Locally finite measure

Builds on Borel measure Open set Compact set Locally compact space

Definition

Let $X$ be a topological space and let $\mu$ be a Borel measure on $X$ . The measure $\mu$ is locally finite if every point $x \in X$ has an open neighbourhood $U$ with $\mu(U) < \infty$ .

Remarks

Example: Lebesgue measure $\lambda$ on $\R^n$ is locally finite. The open ball of radius $1$ around $x$ has finite measure.

Non-example: counting measure on $\R$ is not locally finite. Every open neighbourhood of a point is infinite, so it has measure $\infty$ .

A locally finite measure is finite on every compact set $K$ : finitely many neighbourhoods of finite measure cover $K$ . Conversely, if $X$ is a Hausdorff locally compact space and $\mu$ is finite on compact sets, then $\mu$ is locally finite, since each point has a compact neighbourhood.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Borel measure concept
Compact set concept
Locally compact space concept
Open set concept
Locally finite measure definition

Tag	`MW5P4P4S9A`
URI	`m2://MW5P4P4S9A`
Kind	definition
Writing	W0 stub
MSC	28C15
Defines	Locally finite measure
Uses	Borel measure, Open set, Compact set, Locally compact space
Changed	2 November 2026

Locally finite measure

Definition

Remarks

Reading path

Further reading

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