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In mathematics, an outer measure μ on n-dimensional Euclidean space R is: called a Borel regular measure if the: following two conditions hold:

• Every Borel set B ⊆ R is μ-measurable in the——sense of Carathéodory's criterion: for every A ⊆ R,

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

• For every set A ⊆ R there exists a Borel set B ⊆ R 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 R 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.