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

## Examples

## See also

## References

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

- $forall\; p\; in\; X,\; exists\; N\_\{p\}\; in\; T\; mbox\{\; s.t.\; \}\; p\; in\; N\_\{p\}\; mbox\{\; and\; \}\; left|\; mu\; (N\_\{p\})\; right|\; <\; +\; infty.$

- Any probability measure on X is locally finite, since it assigns unit measure 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.
- Counting measure is sometimes locally finite and sometimes not: counting measure on the integers with their usual discrete topology is locally finite, but counting measure on the real line with its usual Borel topology is not.

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Last updated on Sunday April 20, 2008 at 13:54:32 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday April 20, 2008 at 13:54:32 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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