4 Introduction where T is a measurable map X → Y . The probability measure appearing on the right-hand side of (7) can also be written (Id ×T )#μ it is the probability measure on X ×Y satisfying the following characteristic property: whenever ζ is a nonnegative (say) measurable function on X × Y , then (8) X×Y ζ(x, y) dπT (x, y) = X ζ(x, T (x)) dμ(x). In particular, the associated total transportation cost is I[πT ] = X c(x, T (x)) dμ(x). What is the condition that T in (7) should satisfy for πT to belong to Π(μ, ν)? Well, in view of (8), condition (2) translates into X ϕ(x) + ψ ◦ T (x) dμ(x) = X ϕ(x) dμ(x) + Y ψ(y) dν(y). Cancelling out ϕ dμ on both sides, we recover the condition (9) X (ψ ◦ T ) dμ = Y ψ dν. This identity should hold true for all nonnegative ψ, or equivalently for all ψ ∈ L1(dν), or equivalently for all ψ ∈ L∞(dν). More precisely: for all ψ ∈ L1(dν), the measurable function ψ ◦ T should lie in L1(dμ), and the values of both integrals in (9) should coincide. Equivalently, in terms of measurable subsets, the condition for πT to belong to Π(μ, ν) can be recast as (10) for any measurable set B ⊂ Y , ν[B] = μ[T −1(B)]. Whenever the equivalent conditions (9) or (10) are satisfied, we shall write ν = T #μ and say that ν is the push-forward, or image measure of μ by T or that T transports μ onto ν. Of course, the law of a random variable U defined on a probability space Ω equipped with a probability P is nothing but the image measure U#P. We can now formulate a strengthened version of Kantorovich’s problem. Monge’s optimal transportation problem: Minimize I[T ] = X c(x, T (x)) dμ(x) over the set of all measurable maps T such that T #μ = ν.

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