Completeness of the Space of Separable Measures in the Kantorovich-Rubinshtein Metric

We consider the space M(X) of separable measures on the Borel σ-algebra ℬ(X) of a metric space X. The space M(X) is furnished with the Kantorovich-Rubinshtein metric known also as the “Hutchinson distance” (see [1]). We prove that M(X) is complete if and only if X is complete. We consider applications of this theorem in the theory of selfsimilar fractals.