On the random Dugundji extension theorem (Q1109159)

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 \(g(\omega,F(\omega)).\)