Concept

# Borel regular measure

Summary
In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called a Borel regular measure 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, ::\mu (A) = \mu (A \cap B) + \mu (A \setminus B).
• For every set A ⊆ Rn there exists a Borel set B ⊆ Rn such that A ⊆ B and μ(A) = μ(B).
Notice that the set A need not be μ-measurable: μ(A) is however well defined as μ is an outer measure. 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 (with the Borel set B replaced by a measurable set B) 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)
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

Related people

Related units