Stone-Weierstrass type theorems for large deviations

We give a general version of Bryc’s theorem valid on any topological space and with any algebra A of real-valued continuous functions separating the points, or any wellseparating class. In absence of exponential tightness, and when the underlying space is locally compact regular and A constituted by functions vanishing at infinity, we give a sufficient condition on the functional Λ(·)|A to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on Λ(·)|A.