# Completely I-non-measurable unions in Polish spaces

Problem. Let $$X$$ be a Polish space, $$\mathcal I$$ be a $$\sigma$$-ideal with Borel base, and $$\mathcal A\subset\mathcal I$$ be a point-finite cover of $$X$$. Is it true that $$\mathcal A$$ conatins a subfmaily $$\mathcal B$$ whose union $$\bigcup\mathcal B$$ is completely $$\mathcal I$$-nonmeasurable in the sense that any Borel subset $$B\notin\mathcal I$$ intersects both sets $$\bigcup\mathcal B$$ and $$X\setminus\bigcup\mathcal B$$.

(This problem (attributed to Jacek Cichoń) was written 27.07.2018 by Szymon Żeberski from Wrocław on page 28 of Volume 2 of the Lviv Scottish Book).

• It would be good to have the tag "scottish-book". – user95282 Oct 28 '18 at 14:02
• Unfortunately, creating meta-tags like "scottish-book" is not recommended by the rules of MO. – Lviv Scottish Book Oct 31 '18 at 17:22