Characterization of Null Sets of Variation of Complex Measure

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

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

Let $\size \mu$ be the variation of $\mu$.

Then $A \in \Sigma$ is such that $\map {\size \mu} A = 0$ :


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