Remarks on the extension of random contractions
Let (\(\Omega\),\(\Sigma\)) be a measurable space with \(\Sigma\) a \(\sigma\)-algebra of subsets of \(\Omega\) and let \(F: \Omega \rightarrow 2^{X}\smallsetminus \{ \emptyset \}\) be a multifunction. We will say that \(F(.)\) is separable if there exists a countable set \(D\subset X\) such that \(F(\omega)= \overline{ F(\omega)\cap D}\), for all \(\omega \in \Omega\). Let \(\text{Gr}F\) be a graph of \(F(.)\). Then \(T:\text{Gr}F\rightarrow 2^{Y}\smallsetminus \{ \emptyset \}\) is a random operator with stochastic domain \(F(.)\) if for all \(x\in X\) and all \(U\subset Y\) open, \(\{ \omega\in \Omega : T(\omega ,x)\cap U\neq \emptyset , x\in F(\omega)\} \in \Sigma\). A function \(f\) (defined and assuming its values in metric spaces) is contraction if it is a nonexpansive mapping. The results contained in this paper are connected with the existence of the extension of some contraction functions. In particular, the main theorem of this paper is the following Let (\(\Omega\),\(\Sigma\)) be a measurable space with \(\Sigma\) a Souslin family. Let \(X\) and \(Y\) be separable complete metric spaces such that the pair (\(X,Y\)) has Kirzbraun property. Let \(F:\Omega \rightarrow 2^{X}\smallsetminus \{ \emptyset \}\) be a separable weakly measurable multifunction. Then every random contraction \(f:\text{Gr}F\rightarrow Y\) with stochastic domain \(F(.)\) can be extended to a random contraction on all \(\Omega \times X\).
scientific article; zbMATH DE number 1512515