On an inequality of Edgar (Q1192117)

Let \((\Omega,{\mathcal F},P)\) be a probability space. For an adapted sequence of non-negative random variables \((X_ n\)) Edgar's inequality is \[ \int_ \Omega \limsup_ n X_ n dP\leq\limsup_{\tau\in T} \int_ \Omega X_ \tau dP, \] where \(T\) is the set of bounded stopping times corresponding to the increasing sequence \(({\mathcal F}_ n)\) of sub- \(\sigma\)-algebras of \({\mathcal F}\), see \textit{G. Edgar} [Ann. Inst. Henri Poincaré, N. Sér., Sect. B 15, 197-203 (1979; Zbl 0423.60046)]. In this paper this inequality is proved for \({\mathcal L}_ 1\)-bounded adapted sequences of \({\mathcal K}\)-valued random variables, where \({\mathcal K}\) denotes the space of all non-empty convex, weak\(^*\)-compact subsets of the dual of a separable Banach space \(E\) equipped with the Hausdorff metric. The author obtains from this theorem that any \({\mathcal L}_ 1\)-bounded multivalued uniform amart \((X_ n,{\mathcal F_ n})\) converges to a multivalued random variable \(X_ \infty: \Omega\to{\mathcal K}\) in the following sense: there is a null set \(A\subseteq\Omega\) such that for all \(\omega\in\Omega\setminus A\) and \(x\in E\) the following holds true \[ \lim_{n\to\infty} \sup_{x'\in X_ n(\omega)} \langle x,x'\rangle=\sup_{x'\in X_ \infty(\omega)}\langle x,x'\rangle. \]