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nonmeasurable set; Bernstein set; Polish ideal space
Let $(X,\mathbb I)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R(A^2)$ is completely $\mathbb I $-nonmeasurable, i.e, it is $\mathbb I$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations $(R_\alpha )_{\alpha <2^\omega }.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.
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