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:
““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
“What was that?” inquired Alice.
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—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“In abnormal times like our own, when institutions are changing rapidly in several directions at once and the traditional framework of society has broken down, it becomes more and more difficult to measure any type of behavior against any other.”
—John Dos Passos (1896–1970)