Remarks on nonmeasurable unions of big point families

We show that under some conditions on a family \mathcal{A}\subset\bbi there exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup \mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal \bbi with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line.Comment: 8 pages, accepted to Math. Log. Quar

SPM Bulletin 30

This is the 30th issue of this bulletin, dedicated to mathematical selection principles and related areas. Now in a concise format.Comment: Boaz Tsaban is an editor of this bulleti

Nonmeasurable sets and unions with respect to selected ideals especially ideals defined by trees

In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by Miller trees. With the mentioned ideals one can consider s, l and m-measurablility. We have shown that there exists a subset A of the Baire space which is s, l and m nonmeasurable at the same time. Moreover, A forms m.a.d. family which is also dominating. We show some examples of subsets of the Baire space which are measurable in one sense and nonmeasurable in the other meaning. We also examine terms nonmeasurable and completely nonmeasurable (with respect to several ideals with Borel base). There are several papers about finding (completely) nonmeasurable sets which are the union of some family of small sets. In this paper we want to focus on the following problem: "Let P be a family of small sets. Is it possible that for all A which is a subset of P, union of A is nonmeasurable implies that union of A is completely nonmeasurable?" We will consider situations when P is a partition of R, P is point-finite family and P is point-countable family. We give an equivalent statement to CH using terms nonmeasurable and completely nonmeasurable.Comment: 13 page

Bernstein sets and κ\kappa-coverings

In this paper we study a notion of a $\kappa$-covering in connection with Bernstein sets and other types of nonmeasurability. Our results correspond to those obtained by Muthuvel and Nowik. We consider also other types of coverings.Comment: 12 page