In mathematics, a Loeb space is a type of measure space introduced by Loeb(1975) using nonstandard analysis.

Construction

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

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