In mathematics, an outer measure μ on n-dimensional Euclidean space R<sup>n</sup> is called a Borel regular measure if the following two conditions hold:
- Every Borel set B ⊆ R<sup>n</sup> is μ-measurable in the sense of Carathéodory's criterion: for every A ⊆ R<sup>n</sup>,
::<math>\mu (A) = \mu (A \cap B) + \mu (A \setminus B).</math>
- For every set A ⊆ R<sup>n</sup> there exists a Borel set B ⊆ R<sup>n</sup> 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<sup>n</sup> 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.