Geodesic
On nonmeasurable images
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 423-434

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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.
Classification : 03E35, 03E75, 28A99
Keywords: nonmeasurable set; Bernstein set; Polish ideal space 
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     author = {Ra{\l}owski, Robert and \.Zeberski, Szymon},
     title = {On nonmeasurable images},
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     pages = {423--434},
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     zbl = {1224.03028},
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Rałowski, Robert; Żeberski, Szymon. On nonmeasurable images. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 423-434. http://geodesic.mathdoc.fr/item/CMJ_2010__60_2_a9/