Locally finite measure
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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.
[edit] Definition
Let <math>(X, \mathcal{T})</math> be a topological space, and let <math>\mathcal{F}</math> be a sigma algebra on <math>X</math> that contains the topology <math>\mathcal{T}</math> (so that all open sets are measurable sets). A measure <math>\mu : \mathcal{F} \to [0, + \infty]</math> is called locally finite if, for every point <math>p \in X</math>, there is an open neighbourhood <math>p \in N_{p} \in \mathcal{T}</math> such that <math>\mu (N_{p}) < + \infty.</math>
[edit] Examples
- Any probability measure on <math>X</math> is locally finite, since it assigns the whole space <math>X</math> unit measure.
- Lebesgue measure on Euclidean space is locally finite.
- More generally, any Radon measure is locally finite, by the definition of a Radon measure.
- Counting measure is sometimes locally finite and sometimes not: counting measure on the integers with their discrete topology is locally finite, but counting measure on the real line with its Borel topology is not.
