Added to Favorites

Related Searches

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.

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)

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

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

Copyright © 2014 Dictionary.com, LLC. All rights reserved.