THC Science

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

Construction[edit]

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