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

Follows as a corollary of Countably Additive Function also Finitely Additive.

$\blacksquare$

Sources

• René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $4.3 \ \text{(i)}$