Group action on Polish spaces

In this paper we investigate the action of Polish groups (not necessary abelian) on an uncountable Polish spaces. We consider two main situations. First, when the orbits given by group action are small and the second when the family of orbits are at most countable. We have found some subgroups which are not measurable with respect to a given $\sigma$-ideals on the group and the action on some subsets gives a completely nonmeasurable sets with respect to some $\sigma$-ideals with a Borel base on the Polish space. In most cases the general results are consistent with ZFC theory and are strictly connected with cardinal coefficients. We give some suitable examples, namely the subgroup of isometries of the Cantor space where the orbits are suffitiently small. In a opposite case we give an example of the group of the homeomorphisms of a Polish space in which there is a large orbit and we have found the subgroup without Baire property and a subset of the mentioned space such that the action of this subgroup on this set is completely nonmeasurable set with respect to the $\sigma$-ideal of the subsets of first category.Comment: 9 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

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

On nonmeasurable images

summary:Let $(X,\mathbb I)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R(A^2)$ is completely $\mathbb I$-nonmeasurable, i.e, it is $\mathbb I$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations $(R_\alpha )_{\alpha <2^\omega }.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak