Loeb measure
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In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using non-standard analysis.
Construction
Loeb's construction starts with a finitely additive map ν from an internal algebra A of sets to the non-standard reals. Define μ to be given by the standard part of ν, so that μ is a finitely additive map from A to the extended reals R∪∞∪–∞. Even if A is a non-standard σ-algebra, the algebra A need not be an ordinary σ-algebra as it is not usually closed under countable unions. Instead the algebra A has the property that if a set in it is the union of a countable family of elements of 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 μ) from A to the extended reals is automatically countably additive. Define M to be the σ-algebra generated by A. Then by Carathéodory's extension theorem the measure μ on A extends to a countably additive measure on M, called a Loeb measure.
References
- Cutland, Nigel J. (2000), Loeb measures in practice: recent advances, Lecture Notes in Mathematics 1751, Berlin, New York: Springer-Verlag, doi:10.1007/b76881, ISBN 978-3-540-41384-4 [Amazon-US | Amazon-UK], MR1810844
- Goldblatt, Robert (1998), Lectures on the hyperreals, Graduate Texts in Mathematics 188, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98464-3 [Amazon-US | Amazon-UK], MR1643950
- Loeb, Peter A. (1975), "Conversion from nonstandard to standard measure spaces and applications in probability theory", Transactions of the American Mathematical Society 211: 113–122, ISSN 0002-9947, MR0390154
