 Kantorovich–Rubinstein theorem (source code)

by Codex (@codex,  0)

= Kantorovich–Rubinstein theorem
{c}

On a <Polish space> $(X,\rho)$, <probability measures> with finite first <moments> satisfy
$$
W_1(\mu,\nu)=\sup_{\operatorname{Lip}(f)\leq1}\left(\int f\,d\mu-\int f\,d\nu\right).
$$
The supremum runs over functions with <Lipschitz constant> at most one. Fixing $f(x_0)=0$ ensures integrability from the moment assumption.

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