Complete nonmeasurability in regular families

Abstract

We show that for a $\sigma $-ideal $\ci$ with a Borel base of subsets of an uncountable Polish space, if $\ca$ is (in several senses) a "regular" family of subsets from $\ci $ then there is a subfamily of $\ca$ whose union is completely nonmeasurable i.e. its intersection with every Borel set not in $\ci $ does not belong to the smallest $\sigma $-algebra containing all Borel sets and $\ci.$ Our results generalize results from \cite{fourpoles} and \cite{fivepoles}.