A primal representation of the Monge-Kantorovich norm

TL;DR

This paper demonstrates that the Monge-Kantorovich norm on measures over a compact metric space can be represented in a primal form similar to the Hanin norm, revealing new equivalences and extensions.

Contribution

It introduces a primal representation of the Monge-Kantorovich norm, linking it to the Hanin norm and extending the Kantorovich-Rubinstein norm.

Findings

Monge-Kantorovich and Hanin norms are equivalent.

The Monge-Kantorovich norm extends the Kantorovich-Rubinstein norm.

New results on the structure of measure norms.

Abstract

In this note, following \cite{Chitescuetal2014}, we show that the Monge-Kantorovich norm on the vector space of countably additive measures on a compact metric space has a primal representation analogous to the Hanin norm, meaning that similarly to the Hanin norm, the Monge-Kantorovich norm can be seen as an extension of the Kantorovich-Rubinstein norm from the vector subspace of zero-charge measures, implying a number of novel results, such as the equivalence of the Monge-Kantorovich and Hanin norms.