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| In [[mathematics]], an [[outer measure]] ''μ'' on ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> is called '''Borel regular''' if the following two conditions hold:
| | My name is Sabrina Birdsong. I life in Millbounds (Great Britain).<br><br>Also visit my website; milicainthehat.com ([http://www.milicainthehat.com/ForumRetrieve.aspx?ForumID=2825&TopicID=193044&NoTemplate=False Going in www.milicainthehat.com]) |
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| * Every [[Borel set]] ''B'' ⊆ '''R'''<sup>''n''</sup> is ''μ''-measurable in the sense of [[Carathéodory's criterion]]: for every ''A'' ⊆ '''R'''<sup>''n''</sup>,
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| ::<math>\mu (A) = \mu (A \cap B) + \mu (A \setminus B).</math>
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| * For every set ''A'' ⊆ '''R'''<sup>''n''</sup> (which need not be ''μ''-measurable) there exists a Borel set ''B'' ⊆ '''R'''<sup>''n''</sup> such that ''A'' ⊆ ''B'' and ''μ''(''A'') = ''μ''(''B'').
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| 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 is called a ''[[regular measure]]''.
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| The [[Lebesgue outer measure]] on '''R'''<sup>''n''</sup> is an example of a Borel regular measure.
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| 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.
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| ==References== | |
| *{{cite book
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| | last = Evans
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| | first = Lawrence C.
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| | coauthors = Gariepy, Ronald F.
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| | title = Measure theory and fine properties of functions
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| | publisher = CRC Press
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| | year = 1992
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| | pages =
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| | isbn = 0-8493-7157-0
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| }}
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| *{{cite book
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| | last = [[Angus E. Taylor|Taylor]]
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| | first = Angus E.
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| | title = General theory of functions and integration
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| | publisher = Dover Publications
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| | year = 1985
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| | pages =
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| | isbn = 0-486-64988-1
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| }}
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| *{{cite book
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| | last = Fonseca
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| | first = Irene | authorlink = Irene Fonseca
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| | coauthors = Gangbo, Wilfrid
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| | title = Degree theory in analysis and applications
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| | publisher = Oxford University Press
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| | year = 1995
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| | pages =
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| | isbn = 0-19-851196-5
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| }}
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| [[Category:Measures (measure theory)]]
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My name is Sabrina Birdsong. I life in Millbounds (Great Britain).
Also visit my website; milicainthehat.com (Going in www.milicainthehat.com)