Definition: Finite and Sigma-Finite Measure

Let \((M, {\mathcal {A}})\) be a measurable set with the \(\sigma\)-algebra \(\mathcal{A}\,\). A measure \(\mu\) defined on this \(\sigma\)-algebra is called finite, if for all subsets \(T_i\in\mathcal{A}\) there exists a real number \(t_i\) with \[\mu(T_i) = t_i < \infty.\]

The measure is called \(\sigma\)-finite, if \(M\) can be written as a union of a countable family of subsets \(T_{i}\subseteq {\mathcal {A}}\), \(i\in I\) with a finite measure, formally: \[M=\left(\bigcup _{i\in I}T_{i}\right)\quad\text{with}=\mu(T_i) = t_i < \infty\quad\text{for all }i.\]

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück