Loeb space explained
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
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.
References
- Loeb . Peter A. . Conversion from nonstandard to standard measure spaces and applications in probability theory . 1997222 . 0390154 . 1975 . . 0002-9947 . 211 . 113–22 . 10.2307/1997222 . . free .
External links
