Completely nonmeasurable unions

Assume that no cardinal κ < 2 ω 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...

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Bibliographic Details
Published in:Central European journal of mathematics 2010, Vol.8 (4), p.683-687
Main Authors: Rałowski, Robert, Żeberski, Szymon
Format: Article
Language:English
Subjects:
03E35
03E75
28A99
Algebra
Bernstein set
Boolean algebra
c.c.c. ideal
Geometry
Lie Groups
Mathematics
Mathematics and Statistics
Nonmeasurable set
Number Theory
Polish space
Probability Theory and Stochastic Processes
Quasi-measurable cardinal
Research Article
Topological Groups
Unions
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Summary:Assume that no cardinal κ < 2 ω 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 ⊆ of X contains uncountably many pairwise disjoint subfamilies , with -Bernstein unions ∪ (a subset A ⊆ X is -Bernstein if A and X \ A meet each Borel -positive subset B ⊆ X ). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].
ISSN:1895-1074
2391-5455
1644-3616
2391-5455
DOI:10.2478/s11533-010-0038-z