  Assume that there is no quasi-measurable cardinal smaller than $2^\omega$.
($\kappa$ is quasi measurable if there exists $\kappa $-additive ideal $\ci $
of subsets of $\kappa $ such that the Boolean algebra $P(\kappa)/\ci$ satisfies
c.c.c.) We show that for a c.c.c. $\sigma $-ideal I with a Borel base of
subsets of an uncountable Polish space, if $\cal A$ is a point-finite family of
subsets from I then there is an uncountable collection of pairwise disjoint
subfamilies of $\cal A$ whose union is completely nonmeasurable i.e. its
intersection with every non-small Borel set does not belong to the $\sigma
$-field generated by Borel sets and the ideal I. This result is a
generalization of Four Poles Theorem.
