Two ways to define compatible metrics on the simplex of measures
type: Item
instance of: scholarly article
title: Two ways to define compatible metrics on the simplex of measures
zbMATH Open document ID: 1319.46024
published in: Journal of Mathematical Sciences (New York)
publication date: 2014-09-25T00:00:00Z
review text: In this paper, \((X, r) \) is a metric space, \(\mathrm{Meas}(X)\) is the simplex of all probability measures with the topology of pointwise convergence on the space of continuous functions with compact supports. For \(x \in X\), let \(\delta_{x}\) denote the Dirac measure at \(x\). A metric \(\rho\) on \(\mathrm{Meas}(X)\) is said to be compatible with \(r\) if \(\rho(\delta_{x}, \delta_{y}) =r(x, y)\). In this work, the author discusses two methods of getting compatible metrics, which are generalizations of Kantovorich methods. For the first method, called the direct method by the author, take \(\mu_{1}, \mu_{2}\) in \(\mathrm{Meas}(X) \) and let \(\Psi_{\mu_{1},\mu_{2}}\) be the set of all Borel measures on \(X \times X\) having \(\mu_{1}, \mu_{2}\) as its marginals. For a Borel probability measure \(\psi\) on \(X \times X\), let \(\mathcal{N}_{\;\psi}\) be a norm in the space of real-valued measurable functions on \(X \times X \). Then a compatible metric \(\rho_\mathcal{N}^{r}\) on \(\mathrm{Meas}(X)\) is defined as: \[ \rho_\mathcal{N}^{r} (\mu_{1}, \mu_{2})= \inf_{\psi \in \Psi_{\mu_{1},\mu_{2}}}\mathcal{N}_{\psi}(r(., .)). \] For the second method, let \(\mathrm{Lip}_{1}(X, r)\) be the space of Lipschitz functions with constant \(1\). A compatible metric is defined as \[ \rho_{\mathcal{L}}^{r} (\mu_{1}, \mu_{2}) = \sup_{u \in \mathcal{L}} \left| \int_{X} u d (\mu_{1}- \mu_{2}) \right|, \] \(\mathcal{L}\) being an appropriate subspace of \(\mathrm{Lip}_{1}(X, r).\) Many additional comments and additional results about these compatible metrics are given.
reviewed by: Surjit Singh Khurana
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 6347725
zbMATH Keywords: compatible metrics
zbMATH Keywords: simplex of probability measures
zbMATH Keywords: Lipschitz function
zbMATH Keywords: Kantorovich metrics
author: A. M. Vershik
MaRDI profile type: Publication
full work available at URL: s10958-013-1645-6
OpenAlex ID: W1987878761
cites work: The Monge-Kantorovich problem: achievements, connections, and perspectives
cites work: The Kantorovich metric: the initial history and little-known applications
cites work: Virtual continuity of measurable functions of several variables and embedding theorems
cites work: Mathematical methods of organizing and planning production. English translation by Robert W. Campbell and W. H. Marlow
cites work: Q3250845
cites work: Q4020553
cites work: Linearly rigid metric spaces and the embedding problem
cites work: THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
cites work: Metric aspects of noncommutative homogeneous spaces
cites work: The Existence of Probability Measures with Given Marginals
cites work: Polymorphisms, Markov processes, and quasi-similarity
cites work: Long history of the Monge-Kantorovich transportation problem
cites work: Optimal Transport
DOI: 10.1007/S10958-013-1645-6
description: scientific article; zbMATH DE number 6347725