Measure of Set Difference with Subset/Signed Measure

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

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

Let $S, T \in \Sigma$ be such that $S \subseteq T$ with $\size {\map \mu S} < \infty$.

Then:


 * $\map \mu {T \setminus S} = \map \mu T - \map \mu S$

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

Then, since:


 * $\size {\map \mu S} < \infty$

We have:


 * $\map {\mu^+} S < \infty$ and $\map {\mu^-} S < \infty$

Then, we have: