The Connections Between Consistent Maps and Measures

Abstract

Let Y denote the space of places of the algebraic closure of the rationals as defined in a 2009 article of Allcock and Vaaler. As part of an effort to classify certain dual spaces, the second author defined an object called a consistent map. Every signed Borel measure on Y can be used to construct a consistent map, however, we asserted without proof that not all consistent maps arise in this way. By constructing a counterexample, we show in the present article that not all consistent maps arise from measures, confirming claims made in the second author's earlier work.