Jump to content

Loeb space

From Wikipedia, the free encyclopedia

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

Construction

[edit]

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

[edit]
  • 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.
[edit]