Prohorov's theorem for transition probabilities and its applications to optimal control

We present a criterion for relative weak compactness of sets of transition probabilities that generalizes Prohorov's classical theorem. Our result is closely connected to the theory of relaxed control. By means of relaxation in all variables but one, the result can be used to obtain very general lower closure and lower semicontinuity results for integral functionals. From these, in turn, various existence results for optimal control, both with and without convexity conditions, can be derived.