Completely nonmeasurable unions

Abstract

Abstract:Assume that no cardinal κ < 2 ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal I of subsets of κ such that the Boolean algebra P(κ)/I satisfies c.c.c.). We show that for a metrizable separable space X and a proper c. MSC:03E35, 03E75, 28A99

A few remarks on non-Baire sets in category bases

“…The above definition is somewhat analogous to the notion of a completely I-nonmeasurable set, given in [8].…”

On completely non-Baire union in category bases

“…In the next theorem, we show that even if {X t } t∈T is a family of disjoint sets but not a partition of X, still it is possible to obtain at least #(X) number of mutually disjoint completely non-Baire unions by slightly changing the hypothesis. Here we have adapted the technique used to prove Theorem 2.1 [4].…”

Luzin and Sierpiński sets, some nonmeasurable subsets of the plane and additive properties on the line

“…• a strong Luzin set if A is a Luzin set and every intersection of A and a M-positive Borel set is uncountable, • a Sierpiński set if |S| = c and every intersection of S and a null set is countable, • a strong Sierpiński set if A is a Sierpiński set and every intersection of A and a Npositive Borel set is uncountable, • a Bernstein set if for each perfect set P we have B ∩ P = ∅ and B c ∩ P = ∅. Let us note that a notion of I-nonmeasurability and complete I-nonmeasurability agrees with [21], [22] and [25].…”