Kantorovich-Rubinstein theorem

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments.

Let $$X,Y$$ be topological spaces. Let $$\mathcal P(X), \mathcal P(Y)$$ be the spaces of all Borel probability measures on $$X,Y$$ respectively. Let $$c: X \times Y \to [0, +\infty]$$. Fix $$\mu \in \mathcal P(X)$$ and $$\nu \in \mathcal P(Y)$$.

• $$\Pi(\mu, \nu)$$ is the set of $$\pi \in \mathcal P(X \times Y)$$ such that for all measurable subsets $$A \subset X$$ and $$B \subset Y$$, $$\pi (A \times Y) = \mu (A) \quad \text{and} \quad \pi (X \times B) = \nu (B).$$

• $$\Phi_{c}$$ is the set of all $$(\varphi, \psi) \in L_1 (\mu) \times L_1 (\nu)$$ satisfying $$\varphi(x)+\psi(y) \leq c(x, y)$$ for $$\mu$$-a.e. $$x \in X$$ and $$\nu$$-a.e. $$y \in Y$$.

• For $$\pi \in \mathcal P(X \times Y)$$ and $$(\varphi, \psi) \in L_1 (\mu) \times L_1 (\nu)$$, let $$\mathbb K (\pi) := \int_{X \times Y} c d \pi \quad \text{and} \quad \mathbb J(\varphi, \psi) := \int_{X} \varphi d \mu+\int_{Y} \psi d \nu .$$

Kantorovich-Rubinstein: Let $$X = Y$$ be Polish spaces and and $$c$$ a lower semi-continuous metric. Define $$\|f\|_{\mathrm{Lib}} = \sup_{x\neq y} \frac{|f(x) - f(y)|}{c(x, y)}.$$ Assume that there exists $$(a,a) \in X \times Y$$ such that $$c( \cdot, a) \in L_1(\mu)$$ and $$c(a, \cdot) \in L_1(\nu)$$. Then $$\min_{\pi \in \Pi (\mu, \nu)} \mathbb K (\pi) = \sup \left \{ \int_X f d (\mu - \nu) \,\middle\vert\, f \in L_1 (|\mu- \nu|) , \|f\|_{\mathrm{Lib}} \le 1 \right \}.$$

1. $$c$$ is bounded, i.e., $$\sup_{x, y\in X} c(x, y) < C$$ for some $$C \in \mathbb R$$.

Then $$\|f\|_{\mathrm{Lib}} \le 1$$ implies $$f$$ is bounded, and thus $$f \in L_1 (|\mu- \nu|)$$ and $$(-f, f) \in \Phi_{c}$$. By Kantorovich duality, it suffices to prove $$\sup _{\Phi_{c}} \mathbb J(\varphi, \psi) = \sup \left \{ \int_X f d (\mu - \nu) \,\middle\vert\, f \in L_1 (|\mu- \nu|) , \|f\|_{\mathrm{Lib}} \le 1 \right \}.$$

If $$f \in L_1 (\mu)$$, then $$f^c:Y \to \mathbb R \cup \{\pm \infty\}, y \mapsto \inf_{x \in X} [c(x, y) - f(x)]$$ is such that $$\|f^c\|_{\mathrm{Lib}} \le 1$$ and $$f^{cc} = -f^c$$. By this lemma, we have $$\sup _{\Phi_{c}} \mathbb J(\varphi, \psi) = \sup _{f \in L_1 (\mu)} \mathbb J(f^{cc}, f^c) = \sup _{f \in L_1 (\mu)} \mathbb J(-f^{c}, f^c) \le \sup _{\|f\|_{\mathrm{Lib}} \le 1} \mathbb J(-f, f) \le \sup _{\Phi_{c}} \mathbb J(\varphi, \psi).$$

1. $$c$$ is unbounded.

We have $$\sup \left \{ \int_X f d (\mu - \nu) \,\middle\vert\, f \in L_1 (|\mu- \nu|) , \|f\|_{\mathrm{Lib}} \le 1 \right \} \le \sup _{\Phi_{c}} \mathbb J(\varphi, \psi).$$

It remains to prove $$\sup \left \{ \int_X f d (\mu - \nu) \,\middle\vert\, f \in L_1 (|\mu- \nu|) , \|f\|_{\mathrm{Lib}} \le 1 \right \} \ge \sup _{\Phi_{c}} \mathbb J(\varphi, \psi).$$

Let $$c_n := \min\{n, c\}$$. Then $$c$$ is a bounded metric on $$X$$ such that $$c_n \nearrow c$$. By this result, we have $$\lim_n \inf_{\pi \in \Pi(\mu, \nu)} \int c_n d\pi = \inf_{\pi \in \Pi(\mu, \nu)} \int c d\pi = \sup _{\Phi_{c}} \mathbb J(\varphi, \psi).$$

On the other hand, \begin{align} \inf_{\pi \in \Pi (\mu, \nu)} \int_{X \times Y} c_n d \pi &= \sup \left \{ \int_X f d (\mu - \nu) \,\middle\vert\, f \in L_1 (|\mu- \nu|) , \|f\|_{\mathrm{Lib}, c_n} \le 1 \right \} \\ &\le \sup \left \{ \int_X f d (\mu - \nu) \,\middle\vert\, f \in L_1 (|\mu- \nu|) , \|f\|_{\mathrm{Lib}} \le 1 \right \}. \end{align}

This completes the proof.