An application of a random fixed point theorem to random best approximation (Q1568650)

Let \(X\) be a Banach space and let \((\Omega, \mathcal{A})\) be a measurable space. Functions \(f,g\) from \(\Omega \times X \mapsto X\) are here called random operators. The authors show that if \(f\) anf \(g\) satisfy a long list of complicated conditions (impossible to summarize or explain here) then there is a mapping \(\xi\) from \(\Omega\) to a set \(P \subseteq X\) (\(P\) is the set of best approximations in a set \(S\) to a given point) with the property that \(\xi(\omega) = f(\omega, \xi(\omega)) = g(\omega, \xi(\omega))\) for all \(\omega \in \Omega\). Such a function \(\xi\) is here called a common random fixed point. The reviewer has the impression that, in any given instance, it may be easier to prove the result directly than verify the conditions of the theorem.