In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis.
Loeb's construction starts with a finitely additive map
of sets to the nonstandard reals.
is a finitely additive map from
-algebra as it is not usually closed under countable unions.
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
to the extended reals is automatically countably additive.
Then by Carathéodory's extension theorem the measure
extends to a countably additive measure on
, called a Loeb measure.