Let \((M, {\mathcal {A}})\) be a measurable set with the \(\sigma\)-algebra \(\mathcal{A}\,\). A function mapping \(\mathcal {A}\) to the set of positive real numbers \[\mu \colon \cases{{\mathcal {A}}\longrightarrow {\mathbb {R}}_{+},\cr T\longmapsto \mu (T)},\] is called a measure on \(M\), if \(\mu (\emptyset)=0\) and if for every countable family of mutually disjoint subsets \(T_{i}\subseteq {\mathcal {A}}\), \(i\in I\), 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 measure and a pre-measure is that a measure is defined on a \(\sigma\)-algebra while a pre-measure is defined on a ring of sets.
