Completely nonmeasurable unions

Abstract

Assume that no cardinal κ < 2^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal

of subsets of κ such that the Boolean algebra P(κ)/

satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover

⊆ $\mathbb{I}$ of X contains uncountably many pairwise disjoint subfamilies

, with $\mathbb{I}$-Bernstein unions ∪

(a subset A ⊆ X is $\mathbb{I}$-Bernstein if A and X \ A meet each Borel $\mathbb{I}$-positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].