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:

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

Then, we have:

So $\mu$ is a signed measure.