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4). f. 10). Then we shall apply the extended Fr´echet bounds to solve Problems I and II. 10). We begin with some notation. For two nonnegative Borel measures µ and ν on IR with equal total mass λ denote by M (µ, ν) the set of all nonnegative Borel measures on IR2 with projections µ and ν. Without loss of generality set λ = 1. s of the probability laws in M (µ, ν). 12) F ∈F (A,B) F ∈F (A,B) where B(y) := ν([y, ∞)) and G(x, y) := GX,Y (x, y) := P (X ≤ x, Y ≥ y). Clearly, the laws corresponding to F ∗ and G∗ are in M (µ, ν).

Consider next the special case c(x, y) = |x − y|. f. F . Then by the triangle inequality, E|X − Y | ≤ inf (E|X − a| + E|Y − a|). 4), provided that the sum of the marginal FX + FY is known. 5) 1 G−1 (t) − G−1 (1 − t) p p ≥ 1. 3). 1 Suppose c ≥ 0 is symmetric and satisfies the Monge condition: c(x , y ) − c(x, y ) − c(x , y) + c(x, y) ≤ 0 ∀x ≥ x, y ≥ y. s (couplings) are given by (G−1 (U ), G−1 (U )), resp. (G−1 (U ), G−1 (1 − U )). Proof: Since c is symmetric, we obtain for any F ∈ FA+B , c(x, y) dF (x, y) = = 1 (c(x, y) + c(y, x)) dF (x, y) 2 F (x, y) + F (y, x) .

For two nonnegative Borel measures µ and ν on IR with equal total mass λ denote by M (µ, ν) the set of all nonnegative Borel measures on IR2 with projections µ and ν. Without loss of generality set λ = 1. s of the probability laws in M (µ, ν). 12) F ∈F (A,B) F ∈F (A,B) where B(y) := ν([y, ∞)) and G(x, y) := GX,Y (x, y) := P (X ≤ x, Y ≥ y). Clearly, the laws corresponding to F ∗ and G∗ are in M (µ, ν). 13) σ G (x, y) := σ((−∞, x] × [y, ∞)) F(A, B, F σ ) := {F ∈ F(A, B); F ≤ F σ }, G(A, B, Gσ ) := {GX,Y ; FX,Y ∈ F(A, B), GX,Y ≤ Gσ } .

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