Loeb space

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

• {{Citation | last1=Cutland | first1=Nigel J. |authorlink = Nigel J. Cutland| title=Loeb Measures in Practice: Recent Advances | doi=10.1007/b76881 | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-41384-4 |mr=1810844 | year=2000 | volume=1751}}
• {{Citation | last1=Goldblatt | first1=Robert |authorlink = Robert Goldblatt| title=Lectures on the hyperreals | url=https://books.google.com/books?id=TII-PX_OdloC | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98464-3 |mr=1643950 | year=1998 | volume=188 | doi=10.1007/978-1-4612-0615-6}}
• {{cite journal |last=Loeb |first=Peter A. |title=Conversion from nonstandard to standard measure spaces and applications in probability theory |jstor=1997222 |mr=0390154 |year=1975 |journal=Transactions of the American Mathematical Society |issn=0002-9947 |volume=211 |pages=113–22 |doi=10.2307/1997222 |doi-access=free }}

• [http://www.math.uiuc.edu/~loeb/ Home page of Peter Loeb]

Category:Measure theory

Category:Nonstandard analysis