A characterization of random approximations (Q1283424)

Let \((\Omega, \Sigma)\) be a measurable space with \(\Sigma\) a sigma algebra of subsets of \(\Omega\) and \(M\) be a nonempty convex subset of a complex normed space \(X:A\) map \(T:\Omega\times M\to X\) is called a random operator if for each fixed \(x\in M,\) the map \(T(\cdot, x):\Omega\to X\) is measurable. Let \(\xi:\Omega\to M\) be a measurable map such that \(T(\omega, \xi(\omega)) \notin cl(M).\) The necessary and sufficiently condition for \(\xi\) to be a random best approximation for \(T\) with respect to the metric of \(X\) is established. The obtained result is analogous to the well-known Kolmogorov theorem on a characterization of the best approximation element.