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A note on Olech’s lemma. (English) Zbl 0735.28009
Let \((\Omega,{\mathcal C},\mu)\) be a \(\sigma\)-finite measure space and \((f_n)\) a sequence of integrable functions \(f_n: \Omega\to \mathbb{R}^ q\) such that \((f_n)\) converges weakly to \(f\) in \(L_1((\Omega,{\mathcal C},\mu);\mathbb{R}^q)\). The authors study the question whether there are suitable conditions that ensure the convergence of \((f_ n)\) in the norm topology. One of the first results in this direction was given by J. L. Tartar [Res. Notes Math. 39, 136–212 (1979; Zbl 0437.35004)], who introduced the Young measure into this subject. A. Visintin [Commun. Partial Differ. Equations 9, 439–466 (1984; Zbl 0545.49019)] gives an affirmative answer in terms of extreme points. Visintin’s hypothesis, that prevents the \((f_n(\omega))\) from oscillating around \(f(\omega)\), says that \(f(\omega)\) is a.e. an extreme point of the closed convex hull of the set \(\{f_n(\omega): n\in \mathbb{N}\}\). Extensions have been obtained by Amrani-Castaing-Valadier, Balder, Castaing and Rzeżuchowski.
In this paper the authors consider the following extreme point condition: The sequence \((\int f_n\,d\mu)\) converges to an extreme point of \(\int F(\omega)\,d\mu\), where \(F\) denotes a set-valued function with closed values such that \(f_n(\omega)\in F(\omega)\) a.e. The elegant proof is based on Z. Artstein’s finite-dimensional Fatou lemma [ J. Math. Econ. 6, 277–282 (1979; Zbl 0433.28004)] and his approach to a result due to C. Olech [J. Differ. Equations 6, 512–526 (1969; Zbl 0184.19404)].

MSC:
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
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