Supports of extremal measures with given marginals
Let X and Y be countable measure spaces with positive finite measures \(\mu\) and \(\nu\), respectively. Let \(M(\mu,\nu)\) be the (convex) class of all positive measures on \(X\times Y\) with marginals (projections) \(\mu\) and \(\nu\), respectively. The extreme points of \(M(\mu\),\(\nu)\) have been characterized by Douglas Lindenstrauss by the following theorem: \(\lambda \in M(\mu,\nu)\) is extreme if and only if \(\{p+q:p\in L_ 1(\mu)\), \(q\in L_ 1(\nu)\}\) is norm-dense in \(L_ 1(\lambda).\) The problem of characterizing the supports of the extremes is essentially combinatorial rather than measure-theoretic. The purpose of this paper is to use some simple combinatorial ideas to develop several characterizations of the extremes by their supports. The proofs are elementary, geometrically intuitive, and do not use the Douglas- Lindenstrauss theorem. One result is a strengthened version of the Douglas-Lindenstrauss theorem in this countable case.
supports of extremal measures with given marginals
scientific article; zbMATH DE number 3869655