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

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Abstract

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.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. S. Kravchenko

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  1. A. S. Kravchenko
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 85–96, January–February, 2006.

Original Russian Text Copyright © 2006 Kravchenko A. S.

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Kravchenko, A.S. Completeness of the Space of Separable Measures in the Kantorovich-Rubinshtein Metric. Sib Math J 47, 68–76 (2006). https://doi.org/10.1007/s11202-006-0009-6

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  • DOI: https://doi.org/10.1007/s11202-006-0009-6

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