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

Charles L. Samuels

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arXiv · 2405.06519

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

Charles L. Samuels

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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 $\mathcal R$.

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Charles L. Samuels. 2024-05-10. The correspondence between consistent maps and measures on the places of $\overline{\mathbb Q}$. https://arxiv.org/abs/2405.06519

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