  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 $\mathcal R$.
