Supports of extremal measures with given marginals
type: Item
instance of: scholarly article
title: Supports of extremal measures with given marginals
zbMATH Open document ID: 0546.28002
author: H. G. Mukerjee
published in: Illinois Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 46E27
Mathematics Subject Classification ID: 46A55
zbMATH DE Number: 3869655
zbMATH Keywords: supports of entreme measures
zbMATH Keywords: Douglas-Lindenstrauss theorem
zbMATH Keywords: supports of extremal measures with given marginals
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3869655