Remarks on nonmeasurable unions of big point families

We show that under some conditions on a family $\mathcal{A}\subset \text{bbi}$ there exists a subfamily ${\mathcal{A}}_0\subset \mathcal{A}$ such that $\bigcup $\mathcal{A}$_0$isnonmeasurablewithrespecttoafixedideal$\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.