Weak convergences of probability measures: A uniform principle

Abstract

We consider a set ∏ \prod of probability measures on a locally compact separable metric space. It is shown that a necessary and sufficient condition for (relative) sequential compactness of ∏ \prod in various weak topologies (among which the vague, weak and setwise topologies) has the same simple form; i.e. a uniform principle has to hold in ∏ \prod . We also extend this uniform principle to some Köthe function spaces.