Robert Rałowski

Remarks on nonmeasurable unions of big point families

We show that under some conditions on a family A ⊂ I there exists a subfamily A0 ⊂ A such that ∪ A0 is nonmeasurable with respect to a fixed ideal I 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 (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Bernstein sets and k -coverings

䅢stract. In this paper we study a notion of a �-covering in connection with Bernstein sets and other types of nonmeasurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings. 1. Definitions and notation In 1993 Car汳on 楮 h楳 paper 嬳] 楮troduced a not楯n of �-cover楮gs and used 楴 for 楮vest楧at楮g whether some 楤ea汳 are or are not �-trans污tab汥. Later on �-cover楮gs were stud楥d by other authors, e⹧. Muthuvel (cf. [7 崩 and Now楫 (cf. 嬸崬 嬹崩. In th楳 paper we present new resu汴s on �-cover楮gs 楮 connect楯n w楴h Bernste楮 sets. We a汳o 楮troduce two natural genera汩zat楯ns of the not楯n of �-cover楮gs, name汹 �-S-cover楮gs and �-I-cover楮gs. We use standard set-theoret楣al notat楯n and term楮o汯gy from [ 1崮 Reca汬 that the card楮a汩ty of the set of a汬 real numbers R 楳 denoted by c. The card楮a汩ty of a set A 楳 denoted by jAj. If � 楳 a card楮al number then

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 ⊆

Classifying invariant -ideals with analytic base on good Cantor measure spaces

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On Deformations of Commutation Relation Algebras

of X contains uncountably many pairwise disjoint subfamilies , with

Topologically invariant σ-ideals on the Hilbert cube

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