The Set-Cover game and nonmeasurable unions

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cicho'n, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski–Burstin representable ideals, as well as to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.