1.2. Distance cost functions 35 convergence in distance dBL∗ is equivalent to weak convergence, while con- vergence in distance Td is just slightly stronger when X is unbounded, as we shall discuss in Chapter 7. From the Kantorovich-Rubinstein theorem we will now deduce a corol- lary which is intuitively clear, but not so trivial to prove. To state it in a clear way, it will be convenient to consider mass transportation between nonnegative measures whose mass is not normalized to unity (this extension is straightforward). Corollary 1.16 (Invariance of Kantorovich-Rubinstein distance un- der mass subtraction). Let X = Y be a Polish space and let d be a lower semi-continuous distance on X. Let μ, ν and σ be three Borel (nonnegative) measures on X, such that μ[X] = ν[X] +∞, σ[X] +∞. Then Td(μ + σ, ν + σ) = Td(μ, ν). Of course the bound Td(μ + σ, ν + σ) ≤ Td(μ, ν) is immediate (exercise) however the converse bound would be more tricky to establish! This corol- lary can be reformulated in the following way: whenever μ and ν are two probability measures on X, then Td(μ, ν) = Td μ − [μ − ν]+,ν − [ν − μ]+ . In other words, in a mass transportation problem with a distance cost func- tion, one can assume that all the mass shared between the two probability measures does stay in place. In the above the notation ρ+ stands for the nonnegative part of the Radon measure ρ it is defined by the characteristic property that ρ can be written as the Hahn decomposition ρ+ − ρ−, where ρ+ and ρ− are nonnegative Borel measures which are singular to each other, i.e. concentrated on disjoint sets. We now come to the proof of the Kantorovich-Rubinstein theorem. We note that a more general version (without the completeness assumption) is proven in [119, Section 11.8]. Proof of Theorem 1.14. Let dn = d/(1+n−1d): this is a distance satisfy- ing dn ≤ d, and for all x, y the quantity dn(x, y) converges monotonically to d(x, y) as n → ∞. In particular, the set of 1-Lipschitz functions for dn is in- cluded in the set of 1-Lipschitz functions for d. Reasoning as in Step 3 of the proof of Theorem 1.3, we see that we just have to prove Theorem 1.14 with d replaced by dn. Hence, in the sequel we shall assume that d is bounded. In this case, all Lipschitz functions are bounded, and therefore integrable with respect to μ, ν. So, in view of Theorem 1.3, the only thing to check is

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