Parametrized integral of multifunctions in Banach spaces
type: Item
instance of: scholarly article
title: Parametrized integral of multifunctions in Banach spaces
zbMATH Open document ID: 0980.28006
author: Jean Saint-Pierre
author: Saïd Sajid
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-03-02T00:00:00Z
review text: Let \(T\), \(S\) be complete separable metric spaces, \(E\) a separable Banach space, \(\lambda\) a finite measure on \(T\), \((\beta_t)_{t\in T}\) a family of probability measures on \(S\), and let \(F\) be a measurable multifunction from \(T\times S\) to the space of all nonempty compact subsets of \(E\). Suppose further that for all \(t\), the multifunction \(s\to F(t,s)\) is \(\beta_t\)-integrably bounded and for any \(D\in \mathbb{B}(s)\), \(t\to \beta_t(D)\) is Borel-measurable. It is asserted then that the multifunction \[ t\to \Gamma(t)= \text{cl}\Biggl(\int_s F(t,s) \beta_t(ds)\Biggr) \] is \(\mathbb{B}_\lambda(T)\)-measurable. Moreover, the structure for measurable selections for \(\Gamma\) in terms of integrals of a sequence of measurable selections of \(F\) is obtained. It is asserted that when \(F\) is convex-valued one can find a function \(g\) such that, if \(\gamma\) is a selection of \(\Gamma\), \[ \gamma(t)= \int_s g(t,s)\beta_t(ds). \]
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 1389235
zbMATH Keywords: parametrized integral
zbMATH Keywords: separable Banach space
zbMATH Keywords: measurable multifunction
zbMATH Keywords: measurable selections
reviewed by: Alberto Castejón
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6535
OpenAlex ID: W2043665709
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DOI: 10.1006/JMAA.1999.6535
description: scientific article; zbMATH DE number 1389235