Completeness of certain metric spaces of measures

We prove that the set of finite Borel measures on a separable and directionally limited metric space (X, d) is complete with respect to the metric d(A)(mu, v) = sup (A is an element of A)vertical bar mu (A) - nu(A)vertical bar for all families of Borel sets A that contain every closed ball of X. This allows to prove the existence and uniqueness of the invariant Borel probability measure of certain Markov processes on X. A natural application is a Markov process induced by a random similitude.