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:

\(\ds \map {\mu_A} X\)

\(=\)

\(\ds \map \mu {A \cap X}\)

Definition of Intersection Measure

\(\ds \)

\(=\)

\(\ds \map \mu A\)

Intersection with Subset is Subset

\(\ds \)

\(\le\)

\(\ds \map \mu X\)

Measure is Monotone

\(\ds \)

\(<\)

\(\ds \infty\)

So $\mu_A$ is a finite measure.

$\blacksquare$