Lifting theorems in nonstandard measure theory

Author:
David Ross

Journal:
Proc. Amer. Math. Soc. **109** (1990), 809-822

MSC:
Primary 03H05; Secondary 28A51, 28E05

DOI:
https://doi.org/10.1090/S0002-9939-1990-1019753-0

MathSciNet review:
1019753

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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.

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1019753-0

Article copyright:
© Copyright 1990
American Mathematical Society