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 σ \sigma -algebra, the algebra \( {\mathcal {A}} \) need not be an ordinary σ \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 M {\mathcal {M}} to be the σ \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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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