Lifting theorems in nonstandard measure theory

Abstract

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 pf the general Lusin theorem. Another consequence is the existence of a Loeb measurable function, not admitting a lifting, into a relatively small topological space. IntroductionA fundamental tool in nonstandard measure theory is the 'lifting' construction, in which an (external) measurable function is approximated by an internal one. Most lifting theorems in the literature derive from a result of Anderson, which assumes that the target space is second countable.Nonstandard measure theory is being used to study increasingly exotic structures (random sets, random measures, stochastic processes on infinite-dimensional Banach spaces), where the target spaces for measurable functions are not necessarily second countable (or even topological!), and more general lifting theorems are therefore required. It is natural to conjecture that any measurable function will have a lifting, provided the cardinality of the target space is small relative to the degree of saturation of the nonstandard model. This conjecture turns out to be false (Corollary 6.3 and Example 6.4).The paper reduces questions about a function's liftability to questions about the analytic structure (in the sense of descriptive set theory) of the function's graph. The main tool used is Choquet's theorem from the theory of capacities; this theory is reviewed in §2, where a nonstandard capacity constructionanalogous to Loeb's measure construction-is described.

Uniqueness of unbounded Loeb measure using Choquet’s theorem

“…In this paper we show how Choquet's capacitability theorem can be used when countably determined sets are replaced by Souslin sets. Although the class of countably determined sets is much wider than the class of Souslin sets (the class of countably determined sets contains the class of projective sets of finite rank), which makes the results obtained by Henson more general, we believe that the proof we present here gives an interesting example of the more extensive use of Choquet's theorem in Nonstandard Analysis (see [HR,R,Zl,Z3]). …”