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 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.

ee also

* Inner regular measure
* Strictly positive measure

References