On the random Dugundji extension theorem
type: Item
instance of: scholarly article
title: On the random Dugundji extension theorem
zbMATH Open document ID: 0655.28003
DOI: 10.1016/0022-247X(87)90184-3
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \((\Omega,{\mathcal B})\) be a measurable space and X a metric space. A multifunction \(F: \Omega \to 2^ X\) is called weakly measurable if, for any open \(U\subset X,\) \(\{\omega | F(\omega)\cap U\neq \emptyset \}\in {\mathcal B}.\) Let \(F: \Omega \to 2^ X\) be weakly measurable with graph G and Y another metric space. A function \(g: G\to Y\) is called Carathéodory if, for each \(x\in X,\) \(g(\cdot,x)\) is Borel measurable and, for each \(\omega \in \Omega,\) \(g(\omega,\cdot)\) is continuous. A function \(g: G\to Y\) is called G-Carathéodory if, for each measurable selection \(f: \Omega \to X\) of F, the function \(g(\cdot,f(\cdot))\) is measurable and, for each \(\omega\), \(g(\omega,\cdot)\) is continuous on \(F(\omega)\). Let g be Carathéodory or G-Carathéodory and Y additionally separable normed. Several results are proved on the existence of Carathéodory or G-Carathéodory extensions \(\tilde g\) of g (from G to \(\Omega \times X\) such that, for each \(\omega \in \Omega,\) \(\tilde g(\omega,X)\) is in the convex hull of \(g(\omega,F(\omega)).\)
reviewed by: M.P.Jerschow
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 26E25
zbMATH DE Number: 4069239
zbMATH Keywords: random versions of the Dugundji extension theorem
zbMATH Keywords: G-Carathéodory function
zbMATH Keywords: graph of a weakly measurable multifunction
zbMATH Keywords: measurable extensions
zbMATH Keywords: measurable selections
author: F. S. de Blasi
author: Józef Myjak
MaRDI profile type: Publication
cites work: Measurable extension theorems
cites work: Q5597920
cites work: Convex analysis and measurable multifunctions
cites work: Q5520521
cites work: Measurability of Extensions of Continuous Random Transforms
cites work: Correction to: Precompact contraction of metric uniformities and the continuity of F(t,x)
cites work: Q4177788
description: scientific article; zbMATH DE number 4069239