Szymon Żeberski

On completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal not greater than 2ω. We show that for a c. c. c. σ -ideal with a Borel base of subsets of an uncountable Polish space, if is a point-finite family of subsets from , then there is a subfamily of 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 . This result is a generalization of the Four Poles Theorem (see [1]) and a result from [3]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Completely nonmeasurable unions

AbstractAssume that no cardinal κ < 2ω 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 ⊆

On nonmeasurable unions

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Topologically invariant σ-ideals on Euclidean spaces

of X contains uncountably many pairwise disjoint subfamilies , with

Nonmeasurable sets and unions with respect to tree ideals

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\mathcal{I}

-Bernstein unions ∪ (a subset A ⊆ X is