  We show that under some conditions on a family $\mathcal{A}\subset\bbi$ there
exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup
\mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal $\bbi$ with Borel
base of a fixed uncountable Polish space. Our result applies to the classical
ideal of null subsets of the real line and to the ideal of first category
subsets of the real line.
