Compactness criteria for the weak convergence of vector measures in locally convex spaces

We give some compactness and sequential compactness criteria for a set of Radon vector measures in certain locally convex spaces with respect to the weak convergence of vector measures. Our results contain Prokhorov-LeCam's criteria for real measures and apply to the cases which are not covered by Marz-Shortt's criteria for vector measures in a Banach space. Especially, our criteria apply to the cases that vector measures take values in the space S of all rapidly decreasing, infinitely differentiable functions, the space D of all test functions, and the strong duals of those spaces.