On nonmeasurable unions

• Published in: Topology and its applications Vol. 154; no. 4; pp. 884 - 893
• Main Authors: Cichoń, Jacek, Morayne, Michał, Rałowski, Robert, Ryll-Nardzewski, Czesław, Żeberski, Szymon
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15-02-2007
• Subjects: Algebraic sum; Baire property; Cantor set; Lebesgue measure; secondary; Lebesgue measure; Cantor set; Algebraic sum; Baire property; primary
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2006.09.013

For a Polish space
X and a
σ-ideal I of subsets of
X which has a Borel base we consider families
A
of sets in
I with the union
⋃
A
not in
I. We determine several conditions on
A
which imply the existence of a subfamily
A
′
of
A
whose union
⋃
A
′
is not in the
σ-field generated by the Borel sets on
X and
I. Main examples are
X
=
R
and
I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.