Two ways to define compatible metrics on the simplex of measures
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.
scientific article; zbMATH DE number 6347725
scientific article