Fatou's lemma for Gelfand integrable mappings
type: Item
instance of: scholarly article
title: Fatou's lemma for Gelfand integrable mappings
zbMATH Open document ID: 1027.28016
DOI: 10.1023/A:1020292302718
published in: Positivity
publication date: 2003-03-12T00:00:00Z
review text: The authors give an exact and an approximate version of Fatou's lemma for integrably bounded, Gelfand integrable mappings taking values in the topological dual \(E^*\) of a separable Banach space \(E\) with the weak* topology. An approximate Fatou's lemma is formulated as follows: Let \((\Omega,{\mathcal A},\mu)\) be a complete finite positive measure space and \(E\) a separable Banach space. Let \((f_n)\) be an integrably bounded sequence of Gelfand integrable mappings from \(\Omega\) to \(E^*\) such that the weak* limit \(\lim_n\int_\Omega f_nd\mu\) exists. Then, for every \(w^*\)-neighborhood \(W\) of the origin in \(E^*\), there exists a Gelfand integrable mapping \(f_W : \Omega\to E^*\) such that \(\int_\Omega f_Wd\mu-\lim_n\int_\Omega f_nd\mu\in W\) and \(f_W(\omega)\in\text{Ls}_n\{f_n(\omega)\}\) for a.e.\ \(\omega\in\Omega\), where \(\text{Ls}_n\{x_n\}\) denotes the set of all weak* limit points of a sequence \((x_n)\) in \(E^*\). Their work is directly motivated by an economic model of spatial economies and the above approximate Fatou's lemma forms the central technical tool to prove the existence of general equilibria for such economies.
reviewed by: Jun Kawabe
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 1880387
zbMATH Keywords: Fatou lemma
zbMATH Keywords: Banach space
zbMATH Keywords: dual space
zbMATH Keywords: measure space
zbMATH Keywords: weak-star integral
zbMATH Keywords: Gelfand integral
author: Bernard Cornet
author: Jean-Philippe Médecin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1880387