Loeb space: Difference between revisions
No edit summary |
Vitamindeth (talk | contribs) m Latex. |
||
Line 3: | Line 3: | ||
==Construction== |
==Construction== |
||
Loeb's construction starts with a [[finitely additive]] map |
Loeb's construction starts with a [[finitely additive]] map <math>\nu</math> from an [[internal set|internal]] algebra <math>\mathcal A</math> of sets to the [[hyperreal number|nonstandard real]]s. Define <math>\mu</math> to be given by the standard part of <math>\nu</math>, so that <math>\mu</math> is a finitely additive map from <math>\mathcal A</math> to the [[Extended real number line|extended reals]] <math>\overline\mathbb R</math>. Even if <math>\mathcal A</math> is a nonstandard <math>\sigma</math>[[σ-algebra|-algebra]], the algebra <math>\mathcal A</math> need not be an ordinary <math>\sigma</math>-algebra as it is not usually closed under countable unions. Instead the algebra <math>\mathcal A</math> has the property that if a set in it is the union of a countable family of elements of <math>\mathcal A</math>, then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as <math>\mu</math>) from <math>\mathcal A</math> to the extended reals is automatically countably additive. Define <math>\mathcal M</math> to be the <math>\sigma</math>-algebra generated by <math>\mathcal A</math>. Then by [[Carathéodory's extension theorem]] the measure <math>\mu</math> on ''<math>\mathcal A</math>'' extends to a countably additive measure on <math>\mathcal M</math>, called a Loeb measure. |
||
==References== |
==References== |
Latest revision as of 15:26, 17 November 2021
In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis.
Construction[edit]
Loeb's construction starts with a finitely additive map from an internal algebra of sets to the nonstandard reals. Define to be given by the standard part of , so that is a finitely additive map from to the extended reals . Even if is a nonstandard -algebra, the algebra need not be an ordinary -algebra as it is not usually closed under countable unions. Instead the algebra has the property that if a set in it is the union of a countable family of elements of , then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as ) from to the extended reals is automatically countably additive. Define to be the -algebra generated by . Then by Carathéodory's extension theorem the measure on extends to a countably additive measure on , called a Loeb measure.
References[edit]
- 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.