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.

## 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
* 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, https://books.google.com/books?id=TII-PX_OdloC
* 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.

## External links

* Home page of Peter Loeb

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