On Banach and Kuratowski Theorem and generalized strong sequences

Abstract

In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing $\sigma$-additive finite measure on $\mathbb{R}$ which is defined for every set of reals. In 2003 Bartoszy\'nski and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of $ZFC + \neg CH$. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.