Absolute Value of Signed Measure Bounded Above by Variation

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 $\mu$.

Then:


 * $\size {\map \mu A} \le \map {\size \mu} A$

for each $A \in \Sigma$.

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

Then:


 * $\mu = \mu^+ - \mu^-$

and:


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

We have: