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}.Comment: 7 page