On the basic representation theorem for convex domination of measures
type: Item
instance of: scholarly article
title: On the basic representation theorem for convex domination of measures
zbMATH Open document ID: 0947.28013
author: John H. Elton
author: Theodore P. Hill
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-11-06T00:00:00Z
full work available at URL: https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi
review text: Let \(P\) and \(Q\) be finite measures on \(\mathbb{R}^d\) with finite support. The main result uses a constructive proof to establish three statements, each of which is equivalent to the convex domination of \(Q\) by \(P\) (i.e., \(\int c dP\geq \int c dQ\) for every nonnegative convex function \(c\)). A second result shows how this result can be used to produce representation theorems for measures defined on infinite-dimensional spaces or which have infinite support. The latter result implies a number of well-known results, including a theorem of \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya} [Messenger Math. 58, 145-152 (1929; JFM 55.0740.04)].
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60A10
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 1246329
zbMATH Keywords: fusion of a measure
zbMATH Keywords: dilation
zbMATH Keywords: convex domination
zbMATH Keywords: majorization
zbMATH Keywords: Banach space
zbMATH Keywords: locally convex topological vector space
reviewed by: R. James Tomkins
MaRDI profile type: Publication
OpenAlex ID: W1994123808
cites work: Equivalent Comparisons of Experiments
cites work: Comparaison, des mesures portées par un ensemble convexe compact
cites work: Fusions of a probability distribution
cites work: Balayage Defined by the Nonnegative Convex Functions
cites work: Q4136031
cites work: Majorization of vectors and inequalities for convex functions
cites work: Q5508589
cites work: The Existence of Probability Measures with Given Marginals
DOI: 10.1006/JMAA.1998.6158
description: scientific article; zbMATH DE number 1246329