Unbounded Loeb measures

Henson, C. Ward

doi:10.1090/S0002-9939-1979-0521888-9

Unbounded Loeb measures
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by C. Ward Henson PDF

Proc. Amer. Math. Soc. 74 (1979), 143-150 Request permission

Abstract:

Uniqueness of the Carathéodory extension is proved for the unbounded case of Loeb’s measure space construction in nonstandard analysis and these measures are studied. Methods are developed for handling sets in the $\sigma$-algebra generated by the internal sets. A quantitative theory of degrees of null measure is developed.

References

Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559

Star-finite probability theory

C. Ward Henson, Analytic sets, Baire sets and the standard part map, Canadian J. Math. 31 (1979), no. 3, 663–672. MR 536371, DOI 10.4153/CJM-1979-066-0
H. Jerome Keisler, Hyperfinite model theory, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977, pp. 5–110. MR 0491155
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
A. H. Lightstone and Abraham Robinson, Nonarchimedean fields and asymptotic expansions, North-Holland Mathematical Library, Vol. 13, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0414354
W. A. J. Luxemburg, On a class of valuation fields introduced by A. Robinson, Israel J. Math. 25 (1976), no. 3-4, 189–201. MR 460102, DOI 10.1007/BF02756999
Abraham Robinson, Function theory on some nonarchimedean fields, Amer. Math. Monthly 80 (1973), no. 6, 87–109. Papers in the foundations of mathematics. MR 330126, DOI 10.2307/3038223
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163