A Solution of Ulam's Problem on Relative Measure

Suppose stfis a collection of subsets of the unit interval and, for A e s/, \iA is a Borel measure on A which vanishes on points and gives A measure 1. The system pA (A e jtf) is called a coherent system if pA(C) = nA(B)pB(C) whenever A dSdC are in .Wand all terms are defined. The existence of a coherent system for the collection of perfect sets is shown to be independent of Zermelo-Fraenkel set theory with the axiom of dependent choices. In [4, problem 7, pp. 77-78] Ulam asked how larges/can be if it is the domain of a coherent system. Assuming the axiom of choice, si can be all sets which are not ruled out by some obviously necessary restriction; for example, si cannot contain any countable sets. By restricting Lebesgue measure, a coherent system for the collection of sets of positive Lebesgue measure can be constructed (without using the axiom of choice). In particular, there is a coherent system on the collection of open sets. Without assuming the axiom of choice, the most natural question is whether a coherent system exists for the collection of perfect sets. The main theorem of this paper states that, assuming the principle of dependent choice, rather than the full axiom of choice, one cannot prove the existence of a coherent system for the collection of perfect sets. The proof of the main theorem also shows that the usual axioms for set theory, including the axiom of choice, are not sufficient to prove there is a definable coherent system on the collection of perfect sets.