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 $$\nu$$ 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.