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 formula_1 from an internal algebra formula_2 of sets to the nonstandard reals. Define formula_3 to be given by the standard part of formula_1, so that formula_3 is a finitely additive map from formula_2 to the extended reals formula_7. Even if formula_2 is a nonstandard formula_9-algebra, the algebra formula_2 need not be an ordinary formula_9-algebra as it is not usually closed under countable unions. Instead the algebra formula_2 has the property that if a set in it is the union of a countable family of elements of formula_2, then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as formula_3) from formula_2 to the extended reals is automatically countably additive. Define formula_16 to be the formula_9-algebra generated by formula_2. Then by Carathéodory's extension theorem the measure formula_3 on "formula_2" extends to a countably additive measure on formula_16, called a Loeb measure.