When does a pair correlation function fix the state of an equilibrium system?

The classical demonstrations that a unique single particle external field is associated with each equilibrium single particle density field (Hohenberg and Kohn, Mermin) are reinterpreted in the language of functional Legendre transformations. This picture is readily extended to the pair distribution function problem and the extension offers a context for understanding how the singlet and pair number densities fix the state of a system. It is shown that one can be sure that there are closure relations to integral equations in general and that in principle the correct closure relation fixes not only the distribution functions but also the complete thermodynamic state of a system. It also follows that a correctly closed integral equation possesses a unique solution. Integral equations for the radial distribution function alone, however, are typically produced by projecting out the singlet density field and for this reason they provide an incomplete characterization of the system. The failure to specify a unique state leads to the existence of multiple solutions.