On equivalence of strong and weak convergence in \(L_ 1\)-spaces under extreme point conditions
Under suitable extreme point conditions weak convergence implies strong convergence in \(L^ 1\)-spaces [see \textit{C. Dellacherie} and \textit{P.-A. Meyer}, ``Probabilités et potentiel'' (1975; Zbl 0323.60039), \textit{L. Tartar}, Nonlinear Analysis and Mechanics, Res. Notes Math. 39, 136-212 (1979; Zbl 0437.35004) and \textit{A. Visintin} [Commun. Partial Differ. Equations 9, 439-466 (1984; Zbl 0545.49019)]. Consider the Banach space \(L^ 1(E,\mu)\) of Bochner-integrable functions \(u:\Omega\to E\) built over a \(\sigma\)-finite measure space and a Banach space E. If \(\tau\) denotes a locally convex Hausdorff topology not stronger than the norm topology and not weaker than the weak topology, then \(\text{Ls}_ \tau(A_ n)\) stands for the limes superior in the sense of Kuratowski of a sequence \((A_ n)\) of subsets of \(E\). The main result states the following: Let \((u_ n)\) be a sequence converging weakly in \(L^ 1(E,\mu)\) to \(u_ 0\). If \((\varepsilon_{u_ n})\) is tight with respect to the topology \(\tau\) on \(E\) and if \(u_ 0(\omega)\) is a.e. an extreme point of the closed convex hull of \(\text{Ls}_ \tau(u_ n(\omega))\), then the following convergence property holds: for every function \(g:\Omega\times E\to[0,+\infty]\) such that \(g(\omega,\cdot)\) is l.s.c. on \((E,\tau)\) for a.e. \(\omega\), \[ \liminf_{n\to\infty}\int^*g(\omega,u_ n(\omega))\mu(d\omega)\geq\int^*g(\omega, u_ 0(\omega))\mu(d\omega). \] The main theorem contains the results cited above and theorems of \textit{C. Castaing} [Travaux Sémin. Anal. Convex, Univ. Sci. Tech. Languedoc 18, 2.1-2.18 (1988; Zbl 0698.46024)] and \textit{T. Rzeżuchowski} [Boll. Unione Mat. Ital., VII. Ser. A 6, 71-80 (1992)].
scientific article; zbMATH DE number 63269
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