Summary
LetX be a set,A an algebra of subsets ofX, m andM two mappings fromA to\(\bar {\mathbb{R}_(+)}\). Then there exists a finitely additive measure θ onA such thatm≦θ≦M if and only if for all the sequences (A 1, ...,A p ) and (B 1, ...,B q ) inA such that\(\sum\limits_{i = 1}^p {1_{A_i } \leqq } \sum\limits_{i = 1}^q {1_{Bi} }\) the inequality\(\sum\limits_{i = 1}^p {m(A_i ) \leqq } \sum\limits_{i = 1}^q {M(B_i )}\) is satisfied. This simple condition permits us to deduce and generalize many previous results relating to the “marginal problem”.
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Hansel, G., Troallic, JP. Sur le problème des marges. Probab. Th. Rel. Fields 71, 357–366 (1986). https://doi.org/10.1007/BF01000211
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DOI: https://doi.org/10.1007/BF01000211