Loeb space

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

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• Home page of Peter Loeb