Inmathematics, alocally finite measure is ameasure for which every point of themeasure space has aneighbourhood offinite measure.^[1][2]

Let${\displaystyle (X,T)}$ be aHausdorfftopological space and let${\displaystyle \mathrm{\Sigma }}$ be a${\displaystyle \sigma }$-algebra on${\displaystyle X}$ that contains the topology${\displaystyle T}$ (so that everyopen set is ameasurable set, and${\displaystyle \mathrm{\Sigma }}$ is at least as fine as theBorel${\displaystyle \sigma }$-algebra on${\displaystyle X}$). A measure/signed measure/complex measure${\displaystyle \mu }$ defined on${\displaystyle \mathrm{\Sigma }}$ is calledlocally finite if, for every point${\displaystyle p}$ of the space${\displaystyle X,}$ there is an openneighbourhood${\displaystyle N_p}$ of${\displaystyle p}$ such that the${\displaystyle \mu }$-measure of${\displaystyle N_p}$ is finite.

In more condensed notation,${\displaystyle \mu }$ is locally finiteif and only if

1. Anyprobability measure on${\displaystyle X}$ 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 onEuclidean space is locally finite.
3. By definition, anyRadon measure is locally finite.
4. Thecounting measure is sometimes locally finite and sometimes not: the counting measure on theintegers with their usualdiscrete topology is locally finite, but the counting measure on thereal line with its usual Boreltopology is not.

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• Measures (measure theory)

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