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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: | |||
* 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>, | |||
::<math>\mu (A) = \mu (A \cap B) + \mu (A \setminus B).</math> | |||
* 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''). | |||
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]]''. | |||
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 | |||
| last = Evans | |||
| first = Lawrence C. | |||
| coauthors = Gariepy, Ronald F. | |||
| title = Measure theory and fine properties of functions | |||
| publisher = CRC Press | |||
| year = 1992 | |||
| pages = | |||
| isbn = 0-8493-7157-0 | |||
}} | |||
*{{cite book | |||
| last = [[Angus E. Taylor|Taylor]] | |||
| first = Angus E. | |||
| title = General theory of functions and integration | |||
| publisher = Dover Publications | |||
| year = 1985 | |||
| pages = | |||
| isbn = 0-486-64988-1 | |||
}} | |||
*{{cite book | |||
| last = Fonseca | |||
| first = Irene | authorlink = Irene Fonseca | |||
| coauthors = Gangbo, Wilfrid | |||
| title = Degree theory in analysis and applications | |||
| publisher = Oxford University Press | |||
| year = 1995 | |||
| pages = | |||
| isbn = 0-19-851196-5 | |||
}} | |||
[[Category:Measures (measure theory)]] | |||
Revision as of 05:53, 7 March 2013
In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called Borel regular if the following two conditions hold:
- Every Borel set B ⊆ Rn is μ-measurable in the sense of Carathéodory's criterion: for every A ⊆ Rn,
- For every set A ⊆ Rn (which need not be μ-measurable) there exists a Borel set B ⊆ Rn such that A ⊆ B and μ(A) = μ(B).
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.
The Lebesgue outer measure on Rn 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 countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets.
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534