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 [math]\displaystyle{ \nu }[/math] from an internal algebra [math]\displaystyle{ \mathcal A }[/math] of sets to the nonstandard reals. Define [math]\displaystyle{ \mu }[/math] to be given by the standard part of [math]\displaystyle{ \nu }[/math], so that [math]\displaystyle{ \mu }[/math] is a finitely additive map from [math]\displaystyle{ \mathcal A }[/math] to the extended reals [math]\displaystyle{ \overline\mathbb R }[/math]. Even if [math]\displaystyle{ \mathcal A }[/math] is a nonstandard [math]\displaystyle{ \sigma }[/math]-algebra, the algebra [math]\displaystyle{ \mathcal A }[/math] need not be an ordinary [math]\displaystyle{ \sigma }[/math]-algebra as it is not usually closed under countable unions. Instead the algebra [math]\displaystyle{ \mathcal A }[/math] has the property that if a set in it is the union of a countable family of elements of [math]\displaystyle{ \mathcal A }[/math], then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as [math]\displaystyle{ \mu }[/math]) from [math]\displaystyle{ \mathcal A }[/math] to the extended reals is automatically countably additive. Define [math]\displaystyle{ \mathcal M }[/math] to be the [math]\displaystyle{ \sigma }[/math]-algebra generated by [math]\displaystyle{ \mathcal A }[/math]. Then by Carathéodory's extension theorem the measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ \mathcal A }[/math] extends to a countably additive measure on [math]\displaystyle{ \mathcal M }[/math], 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.
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