The measurability of Carathéodory set-valued mappings and random fixed point theorems (Q1307311)

Let \((\Omega,\mathcal A)\) be a measurable space, \(X\) and \(Y\) sufficiently ``nice'' topological spaces and \(F:\Omega \times X\to 2^Y\) a set-valued mapping. \(F\) is called Carathéodory mapping if \(F(\cdot,x)\) is measurable for each \(x\) and \(F(\omega,\cdot)\) is upper semicontinuous for every \(\omega\). The first result says that each compact Carathéodory set-valued mapping with closed convex values contains a compact Carathéodory set-valued mapping with closed convex values which is jointly measurable. The result is used to derive some new random fixed point theorems (i.e., stating that for \(F:\Omega\times X\to 2^X\) there exists a measurable \(f:\Omega \to X\) such that \(f(\omega)\in F(\omega,f(\omega))\) for every or almost every \(\omega\)).