ON THE EXISTENCE OF PROBABILITY MEASURES WITH GIVEN MARGINALS

As an immediate consequence of a minimax theorem Ky Fan [2] proved a characterization of consistency of a system of convex inequalities. The aim of this paper is to show that by means of Ky Fan's theorem one can obtain simple and unified proofs of some existence theorems on probability measures with given marginals, avoiding in this way approximation and disintegration techniques. 1. Preliminaries. In fact we shall make use of the following special case of Theorem 1 of Ky Fan [2]: THEOREM 1. Let Κ be a compact, convex set in a real topological vector space X. Let (Fg)ses be a family of continuous, linear, real functions on X and ( a s) s 6 s be a family of real numbers. Then the system (1) Fs ( x ) = as, s e S is consistent on Κ (i.e. there exists an xQ 6 Κ satisfying (1)), if and only if for any finite set of indices 6 S and AMS subject classification: 28 A 35, 46 E 27.