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   RePEc:bdi:wptemi:td_1389_22.

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1. A Proofs Before turning to the proofs, we need to introduce notation. Following Agueh & Carlier (2011), we define the space of continuous functions with at most quadratic growth by Cq := (1 + k k2 2)Cb(Rd ) = f ∈ C(Rd ) : f 1 + k k2 is bounded , which will be equipped with the norm kfkq := sup y∈Rd |f(y)| 1 + kyk2 , where k k2 denotes the Euclidean norm in Rd and Cb(Rd ) denotes the space of all bounded and continuous functions. Throughout, we work in the closed subspace Cq,0 of Cq given by Cq,0 := (1 + k k2 2)C0(Rd ) = f ∈ C(Rd ) : lim kyk2→∞ f(y) 1 + kyk2 = 0 ,
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57. Proof of Proposition 1. Lemmas 2 and 3 have shown the existence of solutions to (3) under Assumption 1. Furthermore, Propositions 3.5 in Agueh & Carlier (2011) shows that a solution to (4) exists and is unique under Assumptions 1 and 2. Finally, Remark 3.9 in Agueh & Carlier (2011) shows that if condition (5) holds for some measure P, i.e. where ∇ϕj is the optimal transport map between PYjt and P, then P coincides with the barycenter P(λ) for this given λ. Therefore, if there exists a λ∗ ∈ ∆J−1 such that condition (5) holds where ∇ϕj is the optimal transport map between PYjt and PY1t , then PY1t coincides with the unique barycenter P(λ∗ ) for given λ∗
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58. Proof. The following standard mathematical argument implies that P2(Y) is compact in the weak topology (i.e. the topology with respect to the dual space of Cq(Y)). First, the Banach-Alaoglu theorem (Aliprantis & Border 2006, Theorem 6.21) implies that the closed unit ball in M2(Rd ) is compact in the weak∗ -topology, i.e. the topology with respect to the dual space of Cq,0(Rd ). Second, the cone of probability measures in M2(Rd ) is closed, so that the intersection P2(Rd ) is compact in the weak∗ -topology. Now since we consider a compact subset Yt ⊂ Rd , it follows that Cq(Y) = C0,q(Yt), so that their topologies coincide. We can therefore say that P2(Yt) is compact in the weak topology defined as the topology of the dual space of Cq(Yt).
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70. where C0(Rd ) is the space of all functions with vanishing tails. By a standard Rieszrepresentation (Aliprantis & Border 2006, chapter 14), one can identify dual space of Cq,0 by the space of all measures on Rd with finite second moments, which we define by M2(Rd ). P2(Rd ) is then defined as M2(Rd )∩P(Rd ), i.e. as the intersection of the dual space of Cq,0 with the positive cone of all probability measures on Rd . A.1 Proof of Proposition 1 We split the proof into two lemmas and the main proof: Lemmas 2 and 3 show the existence of a solution of (3) and (4) and the main proof shows the uniqueness of the counterfactual PN Y1t . Lemma 2. Under Assumptions 1 and 2, the minimum in min P∈P2(Rd) J+1 X j=2 λj W2 2 (P, PYjt ), t ≤ T0 is attained by a unique P(λ), which is continuous in λ ∈ ∆J−1 in the weak topology for the dual of Cq.
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