Stoneham number

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,

$\mu (A)=\mu (A\cap B)+\mu (A\setminus B).$

• 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.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534