Hahn-Kolmogorov theorem

Let ${𝒜}_0$ be an algebra of subsets of a set $X$. If a finitely additive measure ${\mu }_0:𝒜\to ℝ\cup \{\mathrm{\infty }\}$ satisfies

$${\mu }_0(\bigcup _{n=1}^{\mathrm{\infty }}{A}_n)=\sum _{n=1}^{\mathrm{\infty }}{\mu }_0(A_n)$$

for any disjoint family $\{A_n:n\in \mathbb{N}\}$ of elements of ${𝒜}_0$ such that ${\cup }_{n=0}^{\mathrm{\infty }}{A}_n\in {𝒜}_0$, then ${\mu }_0$extends to a measure defined on the $\sigma$-algebra $𝒜$ generated by ${𝒜}_0$; i.e. there exists a measure $\mu :𝒜\to ℝ\cup \{\mathrm{\infty }\}$ such that its restriction (http://planetmath.org/RestrictionOfAFunction) to ${𝒜}_0$ coincides with ${\mu }_0$.

If ${\mu }_0$ is $\sigma$-finite (http://planetmath.org/SigmaFinite), then the extension is unique.