Lifting theorems in nonstandard measure theory

Ross, David

doi:10.1090/S0002-9939-1990-1019753-0

Lifting theorems in nonstandard measure theory
HTML articles powered by AMS MathViewer

by David Ross PDF

Proc. Amer. Math. Soc. 109 (1990), 809-822 Request permission

Abstract:

1. A nonstandard capacity construction, analogous to Loeb’s measure construction, is developed. Using this construction and Choquet’s Capacitability theorem, it is proved that a Loeb measurable function into a general (not necessarily second countable) space has a lifting precisely when its graph is ’almost’ analytic. This characterization is used to generalize and simplify some known lifting existence theorems. 2. The standard notion of ’Lusin measurability’ is related to the nonstandard notion of admitting a ’two-legged’ lifting. An immediate consequence is a new and simple proof of the general Lusin theorem. Another consequence is the existence of a Loeb measurable function, not admitting a lifting, into a relatively small topological space.

References

Sergio Albeverio, Raphael Høegh-Krohn, Jens Erik Fenstad, and Tom Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics, Pure and Applied Mathematics, vol. 122, Academic Press, Inc., Orlando, FL, 1986. MR 859372
Robert M. Anderson, Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687. MR 654856, DOI 10.1090/S0002-9947-1982-0654856-1
Nigel J. Cutland, Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), no. 6, 529–589. MR 720746, DOI 10.1112/blms/15.6.529
Nigel Cutland (ed.), Nonstandard analysis and its applications, London Mathematical Society Student Texts, vol. 10, Cambridge University Press, Cambridge, 1988. Papers from a conference held at the University of Hull, Hull, 1986. MR 971063, DOI 10.1017/CBO9781139172110

Probabilities and potential

D. H. Fremlin, Measurable functions and almost continuous functions, Manuscripta Math. 33 (1980/81), no. 3-4, 387–405. MR 612620, DOI 10.1007/BF01798235
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
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
H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. MR 732752, DOI 10.1090/memo/0297
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
D. Landers and L. Rogge, Universal Loeb-measurability of sets and of the standard part map with applications, Trans. Amer. Math. Soc. 304 (1987), no. 1, 229–243. MR 906814, DOI 10.1090/S0002-9947-1987-0906814-1

Measurable transformations in saturated models of analysis

David Ross, Random sets without separability, Ann. Probab. 14 (1986), no. 3, 1064–1069. MR 841605
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
K. D. Stroyan and José Manuel Bayod, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics, vol. 119, North-Holland Publishing Co., Amsterdam, 1986. MR 849100
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