Loeb space
In mathematics, a Loeb space is a type of measure space introduced by  using nonstandard analysis.

Construction

Loeb's construction starts with a finitely additive map  from an internal algebra  of sets to the nonstandard reals. Define  to be given by the standard part of , so that  is a finitely additive map from  to the extended reals . Even if  is a nonstandard -algebra, the algebra  need not be an ordinary -algebra as it is not usually closed under countable unions. Instead the algebra  has the property that if a set in it is the union of a countable family of elements of , then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as ) from  to the extended reals is automatically countably additive. Define  to be the -algebra generated by . Then by Carathéodory's extension theorem the measure  on  extends to a countably additive measure on , called a Loeb measure.

References

External links

Home page of Peter Loeb

Measure theory
Nonstandard analysis