Measure solutions of systems of inequalities (Q1327676)

Let \(X\) and \(Y\) be compact Hausdorff spaces, \(\{f_ i\}\) and \(\{g_ i\}\) two families of real-valued continuous functions defined on \(X\), respectively \(Y\) and indexed by the same set \(I\). The problem is to prove the existence of nonnegative Radon measures \(\mu\) and \(\nu\) not both zero such that \[ \int_ X f_ i d\mu\leq \int_ Y g_ i d\nu,\quad i\in I. \] Using a generalization of the Mazur-Orlicz theorem, the author proves that there exist probability measures \(\mu\) and \(\nu\) such that the relation from above holds if and only if for any positive integer \(n\) one has \[ \text{Min}_{x\in X} \sum^ n_{k= 1} \lambda_ k f_{i_ k}(x)\leq \text{Max}_{y\in Y} \sum^ n_{k= 1} \lambda_ k g_{i_ k}(y) \] for all \(i_ 1,\dots,i_ n\) in \(I\) and all \(\lambda_ k\geq 0\), \(k= 1,\dots, n\). The minimax theorem of Ky Fan for a concave family of l.s.c. convex functions on a compact convex subset of a Hausdorff t.v.s. and a duality theorem of Gale are then obtained as corollaries. The author gives a necessary and sufficient condition for a more general situation and applies it to Lagrangian duality.