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

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

The Set-Cover game and nonmeasurable unions

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cicho\'n, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski--Burstin representable ideals, as well as to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups

Around Eggleston Theorem

The motivation of this work are the two classical theorems on inscribing rectangles and squares into large subsets of the plane, namely Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem we prove that for every Borel subset of the plane with uncountably many positive (with respect to measure or category) vertical section contains a rectangle $P\times B$ where $P$ is perfect and $B$ is Borel and positive. We also obtained a variant of Eggleston Theorem regarding the $\sigma$-ideal $\mathcal(E)$ generated by closed sets of measure zero. Furthermore we proved that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$, where $T$ is a Spinas tree containing a Silver tree and $H$ is comeager (resp. conull). Moreover we obtained a common generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ modulo diagonal, where $T$ is a uniformly perfect tree, $H$ is comeager (resp. conull) and $[T]\subseteq H$

On Commutation Relations for Quons

The model of generalized quons is described in a purely algebraic way. Commutation relations and corresponding consistency conditions for our generalized quons system are studied in terms of quantum Weyl algebras. Fock space representation and corresponding scalar product is also given.Comment: 17 pages in Latex, (corrected missprints in two formulas