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:

\(\ds \map \mu {T \setminus S}\)

\(=\)

\(\ds \map {\mu^+} {T \setminus S} - \map {\mu^-} {T \setminus S}\)

Definition of Jordan Decomposition

\(\ds \)

\(=\)

\(\ds \paren {\map {\mu^+} T - \map {\mu^+} S} - \paren {\map {\mu^-} T - \map {\mu^-} S}\)

Measure of Set Difference with Subset

\(\ds \)

\(=\)

\(\ds \paren {\map {\mu^+} T - \map {\mu^-} T} - \paren {\map {\mu^+} S - \map {\mu^-} S}\)

\(\ds \)

\(=\)

\(\ds \map \mu T - \map \mu S\)

Definition of Jordan Decomposition

$\blacksquare$