Loeb'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 (such as ) 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.
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