29 research outputs found
Note on nonmeasurable unions
In this note we consider an arbitrary families of sets of ideal
introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish
space and under some combinatorial and set theoretical assumptions
(cov(s_0)=\c for example), that for any family \ca\subseteq s_0 with
\bigcup\ca =X, we can find a some subfamily \ca'\subseteq\ca such that the
union \bigcup\ca' is not -measurable. We have shown a consistency of the
cov(s_0)=\omega_1<\c and existence a partition of the size \ca\in
[s_0]^{\omega} of the real line \bbr, such that there exists a subfamily
\ca'\subseteq\ca for which \bigcup\ca' is -nonmeasurable. We also showed
that it is relatively consistent with ZFC theory that \omega_1<\c and
existence of m.a.d. family \ca such that \bigcup\ca is -nonmeasurable in
Cantor space or Baire space . The consistency of
and is proved also.Comment: 12 page
Complete nonmeasurability in regular families
We show that for a -ideal \ci with a Borel base of subsets of an
uncountable Polish space, if \ca is (in several senses) a "regular" family of
subsets from \ci then there is a subfamily of \ca whose union is
completely nonmeasurable i.e. its intersection with every Borel set not in \ci
does not belong to the smallest -algebra containing all Borel sets
and \ci. Our results generalize results from \cite{fourpoles} and
\cite{fivepoles}.Comment: 7 page