Completely nonmeasurable unions

Robert Rałowski; Szymon Żeberski

Open Mathematics (2010)

  • Volume: 8, Issue: 4, page 683-687
  • ISSN: 2391-5455

Abstract

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

How to cite

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Robert Rałowski, and Szymon Żeberski. "Completely nonmeasurable unions." Open Mathematics 8.4 (2010): 683-687. <http://eudml.org/doc/269524>.

@article{RobertRałowski2010,
abstract = {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].},
author = {Robert Rałowski, Szymon Żeberski},
journal = {Open Mathematics},
keywords = {Quasi-measurable cardinal; Nonmeasurable set; Bernstein set; c.c.c. ideal; Polish space; -ideal; nonmeasurable set; ccc ideal; quasi-measurable cardinal; Borel sets},
language = {eng},
number = {4},
pages = {683-687},
title = {Completely nonmeasurable unions},
url = {http://eudml.org/doc/269524},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Robert Rałowski
AU - Szymon Żeberski
TI - Completely nonmeasurable unions
JO - Open Mathematics
PY - 2010
VL - 8
IS - 4
SP - 683
EP - 687
AB - 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].
LA - eng
KW - Quasi-measurable cardinal; Nonmeasurable set; Bernstein set; c.c.c. ideal; Polish space; -ideal; nonmeasurable set; ccc ideal; quasi-measurable cardinal; Borel sets
UR - http://eudml.org/doc/269524
ER -

References

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  1. [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 Zbl0433.28001
  2. [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] Zbl1109.03049
  3. [3] Jech T., Set Theory, 3rd millenium ed., Springer, Berlin, 2003 
  4. [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] 

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