Variation of Signed Measure is Measure

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 $\size \mu$ is a measure.

Proof

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

Then:

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

So $\size \mu$ is a measure from Linear Combination of Measures.

$\blacksquare$