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:
“A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,—the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white man’s treatment of them, and probably have reason to be so.”
—Henry David Thoreau (1817–1862)
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)