Suite

Abstract

In this chapter we shall demonstrate how the tools we developed in the previous chapters can be used to shed new light on a classical problem in Measure Theory. Assuming the Continuum Hypothesis, Banach and Kuratowski proved a combinatorial theorem which implies that a finite measure defined for all subsets of RR vanishes identically if it is zero for points (for the notion of measure we refer the reader to Oxtoby [3, p. 14]). We shall consider this result—which will be called the Banach-Kuratowski Theorem—from a set-theoretical point of view, and among others it will be shown that the Banach-Kuratowski Theorem is equivalent to the existence of a K-Lusin set of size cc and that the existence of such a set is independent of ZFC + ¬CH.