= Kantorovich duality by positive extension
{c}
For compact metric spaces and a continuous cost, define the marginal functional $\ell(f(x)+g(y))=\int f\,dP+\int g\,dQ$. Its majorant envelope $p(h)=\inf_{s\geq h}\ell(s)$ is finite and sublinear. The dual value is $-p(-c)$. Assign that value to the cost direction and use the dominated <Hahn-Banach theorem> to obtain a <positive linear functional> on all continuous functions of two variables. The <Riesz-Markov-Kakutani representation theorem> turns it into an optimal <transport plan> with the prescribed marginals.
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