  In a paper of 1929, Banach and Kuratowski proved, assuming the continuum
hypothesis, a combinatorial theorem which implies that there is no
non-vanishing sigma-additive finite measure on the real line which is defined
for every set of reals. It will be shown that the combinatorial theorem is
equivalent to the existence of a K-Lusin set of size the continuum and that the
existence of such sets is independent of ZFC plus non CH.
