Théorème de Hoffmann-Jørgensen et application aux amarts multivoques. (Hoffmann-Jørgensen's theorem and application to multivalued amarts)
Let \(\Omega\) be a set, \(\Sigma\) be an algebra of subsets of \(\Omega\) and (\(\Omega\),\({\mathcal F},P)\) be a probability space. Moreover, let \({\mathcal X}\) be a locally convex Hausdorff space and Q be a family of seminorms defining the topology on \({\mathcal X}\). Denote by cfb(\({\mathcal X})\) (\({\mathcal K}({\mathcal X}))\) the family of all non-empty convex bounded closed (and compact) subsets of \({\mathcal X}\). The following theorems are the main results of the paper: 1. (An extension of a result proved by \textit{J. Hoffmann-Jørgensen} [Math. Scand. 28, 5-32 (1971; Zbl 0217.380)] in the case of m with relatively compact range and quasi-complete \({\mathcal X}.)\) Let \(\mu\) be a bounded additive and positive set function on \(\Sigma\), and let \(m: \Sigma\to {\mathcal X}\) be finitely additive. Assume that for each \(p\in Q\) there exists \(Z_ p\subset {\mathcal X}'\) such that for every \(x'\in Z_ p\) the set function x'\(\cdot m\) is \(\mu\)-continuous (in the \(\epsilon\)- \(\delta\) sense) and let \(p(x)=\sup \{<x',x>: x'\in Z_ p\}\) for each \(x\in {\mathcal X}\). Then m(\(\Sigma)\) is precompact if and only if for each \(p\in Q\) and each \(\epsilon >0\) there exists a finite \(\Sigma\)-partition \(\pi\) of \(\Omega\) such that \[ p(\int_{A}f d\mu -m(A))\leq \epsilon \] for every \(A\in \Sigma\), where \(f=\sum_{E\in \pi}(m(E)/\mu (E))\chi_ E\) (we put \(0/0=0).\) This result is then generalized to a finitely additive \(m: \Sigma \to cfb({\mathcal X}).\) 2. Assume that \({\mathcal X}\) is quasi-complete and that \({\mathcal X}'\) and \({\mathcal X}\) have countable subsets separating points of \({\mathcal X}\) and \({\mathcal X}'\) respectively. The following theorem is a generalization of a result of \textit{J. F. Uhl} jun. [Z. Wahrscheinlichkeitstheor. Verw. Geb. 37, 291-295 (1977; Zbl 0325.60046)] proved for strongly measurable Pettis integrable functions with values in a Banach space: Let \((X_ n,{\mathcal F}_ n)_{n\in N}\) be an amart of strongly integrable \(X_ n: \Omega \to {\mathcal K}({\mathcal X})\) with (\({\mathcal F}_ n)_{n\in N}\) being an increasing sequence of sub-\(\sigma\)-algebras of \({\mathcal F}\). Then \((X_ n,{\mathcal F}_ n)_{n\in N}\) is Pettis Cauchy if and only if the limit set function M defined on \(\cup_{n}{\mathcal F}_ n\) is scalarly P-continuous and \(\cup \{M(A): A\in \cup_{n}{\mathcal F}_ n\}\) is relatively compact.
scientific article; zbMATH DE number 4040218
scientific article