A 2009 article of Allcock and Vaaler examined the normed Q- ×× ×× vector space Q /Qtors. We consider the related vector space V := Q /Z , × where Z denotes the group of units in the ring of algebraic integers. By ref- erencing certain types of finitely additive measures called consistent maps, we provide representation theorems the algebraic and analytic duals of V. We further explore how consistent maps may be used to establish other dual representation theorems for function spaces on the set of places of Q.