Loeb spaceIn mathematics, a Loeb space is a type of measure space introduced by using nonstandard analysis.ConstructionLoeb's construction starts with a finitely additive map from an internal algebra of sets to the nonstandard reals. Define to be given by the standard part of, so that is a finitely additive map from to the extended reals. Even if is a nonstandard -algebra, the algebra need not be an ordinary -algebra as it is not usually closed under countable unions. Instead the algebra has the property that if a set in it is the union of a countable family of elements of, then the set is the union of a finite number of elements of the family, so in particular any finitely additive map from to the extended reals is automatically countably additive. Define to be the -algebra generated by. Then by Carathéodory's extension theorem the measure on extends to a countably additive measure on, called a Loeb measure.