Measures and Charges on the Places of Q̄

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 Q̄, but is not a Borel measure on this space. We describe the appropriate ring of sets R for which every consistent map arises from a measure on R. We further obtain the conditions under which a consistent map may be extended to a measure on the smallest algebra containing R.