The coupling definition of the 1-Wasserstein cost on a compact metric space

Let $(X,d)$ be a compact metric space. Write $\mathcal{B}(X)$ for the sigma-algebra generated by the open sets of its metric topology. A Borel probability measure on $X$ is a countably additive measure on $\mathcal{B}(X)$ with total mass one. For Borel probability measures $\mu ,\nu$ on $X$, let ${\mathrm{\Pi }}_X(\mu ,\nu )$ be the set of Borel probability measures $\gamma$ on $X\times X$, with its product topology and Borel sigma-algebra, such that for every $A\in \mathcal{B}(X)$, $\gamma (A\times X)=\mu (A),\gamma (X\times A)=\nu (A)\mathrm{.}$ Define $W_1^X(\mu ,\nu ):={\inf }_{\gamma \in {\mathrm{\Pi }}_X(\mu ,\nu )}{\int }_{X\times X}d(x,y)d\gamma (x,y)\in [0,+\mathrm{\infty }],$ where the infimum is taken in the extended real line and $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The integrand is nonnegative and Borel: the triangle inequality and symmetry give $\mathrm{\mid }d(x,y)-d(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\mid }\le d(x,x^{\mathrm{\prime }})+d(y,y^{\mathrm{\prime }}),$ so $(x,y)\mapsto d(x,y)$ is continuous for the product topology; nonnegativity follows from the metric axioms.

Metric space

Compact metric space

Borel sigma-algebra

Probability measure

Product topology

Extended real line

Infimum in the extended real line

Nonnegative Lebesgue integral

Metric nonnegativity