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).

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).