A note on strong convergence for Pettis integrable functions
type: Item
instance of: scholarly article
title: A note on strong convergence for Pettis integrable functions
zbMATH Open document ID: 1070.28004
author: Erik J. Balder
author: A. R. Sambucini
published in: Vietnam Journal of Mathematics
publication date: 2004-09-22T00:00:00Z
review text: Let \((\Omega,\Sigma,\mu)\) be a complete probability space, \(E\) be a separable Banach space and \({\mathcal P}(E)\) be the family of all probability measures on Borel subsets \({\mathcal B}(E)\) of \(E\). A sequence of measurable functions \(f_n:\Omega\to E\) is said to be s-tight if for every \(\varepsilon>0\) there exists a measurable multifunction \(\Gamma_{\varepsilon}:\Omega\to k(E)\) (nonempty, compact convex subsets of \(E\)) such that \[ \sup_n\mu(\{\omega\in\Omega: f_n(\omega)\notin\Gamma_{\varepsilon}(\omega)\})\leq\varepsilon. \] The authors prove the following result that extends the main theorem of \textit{A. Amrani, C. Castaing} and \textit{M. Valadier} published in [Vietnam J. Math. 26, No. 4, 323--335 (1998; Zbl 0965.46028)]: Let \((f_n)_n\) be an \(s\)-tight sequence of \(E\)-valued Pettis integrable functions that is Pettis uniformly integrable and weakly convergent to a Pettis integrable function \(f:\Omega\to E\). Denote by \(s-Ls\{f_n(\omega)\}\) the Kuratowski limes superior of the sequence \((f_n(\omega))_n\) in the norm topology and assume that for \(\mu\)-almost every \(\omega\) the set \(\overline{\text{conv}}(s-Ls\{f_n(\omega)\})\) is weakly locally compact and does not contain any line. If for almost every \(\omega\) the point \(f(\omega)\) is an extreme point of \(\overline{\text{conv}}(s-Ls\{f_n(\omega)\})\), then \((f_n)_n\) is convergent to \(f\) in the Pettis norm.
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 2101845
zbMATH Keywords: Pettis integration
zbMATH Keywords: convergence
zbMATH Keywords: multifunction
zbMATH Keywords: Young measures
zbMATH Keywords: Kuratowski limes superior
reviewed by: Kazimierz Musial
MaRDI profile type: Publication
Recommended article: On equivalence of strong and weak convergence in \(L_ 1\)-spaces under extreme point conditions
Recommended article: Q3165392
Recommended article: Q3775752
Recommended article: Bocce criterion and convergence in Pettis norme in \(P_E^1(\mu)\)
Recommended article: Q4433334
Recommended article: Q4381734
Recommended article: Q4315317
Recommended article: Strong convergence in Henstock-Kurzweil-Pettis integration under an extreme point condition
Recommended article: Q4010026
Recommended article: Convergence in Pettis norm under extreme point condition
description: scientific article; zbMATH DE number 2101845