Intersection Signed Measure is Signed Measure

Theorem

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

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

Let $F \in \Sigma$.

Let $\mu_F$ be the intersection signed measure of $\mu$ by $F$.

Then $\mu_F$ is a signed measure.

Proof

Since $\mu$ is a signed measure it takes values in either $\overline \R \setminus \set \infty$ or $\overline \R \setminus \set {-\infty}$.

That is:

$\map \mu E \in \overline \R \setminus \set \infty$ for each $E \in \Sigma$

or:

$\map \mu E \in \overline \R \setminus \set {-\infty}$ for each $E \in \Sigma$.

In particular:

$\map {\mu_F} E = \map \mu {E \cap F} \in \overline \R \setminus \set \infty$ for each $E \in \Sigma$

or:

$\map {\mu_F} E = \map \mu {E \cap F} \in \overline \R \setminus \set {-\infty}$ for each $E \in \Sigma$.

Now we verify the two conditions required of a signed measure.

We have:

\(\ds \map {\mu_F} \O\)

\(=\)

\(\ds \map \mu {F \cap \O}\)

Definition of Intersection Signed Measure

\(\ds \)

\(=\)

\(\ds \map \mu \O\)

Intersection with Empty Set

\(\ds \)

\(=\)

\(\ds 0\)

Definition of Signed Measure

Now let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint sets in $\Sigma$.

Then, we have:

\(\ds \map {\mu_F} {\bigcup_{n \mathop = 1}^\infty S_n}\)

\(=\)

\(\ds \map \mu {F \cap \bigcup_{n \mathop = 1}^\infty S_n}\)

Definition of Intersection Signed Measure

\(\ds \)

\(=\)

\(\ds \map \mu {\bigcup_{n \mathop = 1}^\infty \paren {F \cap S_n} }\)

Intersection Distributes over Union

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \map \mu {F \cap S_n}\)

using countable additivity of $\mu$

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \map {\mu_F} {S_n}\)

Definition of Intersection Signed Measure

So $\mu$ is a signed measure.

$\blacksquare$