The Monge–Kantorovich metric on multimeasures and self–similar multimeasures

For a metric space (X,d) the classical Monge-Kantorovich metric dM gives a distance between two probability measures on X which is tied to the underlying distance don X in an essential way. In this paper, we extend the Monge-Kantorovich metric to signed measures and set-valued measures (multimeasures) and, in each case, prove completeness of a suitable space of these measures. Using this extension as a framework, we construct self-similar multimeasures by using an IFS-type Markov operator.