Sur le problème des marges (Q1056963)

Let X be a set, A an algebra of subsets of X, m and M two mappings from A to \({\bar {\mathbb{R}}}_+\). Then there exists a finitely additive measure \(\theta\) on A such that \(m\leq \theta \leq M\) if and only if for all the sequences \((A_ 1,...,A_ p)\) and \((B_ 1,...,B_ q)\) in A such that \(\sum^{p}_{i=1}1_{A_ i}\leq \sum^{q}_{i=1}1_{B_ i}\), the inequality \(\sum^{p}_{i=1}m(A_ i)\leq \sum^{q}_{i=1}M(B_ i)\) is satisfied. This simple condition permits us to deduce and generalize many previous results relating to the ''marginal problem''.