Liftings of Pettis integrable functions
type: Item
instance of: scholarly article
title: Liftings of Pettis integrable functions
zbMATH Open document ID: 0970.46031
author: N. D. Nacheras
author: Kazimierz Musial
published in: Hiroshima Mathematical Journal
publication date: 2001-10-16T00:00:00Z
review text: If \((\Omega,\Sigma,\mu)\) is a finite complete measure space and \(\rho\) a lifting on \(L^\infty(\mu)\), then for weakly measurable and scalarly bounded \(f:\Omega\to\) Banach space \(X\) the \(\rho_1(f): \Omega\to X^{**}\) is uniquely defined by \(x^*(\rho_1(f)(\omega))= \rho(x^*f)(\omega)\) for \(\omega\in \Omega\), \(x^*\in X^*\); \(\rho_0(f): \Omega\to X^*\) is defined similarly for \(w^*\)-measurable and \(w^*\)-bounded \(f: \Omega\to X^*\). Extending results of \textit{M. Talagrand} [``Pettis integral and measure theory'', Mem. Am. Math. Soc. 307 (1984; Zbl 0582.46049)] it is shown that if \(f\) is Pettis integrable, scalarly bounded and each \(z\in X^{***}\) is \(\mu\rho_1(f)^{-1}\)-measurable, then \(\rho_1(f)\) is Pettis integrable, too; an analogous result holds for \(\rho_0(f)\). Similarly, if \(f:\Omega\to X\) (resp. \(X^*\)) is McShane integrable [see \textit{D. H. Fremlin}, Ill. J. Math. 39, No. 1, 39-67 (1995; Zbl 0810.28006)] and scalarly bounded (\(w^*\)-bounded), \(\rho_1(f)\) (resp. \(\rho_0(f)\)) is Pettis integrable, provided \(\rho\) is consistent.
Mathematics Subject Classification ID: 46G15
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 28A15
zbMATH DE Number: 1539262
zbMATH Keywords: McShane integrable
reviewed by: Hans Günzler
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206124684
OpenAlex ID: W2170243251
Wikidata QID: Q128870334
DOI: 10.32917/HMJ/1206124684
description: scientific article; zbMATH DE number 1539262