34 1. The Kantorovich Duality 1.2. Distance cost functions When the cost function is a metric: c(x, y) = d(x, y) on X = Y , then there is more structure in the Kantorovich duality principle. Note carefully that this distance need not be the distance defining the topology of the space. 1.2.1. The Kantorovich-Rubinstein theorem. Theorem 1.14 (Kantorovich-Rubinstein theorem). Let X = Y be a Polish space and let d be a lower semi-continuous metric on X. Let Td be the cost of optimal transportation for the cost c(x, y) = d(x, y), Td(μ, ν) = inf π∈Π(μ,ν) X×X d(x, y) dπ(x, y). Let Lip(X) denote the space of all Lipschitz functions on X, and ϕ Lip ≡ sup x=y |ϕ(x) − ϕ(y)| d(x, y) . Then Td(μ, ν) = sup X ϕ d(μ − ν) ϕ ∈ ∩L1(d|μ − ν|) ϕ Lip ≤ 1 . Moreover, it does not change the value of the supremum above to impose the additional condition that ϕ be bounded. Remarks 1.15. (i) When d is bounded, it is even possible to restrict the supremum to those ϕ’s satisfying 0 ≤ ϕ ≤ d ∞. (ii) Let P1(X) be the space of those probability measures μ such that d(x0,x) dμ(x) +∞ for some (and thus any) x0, and let M1(X) be the vector space generated by P1(X). On M1(X) we can define the norm (1.20) σ KR = sup X ϕ dσ ϕ ∈ ∩L1(d|σ|) ϕ Lip ≤ 1 . Then the Kantorovich-Rubinstein theorem states that Td(μ, ν) = μ−ν KR for all probability measures μ, ν in P1(X). (iii) When d is bounded, Td and ·KR are well-defined on P (X) and M(X) respectively then Td is sometimes called the “bounded Lipschitz distance”, and · KR the “bounded Lipschitz norm”. However it is more standard to define the “bounded Lipschitz distance” in a slightly different way [119]: dBL∗ (μ, ν) = μ − ν BL∗ ≡ sup ϕ BL ≤1 ϕ d(μ − ν), where ϕ BL ≡ ϕ ∞ + ϕ Lip. Obviously, · BL∗ ≤ · KR, but there is no equality in general: in particular, · BL∗ is defined on the whole of M(X) even when d is unbounded. For sequences of probability measures,

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