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

External links

• Home page of Peter Loeb