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,
For every set A ⊆ Rⁿ there exists a Borel set B ⊆ Rⁿ such that A ⊆ B and μ = μ.

Notice that the set A need not be μ-measurable: μ 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 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, becomes a full measure if restricted to the Borel sets.