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EspañolLOCALLY 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.
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
:
# 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.
★ Inner regular measure
★ Strictly positive measure
| Contents |
| Definition |
| Examples |
| See also |
| References |
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.
See also
★ Inner regular measure
★ Strictly positive measure
References
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