Extended Bochner measurable selectors
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) F is extended Bochner measurable and upper \({\mathcal M}\)- measurable. Similar results for ''Bochner measurability'' are also obtained.
first class selectors for semi-continuous multifunctions
scientific article; zbMATH DE number 3964285
scientific article