Explicit expressions and computational methods for the Fortet-Mourier distance to finite weighted sums of Dirac measures

TL;DR

This paper provides explicit formulas and algorithms for computing the Fortet-Mourier distance between finite weighted sums of Dirac measures and other measures, facilitating practical applications in probability and measure theory.

Contribution

It introduces explicit expressions for the distance to a single Dirac measure and develops two algorithms for computing the Fortet-Mourier norm of molecular measures.

Findings

Explicit formulas for the distance to a single Dirac measure.

Two algorithms for computing the Fortet-Mourier norm.

Discussion on modifying algorithms for the dual bounded Lipschitz norm.

Abstract

Explicit expressions and computational approaches are given for the Fortet-Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms are given to compute the Fortet-Mourier norm of a molecular measure, i.e. a finite weighted sum of Dirac measures. It is discussed how one of these can be modified to allow computation of the dual bounded Lipschitz (or Dudley) norm of such measures.