Title: Set theoretic properties of Loeb measure
Author: Arnold W. Miller,
Abstract
In this paper we ask the question: to what extent do basic set
theoretic properties of Loeb measure depend on the nonstandard universe
and on properties of the model of set theory in which it lies?
We show that assuming Martin's axiom and k-saturation the
smallest cover by Loeb measure zero sets must have cardinality less
than k. In contrast to this we show that the additivity of
Loeb measure cannot be greater than omega_1. Define cof(H) as the
smallest cardinality of a family of Loeb measure zero sets which cover
every other Loeb measure zero set. We show that
card(\log_2(H)) < cof(H)