Nonstandard Measure Theory and its Applications

Cutland, Nigel J.

doi:10.1112/blms/15.6.529

Cited by 146 publications

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“…Proposals to construct non-standard measures on infinite sample spaces have been made in the literature (Cutland 1983). But instead of taking the natural numbers as their domain, these probability functions operate on a non-standard extension of the natural numbers.…”

Section: Resultsmentioning

confidence: 99%

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Fair infinite lotteries

Wenmackers

Horsten

2010

Synthese

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Section: Resultsmentioning

confidence: 99%

“…We do not aim at completeness here, but restrict our attention to those ingredients that we will need in the course of this paper. The information of this section is mainly based on Cutland (1983) and Benci et al (2006).…”

Section: Infinitesimalsmentioning

confidence: 99%

Fair infinite lotteries

Wenmackers

Horsten

2010

Synthese

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“…Besides the original Robinson's book [25], the standard general reference are the monographs [1] and [28]. For Loeb measure also the original papers by Loeb [18][19][20][21], as well as [4], [5], [13], [14] and the survey [6] by Cutland can be consulted. Some recent developments are reflected in [17], [3] and the survey [26] by Ross.…”

Section: Introductionmentioning

confidence: 99%

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Ziman

Zlatoš

2006

Mh Math

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Abstract. Let X be a locally compact topological space and (X, E, X ω ) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X ω ⊆ X, such that all internal subsets of X ω are relatively compact in the induced topology and X is homeomorphic to the quotient X ω /E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X → * C. The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient M ω (X)/M 0 (X), for certain external subspaces M 0 (X), M ω (X) of the hyperfinite dimensional Banach space * C X , with the norm f 1 = x∈X |f (x)|. If additionally X = G is a hyperfinite group, X ω = G ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G 0 of G ω , and G is isomorphic to the locally compact group G ω /G 0 , then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M ω (G)/M 0 (G) are isometrically isomorphic as Banach algebras. IntroductionA great deal of methods of nonstandard analysis is based on embedding classical mathematical structures into somehow related hyperfinite ones. As a rule, the hyperfinite set (topological space, measure space, etc.) or hyperfinite dimensional vector space X, extending the classical object X, is subject to the inclusions X ⊆ X ⊆ * X, i.e., it is singled out from the nonstandard extension * X of X. This has the additional advantage of X naturally inheriting the structure from X, via the extension * X, and applicability of the transfer principle.The method, however, may fail to work that way in presence of some already a little bit more complex algebraic structure on X. For instance, given a group G, there need not be any hyperfinite group G subject to G ≤ G ≤ * G. On the other hand, some hyperfinite group G extending G all the same may (though still need not) exist. And similarly for associative linear algebras over some field. The situation becomes even more complicated for topological groups and Banach algebras. On the other hand, especially for the sake of applications of nonstandard methods to the study of spaces and algebras of functions or measures over G, it is desirable to have G embedded into some hyperfinite group G and relate somehow the just 1991 Mathematics Subject Classification. Primary 28E05, 43A10; Secondary 03H05, 22D15, 46S20, 54J05.

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“…Thereby, we show that our hyperfinite construction is the appropriate nonstandard analogue of the G-expectation. For details on nonstandard analysis, we refer the reader to Albeverio et al [1], Cutland [5], Loeb and Wolff [13] and Stroyan and Luxemburg [16].…”

Section: Introductionmentioning

confidence: 99%

Hyperfinite Construction of G-Expectation

Fadina

Herzberg

2015

SSRN Journal

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. We prove a lifting theorem, in the sense of Robinsonian nonstandard analysis, for the G-expectation. Herein, we use an existing discretization theorem for the G-expectation by T. Fadina and F. Terms of use: Documents in EconStor may

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Nonstandard Measure Theory and its Applications

Cited by 146 publications

References 0 publications

Fair infinite lotteries

Fair infinite lotteries

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Hyperfinite Construction of G-Expectation

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