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Florian, Gunsilius. (2021) Distributional synthetic controls.
In: Papers. RePEc:arx:papers:2001.06118.
Full description at Econpapers
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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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