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:

  • Every Borel set BRn is μ-measurable in the sense of Carathéodory's criterion: for every ARn,
  • For every set ARn (which need not be μ-measurable) there exists a Borel set BRn such that AB and μ(A) = μ(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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