Ensemble averages, Poisson processes and Microstates

We consider ensemble averaged theories with discrete random variables. We propose a suitable measure to do the ensemble average. We also provide a mathematical description of such ensemble averages of theories in terms of Poisson point processes. Moreover, we demonstrate that averaging theories of this type has an equivalent description as tracing over parts of the microscopic degrees of freedom in a suitable continuous limit of a single microscopic theory. The results from both approaches can be identified with Liouville gravity, of which we further address some implications on the microscopic theory, including venues to look for quantum effects from the view point of the averaged theory. Generalizations to other point processes are also discussed.