On topological support of Dirichlet prior

Ferguson (1973) states that the topological support of a Dirichlet prior on the Borel [sigma]-field (with respect to the weak convergence topology) of the set of all probability measures on the lines is the set of all probability measures with their topological supports contained in that of the parameter measure of the Dirichlet prior. We prove that statement with the line replaced by an arbitrary separable metric space and cite an application of our result to decision theory.