Editing Borel regular measure

{{Short description|Type of measure on Euclidean spaces}}
{{Use American English|date = February 2019}}
In [[mathematics]], an [[outer measure]] ''μ'' on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> is called a '''Borel regular measure''' if the following two conditions hold:

* Every [[Borel set]] ''B''&nbsp;⊆&nbsp;'''R'''<sup>''n''</sup> is ''μ''-measurable in the sense of [[Carathéodory's criterion]]: for every ''A''&nbsp;⊆&nbsp;'''R'''<sup>''n''</sup>,
::<math>\mu (A) = \mu (A \cap B) + \mu (A \setminus B).</math>
* For every set ''A''&nbsp;⊆&nbsp;'''R'''<sup>''n''</sup> there exists a Borel set ''B''&nbsp;⊆&nbsp;'''R'''<sup>''n''</sup>  such that ''A''&nbsp;⊆&nbsp;''B'' and ''μ''(''A'')&nbsp;=&nbsp;''μ''(''B'').

Notice that the set ''A'' need not be ''μ''-measurable: ''μ''(''A'') is however well defined as ''μ'' is an outer measure.
An outer measure satisfying only the first of these two requirements is called a ''[[Borel measure]]'', while an outer measure satisfying only the second requirement (with the Borel set B replaced by a measurable set B) is called a ''[[regular measure]]''.

The [[Lebesgue outer measure]] on '''R'''<sup>''n''</sup> is an example of a Borel regular measure.

It can be proved that a Borel regular measure, although introduced here as an ''outer'' measure (only [[outer measure|countably ''sub''additive]]), becomes a full [[measure (mathematics)|measure]] ([[countably additive]]) if restricted to the [[Borel set]]s.

==References==
*{{cite book
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 | publisher  = CRC Press
 | year       = 1992
 | pages      = 
 | isbn       = 0-8493-7157-0
}}
*{{cite book
 | last       = Taylor
 | author-link       = Angus E. Taylor
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 | title       = General theory of functions and integration
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 | isbn       = 0-486-64988-1
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 | url       = https://archive.org/details/generaltheoryoff00tayl
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*{{cite book
 | last       = Fonseca
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 |author2=Gangbo, Wilfrid
 | title      = Degree theory in analysis and applications
 | publisher  = Oxford University Press
 | year       = 1995
 | pages      = 
 | isbn       = 0-19-851196-5
}}

{{Measure theory}}

[[Category:Measures (measure theory)]]