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,

$\mu (A)=\mu (A\cap B)+\mu (A\setminus B).$

• 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 (differing from the usual Borel measure), while an outer measure satisfying only the second requirement (with the Borel set B replaced by a μ-measurable set B) is called a regular outer 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.