Borel regular measure

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In mathematics, an outer measure μ on \mathbb R^n is called Borel regular if the following two conditions hold:

  1. every Borel set B \subset \mathbb R^n is μ-measurable
  2. for every set A \subset \mathbb R^n (which need not be μ-measurable) there exists a Borel set B \subset \mathbb R^n such that A \subset B and \mu (A) = \mu (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 \mathbb R^n is an example of a Borel regular measure.

[edit] References

  • Evans, Lawrence C.; Gariepy, Ronald F. (1992). Measure theory and fine properties of functions. CRC Press. ISBN 0849371570. 
  • Taylor, Angus E. (1985). General theory of functions and integration. Dover Publications. ISBN 0486649881. 
  • Fonseca, Irene; Gangbo, Wilfrid (1995). Degree theory in analysis and applications. Oxford University Press. ISBN 0198511965. 
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