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,
- {\displaystyle \mu (A)=\mu (A\cap B)+\mu (A\setminus B).}
- For every set A ⊆ Rn there exists a Borel set B ⊆ Rn such that A ⊆ B and μ(A) = 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 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
[edit ]- 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.