Intersection Measure is Measure

Theorem
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

Let $F \in \mathcal A$ be a measurable set.

Then the intersection measure $\mu_F$ is a measure on the measurable space $\left({X, \mathcal A}\right)$.

Proof
Verify the axioms for a measure in turn for $\mu_F$:

$(1)$
For every $A \in \mathcal A$ have:

$(2)$
For every sequence $\left({A_n}\right)_{n \in \N}$ in $\mathcal A$, have:

$(3')$
By Intersection with Null, $\varnothing \cap F = \varnothing$. Hence:


 * $\mu_F \left({\varnothing}\right) = \mu \left({\varnothing \cap F}\right) = 0$

because $\mu$ is a measure.

Having verified explicitly the conditions, conclude that $\mu_F$ is a measure.