Completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal smaller than $2^ω$. ($κ$ is quasi measurable if there exists $κ$-additive ideal $\ci $ of subsets of $κ$ such that the Boolean algebra $P(κ)/\ci$ satisfies c.c.c.) We show that for a c.c.c. $σ$-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 $σ$-field generated by Borel sets and the ideal I. This result is a generalization of Four Poles Theorem.

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