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A general duality theorem for marginal problems. (English) Zbl 0818.60001

Summary: Given probability spaces \((X_ i, {\mathcal A}_ i, P_ i)\), \(i = 1,2\), let \({\mathcal M} (P_ 1, P_ 2)\) denote the set of all probabilities on the product space with marginals \(P_ 1\) and \(P_ 2\) and let \(h\) be a measurable function on \((X_ 1 \times X_ 2, {\mathcal A}_ 1 \otimes {\mathcal A}_ 2)\). In order to determine \(\sup \int hdP\) where the supremum is taken over \(P\) in \({\mathcal M} (P_ 1, P_ 2)\) a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.

MSC:

60A10 Probabilistic measure theory
28A35 Measures and integrals in product spaces
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