Completeness of certain metric spaces of measures

Abstract

We prove that the set of finite Borel measures on a separable and directionally limited metric space $$(X,\mathtt {d})$$ ( X , d ) is complete with respect to the metric $${\mathbf {d}}_{{\mathcal {A}}}(\mu ,\nu )=\sup _{A\in {\mathcal {A}}}\left| \mu (A)-\nu (A)\right| $$ d A ( μ , ν ) = sup A ∈ A μ ( A ) - ν ( A ) for all families of Borel sets $${\mathcal {A}}$$ 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.