Intersection Complex Measure is Complex Measure

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

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

Let $F \in \Sigma$.

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

Then $\mu_F$ is a complex measure.

Proof
Since $\mu$ is a complex measure, we have:


 * $\map \mu E \in \C$

for each $E \in \Sigma$.

So, in particular:


 * $\map \mu {E \cap F} \in \C$

for all $E \in \Sigma$.

That is:


 * $\map {\mu_F} E \in \C$

for all $E \in \Sigma$.

We verify the two conditions required of a complex 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 complex measure.