On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences

In 2003 Bartoszynski and Halbeisen published the results on various equivalences of Kuratowski and Banach theorem from 1929 concerning some aspect of measure theory. They showed that the existence of the so called BK-matrix related to Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of cardinality continuum. On the other hand, in 1965 Efimov introduced the strong sequences method and using this method proved some well-known theorems in dyadic spaces. The goal of this paper is to show that the existence of such a K-Lusin set is equivalent to the existence of strong sequences of the same cardinality. Some applications of this results are also shown.