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Kantorovich–Rubinstein duality on compact metric spaces under countable choice

Statement

Assume the Axiom of Countable Choice. Let $(X, d)$ be a nonempty compact metric space, let $μ, ν$ be countably additive Borel probability measures on $X$ , and fix $x_{0} \in X$ . Define $Π_{X} (μ, ν)$ to be the Borel probability measures $γ$ on the product-topology space $X \times X$ satisfying $γ (A \times X) = μ (A)$ and $γ (X \times A) = ν (A)$ for every Borel set $A \subseteq X$ , and define $W_{1}^{X} (μ, ν) = \inf_{γ \in Π_{X} (μ, ν)} \int_{X \times X} d (x, y) d γ (x, y)$ . Then $Π_{X} (μ, ν)$ is nonempty and $W_{1}^{X} (μ, ν) = \sup {\int_{X} f d μ - \int_{X} f d ν : f : X \to R, f (x_{0}) = 0, ∣ f (x) - f (y) ∣ \leq d (x, y) for all x, y \in X} .$

Facts & Assumptions

Given: The Axiom of Countable Choice and the compact metric space and measures in the Statement.

[A1]

The Axiom of Countable Choice selects a member from each set in a countable family of nonempty sets. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Under the Axiom of Countable Choice, every compact metric space is second-countable. A compact metric space is second-countable under countable choice

[F2]

On a nonempty compact second-countable metric space, the Borel coupling set is nonempty and the 1-Wasserstein cost equals the supremum over basepoint-normalized 1-Lipschitz functions. Kantorovich–Rubinstein duality on compact second-countable metric spaces

Proof

technique · direct

1.1givenA1F1F2

By [A1] and [F1], the metric topology of $X$ has an at-most-countable basis, so the hypotheses of [F2] hold. The countable-choice use is exactly the one recorded in [F1]; no other selection is needed.

2.1step 1.1F2∎

Apply [F2] to $X, μ, ν$ and the fixed point $x_{0}$ . It gives $Π_{X} (μ, ν) \neq \emptyset$ and the displayed Kantorovich–Rubinstein equality.

Depends on

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Sources

Dmitriy Panchenko, Lecture Notes on Probability Theory, §4.4, Theorem 4.14 and proof, pp. 121–124 (standard reference, not scraped)
Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1–1.3, PDF pp. 1–7 (standard reference, not scraped)
Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, §§2–3, PDF pp. 3–6 (standard reference, not scraped)