The correspondence between consistent maps and measures on the places of $\overline{\mathbb Q}$
Charles L. Samuels
TL;DR
The paper addresses representing consistent maps, defined on the place data of $\overline{\mathbb Q}$, as measures and extends these representations to larger algebras. It constructs a minimal ring $\mathcal{R}$ of sets on the place space $Y$ so that every consistent map corresponds to a signed measure $\mu$ on $\mathcal{R}$, yielding $c(K,v)=\mu(Y(K,v))$ and $\Phi_c(\alpha)=\int_Y f_\alpha \, d\mu$. It then identifies a refinement to an algebra $\mathcal{A}$ and introduces global consistency $\overline{\mathcal{J}^*}$, proving a bijection between measures on $\mathcal{A}$ and globally consistent maps, thus characterizing exactly which consistent maps extend to countably additive measures. The work thus provides a rigorous integral representation framework for extensions of arithmetic functionals to $\mathcal{G}$ via ring and algebra measures, clarifying the limitations of Borel measures on the place space and linking to Omega-type extensions. Overall, it offers a precise measure-theoretic backbone for dual representations of function spaces tied to the places of $\overline{\mathbb Q}$ and the extensions of additive arithmetic functions.
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 $\mathcal R$.