Nonmeasurable sets and unions with respect to tree ideals

Abstract: In this paper we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, and $cl_0$. We show that there exists a subset $A$ of the Baire space $\omega^\omega$ which is $s$-, $l$-, and $m$-nonmeasurable, that forms dominating m.e.d. family. We introduce and investigate a notion of $\mathbb{T}$-Bernstein sets - sets that intersect but does not containt any body of a tree from a given family of trees $\mathbb{T}$. We also acquire some results on $\mathcal{I}$-Luzin sets, namely we prove that there are no $m_0$-, $l_0$-, and $cl_0$-Luzin sets and that if $\mathfrak{c}$ is a regular cardinal, then the algebraic sum (considered on the real line $\mathbb{R}$) of a generalized Luzin set and a generalized Sierpi\'nski set belongs to $s_0, m_0$, $l_0$ and $cl_0$.