Borel regular measure
In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called a Borel regular measure 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
- Evans, Lawrence C.; Gariepy, Ronald F. (1992). Measure theory and fine properties of functions. CRC Press. ISBN 0-8493-7157-0.
- Taylor, Angus E. (1985). General theory of functions and integration. Dover Publications. ISBN 0-486-64988-1.
- Fonseca, Irene; Gangbo, Wilfrid (1995). Degree theory in analysis and applications. Oxford University Press. ISBN 0-19-851196-5.
This article is issued from Wikipedia - version of the 4/11/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.