Nonmeasurable sets and unions with respect to tree ideals

In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals , , , , and . We show that there exists a subset of the Baire space which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of -Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees . We also obtain a result on -Luzin sets, namely, we prove that if is a regular cardinal, then the algebraic sum (considered on the real line ) of a generalized Luzin set and a generalized Sierpinski set belongs to , and .