  Some historical remarks are in order here. In the 1930s mathematicians investigated various versions of the so-called measure problem for sets, a problem which arose in connection with the theory of Lebesgue measure on the continuum. In particular, the following very simple notion of measure was formulated. If X is a set, a (countably additive two-valued nontrivial) measure on X is a map μ on the power set PX to the set {0, 1} satisfying: