Borel regular measure

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

\mu (A) = \mu (A \cap B) %2B \mu (A \setminus B).

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 (only countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets.

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