The correspondence between consistent maps and measures on the places of $\overline{\mathbb Q}$

TL;DR

This paper explores the relationship between consistent maps and measures on the set of places of algebraic numbers, establishing conditions for when consistent maps correspond to actual measures.

Contribution

It characterizes the ring of sets where consistent maps are measures and provides conditions for extending these maps to measures on larger algebras.

Findings

Identification of the ring of sets for consistent maps

Conditions for extending consistent maps to measures

Dual representation theorems for vector spaces related to places

Abstract

Recent work of the author established dual representation theorems for certain vector spaces that arise in an important article of Allcock and Vaaler. These results constructed an object called a consistent map which acts like a measure on the set of places of $\overline{\mathbb Q}$, but is not a Borel measure on this space. We describe the appropriate ring of sets $\mathcal R$ for which every consistent map arises from a measure on $\mathcal R$. We further obtain the conditions under which a consistent map may be extended to a measure on the smallest algebra containing $\mathcal R$.