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Monge-Kantorovich norms on spaces of vector measures. (English) Zbl 1351.28025

Summary: One considers Hilbert space valued measures on the Borel sets of a compact metric space. A natural numerical valued integral of vector valued continuous functions with respect to vector valued measures is defined. Using this integral, different norms (we called them Monge-Kantorovich norm, modified Monge-Kantorovich norm and Hanin norm) on the space of measures are introduced, generalizing the theory of (weak) convergence for probability measures on metric spaces. These norms introduce new (equivalent) metrics on the initial compact metric space.

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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