Note on s0s_0 nonmeasurable unions

In this note we consider an arbitrary families of sets of $s_0$ ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space $X$ 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 $s$-measurable. We have shown a consistency of the cov(s_0)=\omega_1<\c and existence a partition of the size $\omega_1$ \ca\in [s_0]^{\omega} of the real line \bbr, such that there exists a subfamily \ca'\subseteq\ca for which \bigcup\ca' is $s$-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 $s$-nonmeasurable in Cantor space $2^\omega$ or Baire space $\omega^\omega$. The consistency of $a<cov(s_0)$ and $cov(s_0)<a$ is proved also.Comment: 12 page

Complete nonmeasurability in regular families

We show that for a $\sigma$-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 $\sigma$-algebra containing all Borel sets and \ci. Our results generalize results from \cite{fourpoles} and \cite{fivepoles}.Comment: 7 page

Completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal smaller than $2^\omega$. ($\kappa$ is quasi measurable if there exists $\kappa$-additive ideal \ci of subsets of $\kappa$ such that the Boolean algebra P(\kappa)/\ci satisfies c.c.c.) We show that for a c.c.c. $\sigma$-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 $\sigma$-field generated by Borel sets and the ideal I. This result is a generalization of Four Poles Theorem.Comment: 6 page