  If a bulk gravitational path integral can be identified with an average of
partition functions over an ensemble of boundary quantum theories, then a
corresponding moment problem can be solved. We review existence and uniqueness
criteria for the Stieltjes moment problem, which include an infinite set of
positivity conditions. The existence criteria are useful to rule out an
ensemble interpretation of a theory of gravity, or to indicate incompleteness
of the gravitational data. We illustrate this in a particular class of 2D
gravities including variants of the CGHS model and JT supergravity. The
uniqueness criterium is relevant for an unambiguous determination of quantities
such as $\overline{\log Z(\beta)}$ or the quenched free energy. We prove in JT
gravity that perturbation theory, both in the coupling which suppresses
higher-genus surfaces and in the temperature, fails when the number of
boundaries is taken to infinity. Since this asymptotic data is necessary for
the uniqueness problem, the question cannot be settled without a
nonperturbative completion of the theory.
