A solution of Ulam’s problem on relative measure

Abstract:

Suppose $\mathcal {A}$ is a collection of subsets of the unit interval and, for $A \in \mathcal {A}$, ${\mu _A}$ is a Borel measure on $A$ which vanishes on points and gives $A$ measure 1. The system ${\mu _A}(A \in \mathcal {A})$ is called a coherent system if ${\mu _A}(C) = {\mu _A}(B){\mu _B}(C)$ whenever $A$ $B \supseteq C$ are in $\mathcal {A}$ and 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.