Completely nonmeasurable unions

Robert Ralowski; Szymon Zeberski

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arXiv · 1003.0918

Completely nonmeasurable unions

Robert Ralowski·Szymon Zeberski

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Abstract

Assume that there is no quasi-measurable cardinal smaller than $2^ω$. ($κ$ is quasi measurable if there exists $κ$-additive ideal $\ci $ of subsets of $κ$ such that the Boolean algebra $P(κ)/\ci$ satisfies c.c.c.) We show that for a c.c.c. $σ$-ideal I with a Borel base of subsets of an uncountable Polish space, if $\cal A$ is a point-finite family of subsets from I then there is an uncountable collection of pairwise disjoint subfamilies of $\cal A$ whose union is completely nonmeasurable i.e. its intersection with every non-small Borel set does not belong to the $σ$-field generated by Borel sets and the ideal I. This result is a generalization of Four Poles Theorem.

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BibTeX RIS

Robert Ralowski, Szymon Zeberski. 2010-03-03. Completely nonmeasurable unions. https://arxiv.org/abs/1003.0918

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