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 ⊆ 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].
@article{bwmeta1.element.doi-10_2478_s11533-010-0038-z,
author = {Robert Ra\l owski and Szymon \.Zeberski},
title = {Completely nonmeasurable unions},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {683-687},
zbl = {1207.03056},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0038-z}
}
Robert Rałowski; Szymon Żeberski. Completely nonmeasurable unions. Open Mathematics, Tome 8 (2010) pp. 683-687. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0038-z/
[1] Brzuchowski J., Cichon J., Grzegorek E., Ryll-Nardzewski C., On the existence of nonmeasurable unions, Bull. Acad. Polon. Sci. Sér. Sci. Math., 1979, 27(6), 447–448 | Zbl 0433.28001
[2] Cichon J., Morayne M., Rałowski R., Ryll-Nardzewski C., Żeberski S., On nonmeasurable unions, Topol. Appl., 2007, 154(4), 884–893 http://dx.doi.org/10.1016/j.topol.2006.09.013[Crossref] | Zbl 1109.03049
[3] Jech T., Set Theory, 3rd millenium ed., Springer, Berlin, 2003
[4] Zeberski S., On completely nonmeasurable unions, MLQ Math. Log. Q., 2007, 53(1), 38–42 http://dx.doi.org/10.1002/malq.200610024[Crossref]