Locally finite measure
Let (X, T) be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T (so that every open set is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). A measure/signed measure/complex measure μ defined on Σ is called locally finite if, for every point p of the space X, there is an open neighbourhood Np of p such that the μ-measure of Np is finite.
In more condensed notation, μ is locally finite if and only if
- Any probability measure on X is locally finite, since it assigns unit measure to the whole space. Similarly, any measure that assigns finite measure to the whole space is locally finite.
- Lebesgue measure on Euclidean space is locally finite.
- By definition, any Radon measure is locally finite.
- The counting measure is sometimes locally finite and sometimes not: the counting measure on the integers with their usual discrete topology is locally finite, but the counting measure on the real line with its usual Borel topology is not.