We consider a class of convex optimization problems over the space of regular Borel measures on a compact subset of n dimensional Euclidean space where the measures are restricted to be probability measures. Applications of this class of problems are discussed including mixing density estimation, maximum entropy, and optimal design. We provide a complete duality theory using perturbational techniques, establish the equivalence of these problems to associated nonconvex finite dimensional problems, and then establish the equivalence between the finite and infinite dimensional optimality conditions.