On Banach and Kuratowski Theorem and generalized strong sequences

In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing $σ$-additive finite measure on $\mathbb{R}$ which is defined for every set of reals. In 2003 Bartoszyński and Halbeisen proved that Banach and Kuratowski theorem is equivalent to