The Metric of Large Deviation Convergence

We construct a metric space of set functions (, d) such that a sequence {P_n} of Borel probability measures on a metric space (, d*) satisfies the full Large Deviation Principle (LDP) with speed {a_n} and good rate function I if and only if the sequence converges in (, d) to the set function e^–I. Weak convergence of probability measures is another special case of convergence in (, d). Properties related to the LDP and to weak convergence are then characterized in terms of (, d).