Intersection Measure of Finite Measure is Finite Measure

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

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

Let $A \in \Sigma$.

Let $\mu_A$ be the intersection measure of $\mu$ by $A$.

Then $\mu_A$ is a finite measure.

Proof
From Intersection Measure is Measure, $\mu_A$ is a measure.

Since $\mu$ is a finite measure, we have:


 * $\map \mu X < \infty$

Then, we have:

So $\mu_A$ is a finite measure.