Loeb space

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

Construction

Loeb's construction starts with a finitely additive map ν from an internal algebra A of sets to the nonstandard reals. Define μ to be given by the standard part of ν, so that μ is a finitely additive map from A to the extended reals R∪∞∪–∞. Even if A is a nonstandard σ-algebra, the algebra A need not be an ordinary σ-algebra as it is not usually closed under countable unions. Instead the algebra A has the property that if a set in it is the union of a countable family of elements of A, then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as μ) from A to the extended reals is automatically countably additive. Define M to be the σ-algebra generated by A. Then by Carathéodory's extension theorem the measure μ on A extends to a countably additive measure on M, called a Loeb measure.

gollark: Oh, and it goes for looking simple over actually being simple and consistent, hence the horrible verbosity.
gollark: *cough*generics*cough*
gollark: I agree with the Go thing. It seems to be designed so that you can pick it up quickly, but also seems to have the mentality that the people actually using it (as opposed to the implementors) can't be trusted with any advanced features.
gollark: People *use* LISPs?
gollark: I still don't like it. It's just less bad.

References
  • Cutland, Nigel J. (2000), Loeb measures in practice: recent advances, Lecture Notes in Mathematics, 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, 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.