Let \(M\) be a set and let \({\mathcal {P}}\) be a ring of sets defined on \(M\). A pre-measure \(\mu\) defined on this ring of sets is called finite, if for all subsets \(T_i\in\mathcal{P}\) there exists a real number \(t_i\) with \[\mu(T_i) = t_i < \infty.\]
The pre-measure is called \(\sigma\)-finite, if \(M\) can be written as a union of a countable family of subsets \(T_{i}\subseteq {\mathcal {P}}\), \(i\in I\) with a finite pre-measure, formally: \[M=\left(\bigcup _{i\in I}T_{i}\right)\quad\text{with}=\mu(T_i) = t_i < \infty\quad\text{for all }i.\]