On nonmeasurable images

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