Weak convergence of random sets in Banach spaces
type: Item
instance of: scholarly article
title: Weak convergence of random sets in Banach spaces
zbMATH Open document ID: 0784.46010
DOI: 10.1016/0022-247X(92)90136-2
author: Nikolaos S. Papageorgiou
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \((\Omega,\Sigma,\mu)\) be a complete finite measure space, let \(X\) be any separable Banach space. A ``random set'' \(F\) in \(X\) is a set-valued function \(F: \Omega\to \{A\subseteq X: \emptyset\neq A\) closed\} with an \(\Omega\times\text{Borel}(X)\)-measurable graph \(\text{Gr}(F)=\bigl\{(\omega,x): x\in F(\omega)\bigr\}\). A random set \(F\) is called to be integrably bounded if the function \(\psi(\omega)=\sup\bigl\{\| x\|: x\in F(\omega)\bigr\}\) is an integrable function. The vector space of all integrably bounded random sets is denoted by \({\mathcal L}^ 1(X)\). With this notions, certain additional conditions (e.g. compactness or w-compactness of the sets \(F(\omega)\) \(\mu\)-a.e.) and with help of \(L^ 1(X)\)-selectors \(f\) of \(F\) the author proves -- Dunford-Pettis type criteria for the w-compactness of subsets in \({\mathcal L}^ 1(X)\) -- a version of Mazur's Lemma concerning the approximation of weak limits of sequences by convex combinations -- a w-convergence theorem for w-compact martingales. The results are applied to the approximation theory and to the control theory as well.
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 49J24
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 62058
zbMATH Keywords: complete finite measure space
zbMATH Keywords: integrably bounded random sets
zbMATH Keywords: Dunford- Pettis type criteria
zbMATH Keywords: Mazur's Lemma
zbMATH Keywords: w-convergence theorem for w-compact martingales
zbMATH Keywords: control theory
reviewed by: Heinz Junek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 62058