Measure is Finitely Additive Function

Theorem
Let $\Sigma$ be a $\sigma$-algebra on a set $X$.

Let $\mu: \Sigma \to \overline {\R}$ be a measure on $\Sigma$.

Then $\mu$ is finitely additive.

Proof
From the definition of a measure, $\mu$ is countably additive.

From Countably Additive Function also Finitely Additive, $\mu$ is finitely additive.