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- Proof. Let A L B. We show there exists a finite partition {C1, . . . , Cn} of S such that A L B ∪ Ci for all i = 1, . . . , n. The order <L is tight if A ∪ C <L B for all disjoint C L ∅ and B ∪ D <L A for all disjoint D L ∅ jointly imply A ∼L B. The order is fine if, for all A L ∅, there exists a finite partition {C1, . . . , Cn} of S such that A L Ci for all i = 1, . . . , n. An equivalent formulation, due to Theorem 4 of Savage (1954: 38), of the finite partition condition is that <L is fine and tight. If A L B, then the existence of disjoint C such that A B ∪ C can be established using convex-valuedness of ν. Then, A∪C <L B for all disjoint C L ∅ with A L B is excluded and <L is tight.
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