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 B ⊆ Rn is μ-measurable in the sense of Carathéodory's criterion: for every A ⊆ Rn,
- For every set A ⊆ Rn (which need not be μ-measurable) there exists a Borel set B ⊆ Rn such that A ⊆ B 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.
Famous quotes containing the words regular and/or measure:
“This is the frost coming out of the ground; this is Spring. It precedes the green and flowery spring, as mythology precedes regular poetry. I know of nothing more purgative of winter fumes and indigestions. It convinces me that Earth is still in her swaddling-clothes, and stretches forth baby fingers on every side.”
—Henry David Thoreau (18171862)
“Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation appled to life.”
—Henri-Frédéric Amiel (18211881)