Let be an algebra of subsets of a set $X$ . If a finitely additive measure satisfies $$\mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)$$ for any disjoint family $\{A_n:n\in \N\}$ of elements of such that , then $\mu_0$ extends to a measure defined on the $\sigma$ -algebra generated by ; i.e. there exists a measure such that its restriction to coincides with $\mu_0$ .

If $\mu_0$ is $\sigma$ -finite, then the extension is unique.