In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.

Definition

Let (X, T) be a Hausdorff topological space and let \Sigma be a \sigma-algebra on X that contains the topology T (so that every open set is a measurable set, and \Sigma is at least as fine as the Borel \sigma-algebra on X). A measure/signed measure/complex measure \mu defined on \Sigma is called locally finite if, for every point p of the space X, there is an open neighbourhood N_p of p such that the \mu-measure of N_p is finite. In more condensed notation, \mu is locally finite if and only if

Examples