Weak convergence of random sets in Banach spaces
Let \((\Omega,\Sigma,\mu)\) be a complete finite measure space, let \(X\) be any separable Banach space. A ``random set'' \(F\) in \(X\) is a set-valued function \(F: \Omega\to \{A\subseteq X: \emptyset\neq A\) closed\} with an \(\Omega\times\text{Borel}(X)\)-measurable graph \(\text{Gr}(F)=\bigl\{(\omega,x): x\in F(\omega)\bigr\}\). A random set \(F\) is called to be integrably bounded if the function \(\psi(\omega)=\sup\bigl\{\| x\|: x\in F(\omega)\bigr\}\) is an integrable function. The vector space of all integrably bounded random sets is denoted by \({\mathcal L}^ 1(X)\). With this notions, certain additional conditions (e.g. compactness or w-compactness of the sets \(F(\omega)\) \(\mu\)-a.e.) and with help of \(L^ 1(X)\)-selectors \(f\) of \(F\) the author proves -- Dunford-Pettis type criteria for the w-compactness of subsets in \({\mathcal L}^ 1(X)\) -- a version of Mazur's Lemma concerning the approximation of weak limits of sequences by convex combinations -- a w-convergence theorem for w-compact martingales. The results are applied to the approximation theory and to the control theory as well.
w-convergence theorem for w-compact martingales
scientific article; zbMATH DE number 62058
scientific article