On the parametrized integral of a multifunction: the unbounded case
type: Item
instance of: scholarly article
title: On the parametrized integral of a multifunction: the unbounded case
zbMATH Open document ID: 1127.28009
published in: Set-Valued Analysis
publication date: 2007-04-26T00:00:00Z
review text: Consider a measure space \((T,\tau,\lambda)\) with \(\lambda\) positive \(\sigma\)-finite, a measurable space \((\Omega,{\mathfrak A})\) such that \(\tau\) and \({\mathfrak A}\) are countably generated, a transition probability \(\beta: T\times{\mathfrak A}\to [0,a]\), a separable Banach space \(Y\), a \(\tau\otimes{\mathfrak A}\)-measurable closed valued multifunction \(F: T\times\Omega\to 2^Y\) such that \(F(t,\cdot)\) is \(\beta(t,\cdot)\)-integrable for \(\lambda\)-almost all \(t\in T\). Assume further that there exist a \(\lambda\)-null set \(N\) and a probability measure \(Q\) on \((\Omega,{\mathfrak A})\) such that \(Q\) dominates \(\beta(t,\cdot)\) for all \(t\in T\setminus N\). The author states the following selection theorem. Theorem A: (a) The multifunction \(G:t \mapsto\int_\Omega F(t,\omega)\beta_t(d\omega)\) is \(\tau\)-mesurable. (b) If \(g: T\to Y\) is \(\tau\)-measurable such that \(g(t)\in G(t)\) for \(\lambda\)-almost \(t\in T\), then there exists a \(\tau\otimes{\mathfrak A}\)-measurable selection \(f: T\times\Omega\to Y\) of \(F\) such that for \(\lambda\)-almost all \(t\in T\), \(f(t,\cdot)\in L^1_Y(\Omega,{\mathfrak A},\beta_t)\) and satisfies \(g(t)= \int_\Omega f(t,\omega)\beta_t(d\omega)\) for almost all \(t\in T\). A variant of the preceding result concerns the approximate representation of a measurable selection of \(\text{cl}(G(t))\) with the assumption: \(F\) is not necessarily closed valued and is graph measurable, that is, \(Gr(F)\in \tau\times{\mathfrak A}\otimes\beta(Y)\), \(\beta(Y)\) being the Borel tribe of \(Y\). Theorem B: (a) \(G\) is \(\tau_X\)-measurable; (b) if \(h\) is a \(\tau_X\)-measurable selection of \(\text{cl}(G)\), then there exists a sequence \((f_n)_{n> 1}\) of \(\tau\otimes{\mathfrak A}\)-measurable selections of \(F\) such that for all \(N\geq 1\) and for \(\lambda\)-almost all \(t\in T\), \(f_n(t,\cdot)\in L^1_Y(\Omega,{\mathfrak A}\beta_t)\) and \(h(t)= \lim_{n\to\infty}\, \int_\Omega f_n(t,\omega) \beta_t(d…
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 49N99
zbMATH DE Number: 5146251
author: Christian Hess
MaRDI profile type: Publication
full work available at URL: s11228-006-0032-6
OpenAlex ID: W2008861392
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DOI: 10.1007/S11228-006-0032-6
description: scientific article; zbMATH DE number 5146251