Uniform integrability and mean convergence for the vector-valued McShane integral
type: Item
instance of: scholarly article
title: Uniform integrability and mean convergence for the vector-valued McShane integral
zbMATH Open document ID: 0943.28015
published in: Real Analysis Exchange
publication date: 2000-05-21T00:00:00Z
review text: If \(X\) is a Banach space and \(f_k:\mathbb{R} \to X\) is McShane integrable for every \(k \in \mathbb{N}\), \(f_k(x) \to f(x)\) for every \(x\in \mathbb{R}\), the sequence \((f_k)\) is uniformly integrable, then \(f\) is McShane integrable and \(\|f_k-f\|_1 = \sup \{\int _{\mathbb{R}}|x'f|; x' \in X',\|x'\|\leq 1\}\to 0\). By \(X'\) the dual to \(X\) is denoted. Uniform integrability holds in an \(X\)-valued generalization of the B. Levi type monotone convergence theorem.
reviewed by: Štefan Schwabik
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 26A39
zbMATH DE Number: 1449425
zbMATH Keywords: uniform integrability
zbMATH Keywords: McShane integral
zbMATH Keywords: mean convergence
author: Charles Swartz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1449425