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 Ã¢ÂŠÂ† Rⁿ is μ-measurable in the sense of Carathéodory's criterion: for every A Ã¢ÂŠÂ† Rⁿ,

• For every set A Ã¢ÂŠÂ† Rⁿ there exists a Borel set B Ã¢ÂŠÂ† Rⁿ such that A Ã¢ÂŠÂ† B and μ(A) = ÃŽÂ¼(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.

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