Completely nonmeasurable unions

الأصل · EN

Assume that there is no quasi-measurable cardinal smaller than 2ω. (κ is quasi measurable if there exists κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.) We show that for a c.c.c. σ-ideal I with a Borel base of subsets of an uncountable Polish space, if A is a point-finite family of subsets from I then there is an uncountable collection of pairwise disjoint subfamilies of A whose union is completely nonmeasurable i.e. its intersection with every non-small Borel set does not belong to the σ-field generated by Borel sets and the ideal I. This result is a generalization of Four Poles Theorem.