$