Extended Bochner measurable selectors
type: Item
instance of: scholarly article
title: Extended Bochner measurable selectors
zbMATH Open document ID: 0598.28019
DOI: 10.1007/BF01457279
author: R. W. Hansell
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164236
review text: Let (T,\({\mathcal M})\) be a measurable space, \(2^ X\) the nonempty closed subsets of a complete metric space X, and let \(F: T\to 2^ X\) be lower \({\mathcal M}\)-measurable. If X is nonseparable, F need not have any Borel selector even when T is completely metrizable and F is of class 1. Call \({\mathcal E}\subset {\mathcal M}\) \({\mathcal M}\)-discrete if \({\mathcal E}\) is disjoint and, whenever \(M_ E\subset E\) belongs to \({\mathcal M}\) for each \(E\in {\mathcal E},\) then \(\cup_{E\in {\mathcal E}}M_ E\in {\mathcal M}.\) F is \({\mathcal M}\)-discrete if its sets of constancy form an \({\mathcal M}\)-discrete collection. Consider the properties: (A) F has a \(\sigma\)-\({\mathcal M}\)- discrete function base \({\mathcal B}\) if \({\mathcal B}\) is a countable union of \({\mathcal M}\)-discrete collections and each \(\{t: F(t)\cap U\neq \emptyset \},\) U open in X, is a union of sets from \({\mathcal B}\); (B) F is extended Bochner measurable if F is the pointwise limit, relative to the Vietoris topology, of a sequence of \({\mathcal M}\)-discrete functions from T to \(2^ X\). Then (A) implies (B), and the converse holds whenever the range of F has a point-countable open base; in particular, if F is separable-valued. We prove that (A) implies the existence of a selector with a \(\sigma\)- \({\mathcal M}\)-discrete function base. The Kuratowski and Ryll-Nardzewski theorem follows easily from this, and we also deduce sharper forms of several known results on first class selectors for semi-continuous multifunctions. Furthermore, it is shown that F will have property (A) in each of the following cases: (i) for some uniformity \({\mathcal U}\) on X (not necessarily metrizable), F is a \(2^{{\mathcal U}}\)-uniform limit of a sequence of \({\mathcal M}\)-discrete functions from T to \(2^ X\); (ii) F is the pointwise limit, relative to the lower topology on \(2^ X\), of a sequence of separable-valued \({\mathcal M}\)-discrete functions from T to \(2^ X\); (iii) …
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 3964285
zbMATH Keywords: Borel selectors
zbMATH Keywords: Bochner measurability
zbMATH Keywords: extended Bochner measurable selectors
zbMATH Keywords: first class selectors for semi-continuous multifunctions
zbMATH Keywords: Kuratowski and Ryll-Nardzewski theorem
MaRDI profile type: Publication
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OpenAlex ID: W2062122025
description: scientific article; zbMATH DE number 3964285