• Source: 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,




μ
(
A
)
=
μ
(
A
∩
B
)
+
μ
(
A
∖
B
)
.


{\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


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.

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