Equivalence of Definitions of Concentration of Complex Measure on Measurable Set

Theorem
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 $\mu$.

Let $E \in \Sigma$.

Proof
From Characterization of Null Sets of Variation of Complex Measure, we have that:


 * $\map {\size \mu} {E^c} = 0$ :


 * for each $\Sigma$-measurable set $A \subseteq E^c$, we have $\map \mu A = 0$.

Hence the desired equivalence.