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Completely nonmeasurable unions

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Rałowski R. , Żeberski S.

Open Mathematics

2010

tom 8

nr 4

683-687

Assume 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 ⊆ $$ \mathbb{I} $$ of X contains uncountably many pairwise disjoint subfamilies , with $$ \mathbb{I} $$-Bernstein unions ∪ (a subset A ⊆ X is $$ \mathbb{I} $$-Bernstein if A and X \ A meet each Borel $$ \mathbb{I} $$-positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].

Some Non-Measurable Sets

86%

Kierus A. , Institute of Mathematics T. U. o. Ł. W. , alicja.kierus@gmail.com

2010

tom vol. 17

This paper contains constructions of some non-measurable sets, based on classical Vitali’s and Bernstein’s constructions (see for example [6]). This constructions probably belong to mathematical folklore, but as far as we know they are rather hard to be found in literature. It seems that the constructed sets can be used as examples in some interesting situations.