Nets and sequences of \(p\)-Bochner, \(p\)-Dunford and \(p\)-Pettis integrable functions with values in a Banach space
type: Item
instance of: scholarly article
title: Nets and sequences of \(p\)-Bochner, \(p\)-Dunford and \(p\)-Pettis integrable functions with values in a Banach space
zbMATH Open document ID: 1469.46035
author: Pratikshan Mondal
author: Lakshmi Kanta Dey
author: Sk. Jaker Ali
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2021-05-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1573268433
review text: Let \((\Omega,\Sigma,\mu)\) be a finite measure space and \(X\) a real Banach space. For \(p\in[1,\infty)\), a function \(f:\Omega\rightarrow X\) is said to be \(p\)-Bochner (resp. \(p\)-Dunford) integrable if it is measurable (resp. scalarly measurable) and \(\Vert f\Vert\in L_p(\mu)\) (resp. \(x^*f\in L_p(\mu)\) for each \(x^*\in X^*\)); it is said to be \(p\)-Pettis integrable if it is \(p\)-Dunford integrable and Pettis integrable. A net \(\{f_\alpha\}_{\alpha\in I}\) of \(p\)-Bochner (resp. \(p\)-Dunford) integrable functions is \(\delta\)-Cauchy if for each \(\varepsilon > 0\), there exist \(\gamma\in I\) and \(\delta > 0\) such that \(\int_E\Vert f_\alpha-f_\beta\Vert^p \,d\mu <\varepsilon\) (resp. \(\sup\{ \int_E|x^*(f_\alpha-f_\beta)|^p \,d\mu: x^*\in X^*,\,\Vert x^*\Vert\leq 1\} <\varepsilon\)) for all \(\alpha,\beta\in I\) with \(\alpha,\beta\geq \gamma\) and for all \(E\in\Sigma\) with \(\mu(E) < \delta\). The authors study for nets in detail the relationship between different types of convergence, in particular between \(\delta\)-Cauchy convergence, (Cauchy) convergence in measure, scalar equi-convergence or equi-Cauchy convergence in measure, (Cauchy) convergence in \(p\)-Bochner or \(p\)-Pettis norm, and uniform integrability. Moreover, a criterion is given for a net \(\{f_\alpha\}_{\alpha\in I}\) of \(p\)-Pettis integrable functions scalarly convergent in measure to \(f\) so that \(f\) is \(p\)-Pettis integrable and \(\{\text{Pettis-}\int_Ef_\alpha \,d\mu\}_{\alpha\in I}\) converges weakly to \(\text{Pettis-}\int_Ef\,d\mu\) for \(E\in\Sigma\).
reviewed by: Hans Weber
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 40A10
Mathematics Subject Classification ID: 40A30
zbMATH DE Number: 7344686
zbMATH Keywords: \(p\)-Bochner integrable functions
zbMATH Keywords: \(p\)-Dunford integrable functions
zbMATH Keywords: \(p\)-Pettis integrable functions
zbMATH Keywords: scalar equi-convergence in measure
zbMATH Keywords: \(p\)-Bochner \(\delta\)-Cauchy
zbMATH Keywords: \(p\)-Pettis \(\delta\)-Cauchy
MaRDI profile type: Publication
OpenAlex ID: W2984490884
cites work: Linear functional analysis. An application-oriented introduction. Translated from the 6th German edition by Robert Nürnberg
cites work: Vitali type convergence theorems for Banach space valued integrals
cites work: On compactness in \(L_p(\mu,X)\) in the weak topology and in the topology \(\sigma(L_p(\mu,X),L_q(\mu,X'))\)
cites work: Hölder inequality for functions that are integrable with respect to bilinear maps
cites work: Q3415147
cites work: On \(p\)-Dunford integrable functions with values in Banach spaces
cites work: An introductory course in Lebesgue spaces
cites work: Q4148368
cites work: Q4283366
cites work: Young measures and compactness in measure spaces
cites work: A~note on convergence in measure
cites work: Another look at a convergence theorem for the Henstock integral
cites work: Q4080169
cites work: Solutions of differential equations in B-spaces
cites work: Q3871120
cites work: Q3794730
cites work: Q4273174
cites work: Compactness in the Lebesgue-Bochner spaces \(L^p({\mu};X)\)
cites work: Pettis integration in locally convex spaces
cites work: Q3463654
DOI: 10.7169/FACM/1818
description: scientific article; zbMATH DE number 7344686