Monge-Kantorovich norms on spaces of vector measures

TL;DR

This paper introduces new Monge-Kantorovich type norms on spaces of vector measures, generalizing weak convergence theory for probability measures and defining new metrics on compact metric spaces.

Contribution

It defines and analyzes Monge-Kantorovich, modified Monge-Kantorovich, and Hanin norms for vector measures, extending convergence concepts in measure theory.

Findings

Defined new norms on vector measures.

Established properties of these norms and their induced metrics.

Extended weak convergence theory to vector measures.

Abstract

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 functions 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.