An axiom for nonseparable Borel theory

Abstract

Kuratowski asked whether the Lebesgue-Hausdorff theorem held for metrizable spaces. A. Stone asked whether a Borel isomorphism between metrizable spaces must be a generalized homeomorphism. The existence of a Q set refutes the generalized Lebesgue-Hausdorff theorem. In this paper we discuss the consequences of the axiom of the title, among which are “yes” answers to both Kuratowski’s and Stone’s questions. The axiom states that a point finite analytic additive family is σ \sigma discretely decomposable. We show that this axiom is valid in the model constructed by collapsing a supercompact cardinal to ω 2 {\omega _2} using Lévy forcing. Our proof displays relationships between σ \sigma discretely decomposable families, analytic additive families and d families.