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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.

External links

• Home page of Peter Loeb

• Measure theory
• Nonstandard analysis