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

## References

• 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.