In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis.

ConstructionEdit

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.

ReferencesEdit

  • Cutland, Nigel J. (2000), Loeb measures in practice: recent advances, Lecture Notes in Mathematics, vol. 1751, Berlin, New York: Springer-Verlag, doi:10.1007/b76881, ISBN 978-3-540-41384-4, MR 1810844
  • Goldblatt, Robert (1998), Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0615-6, ISBN 978-0-387-98464-3, MR 1643950
  • Loeb, Peter A. (1975). "Conversion from nonstandard to standard measure spaces and applications in probability theory". Transactions of the American Mathematical Society. 211: 113–22. doi:10.2307/1997222. ISSN 0002-9947. JSTOR 1997222. MR 0390154 – via JSTOR.

External linksEdit