Completeness of the space of separable measures in the Kantorovich–Rubinshtein metric

We consider the space $M(X)$ of separable measures on the Borel $\sigma$-algebra $\mathscr{B}(X)$ 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.