# Locally finite measure

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 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 *N*_{p} of *p* such that the *μ*-measure of *N*_{p} is finite.

In more condensed notation, *μ* is locally finite if and only if

## Examples

- 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.

## References

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