Uniqueness of unbounded Loeb measure using Choquet’s theorem

Živaljević, Boško

doi:10.1090/S0002-9939-1992-1094509-3

Uniqueness of unbounded Loeb measure using Choquet’s theorem
HTML articles powered by AMS MathViewer

by Boško Živaljević PDF

Proc. Amer. Math. Soc. 116 (1992), 529-533 Request permission

Abstract:

Uniqueness of the Carathéodory extension of the standard part map of an internal unbounded measure $\mu$ defined on an internal algebra $\mathcal {A}$ of subsets of an internal set $\Omega$ has been proved by Henson using the notion of a countably determined set. Here we show how Choquet’s capacitability theorem can be used in the proof of the same result.

References

D. W. Bressler and M. Sion, The current theory of analytic sets, Canadian J. Math. 16 (1964), 207–230. MR 163854, DOI 10.4153/CJM-1964-021-7
Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953/54), 131–295 (1955). MR 80760, DOI 10.5802/aif.53
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
C. Ward Henson, Unbounded Loeb measures, Proc. Amer. Math. Soc. 74 (1979), no. 1, 143–150. MR 521888, DOI 10.1090/S0002-9939-1979-0521888-9
C. Ward Henson and David Ross, Analytic mappings on hyperfinite sets, Proc. Amer. Math. Soc. 118 (1993), no. 2, 587–596. MR 1126195, DOI 10.1090/S0002-9939-1993-1126195-9
Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
Analytic sets, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. Lectures delivered at a Conference held at University College, University of London, London, July 16–29, 1978. MR 608794
H. Jerome Keisler, Kenneth Kunen, Arnold Miller, and Steven Leth, Descriptive set theory over hyperfinite sets, J. Symbolic Logic 54 (1989), no. 4, 1167–1180. MR 1026596, DOI 10.2307/2274812
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
David Ross, Lifting theorems in nonstandard measure theory, Proc. Amer. Math. Soc. 109 (1990), no. 3, 809–822. MR 1019753, DOI 10.1090/S0002-9939-1990-1019753-0
Boško Živaljević, Every Borel function is monotone Borel, Ann. Pure Appl. Logic 54 (1991), no. 1, 87–99. MR 1130220, DOI 10.1016/0168-0072(91)90010-J
Boško Živaljević, Some results about Borel sets in descriptive set theory of hyperfinite sets, J. Symbolic Logic 55 (1990), no. 2, 604–614. MR 1056374, DOI 10.2307/2274650
Boško Živaljević, $U$-meager sets when the cofinality and the coinitiality of $U$ are uncountable, J. Symbolic Logic 56 (1991), no. 3, 906–914. MR 1129155, DOI 10.2307/2275060