locally finite measure

Let $(X,\; T)$ be a Hausdorff topological space and let $\backslash Sigma$ be a $\backslash sigma$-algebra on $X$ that contains the topology $T$ (so that every open set is a measurable set, and $\backslash Sigma$ is at least as fine as the Borel $\backslash sigma$-algebra on $X$). A measure/signed measure/complex measure $\backslash mu$ defined on $\backslash Sigma$ 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 $\backslash mu$-measure of $N\_p$ is finite.

In more condensed notation, $\backslash mu$ is locally finite if and only if

1. 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.
2. Lebesgue measure on Euclidean space is locally finite.
3. By definition, any Radon measure is locally finite.
4. 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.

• {{annotated link|Inner regular measure}}
• {{annotated link|Strictly positive measure}}

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{{Measure theory}}

Category:Measures (measure theory)