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