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

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 $\bar{\mathbb{Q}}$, 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 $\mathcal{R}$.