Let $R$ be a ring of subsets of $X$ and let $\mu_0:[0,\infty]$. Then $\mu_0$ is a pre-measure if $\mu_0(\emptyset) = 0$ and for every sequence ${A_n}_{n\in\mathbb N}\subset R$ of pairwise disjoint subsets whose union is in $R$, \(\mu_0\left(\bigcup_{n\in\mathbb N}A_n\right) = \sum_{n\in\mathbb N} \mu_0(A_n).\) That is, $\mu_0$ is σ-additive.
Wikidata ID: Q1393014