Let \(M\) be a set and let \({\mathcal {P}}\) be a ring of sets defined on \(M\). A function mapping \(\mathcal {P}\) to the set of positive real numbers \[\mu\colon \cases{{\mathcal {P}}\longrightarrow \mathbb {R_{+}},\cr T\longmapsto \mu (T),}\] is called a pre-measure on \(M\), if \(\mu (\emptyset)=0\) and if for every countable family of mutually disjoint subsets \(T_{i}\subseteq {\mathcal {P}}\), \(i\in I\), the measure of the union of these subsets equals the sum of the measures of each subset, formally: \[\mu \left(\bigcup _{i\in I}T_{i}\right)=\sum _{i\in I}\mu (T_{i}).\] The second property is called \(\sigma\)-additivity.
Please note that the only difference between a pre-measure and a measure is that a pre-measure is defined on a ring of sets, while a measure is defined on a \(\sigma\)-algebra.
