  In this paper we consider nonmeasurablity with respect to sigma-ideals
defined be trees. First classical example of such ideal is Marczewski ideal
s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by
Miller trees. With the mentioned ideals one can consider s, l and
m-measurablility.
  We have shown that there exists a subset A of the Baire space which is s, l
and m nonmeasurable at the same time. Moreover, A forms m.a.d. family which is
also dominating. We show some examples of subsets of the Baire space which are
measurable in one sense and nonmeasurable in the other meaning.
  We also examine terms nonmeasurable and completely nonmeasurable (with
respect to several ideals with Borel base). There are several papers about
finding (completely) nonmeasurable sets which are the union of some family of
small sets. In this paper we want to focus on the following problem: "Let P be
a family of small sets. Is it possible that for all A which is a subset of P,
union of A is nonmeasurable implies that union of A is completely
nonmeasurable?"
  We will consider situations when P is a partition of R, P is point-finite
family and P is point-countable family. We give an equivalent statement to CH
using terms nonmeasurable and completely nonmeasurable.
