A note on strong convergence for Pettis integrable functions
Let \((\Omega,\Sigma,\mu)\) be a complete probability space, \(E\) be a separable Banach space and \({\mathcal P}(E)\) be the family of all probability measures on Borel subsets \({\mathcal B}(E)\) of \(E\). A sequence of measurable functions \(f_n:\Omega\to E\) is said to be s-tight if for every \(\varepsilon>0\) there exists a measurable multifunction \(\Gamma_{\varepsilon}:\Omega\to k(E)\) (nonempty, compact convex subsets of \(E\)) such that \[ \sup_n\mu(\{\omega\in\Omega: f_n(\omega)\notin\Gamma_{\varepsilon}(\omega)\})\leq\varepsilon. \] The authors prove the following result that extends the main theorem of \textit{A. Amrani, C. Castaing} and \textit{M. Valadier} published in [Vietnam J. Math. 26, No. 4, 323--335 (1998; Zbl 0965.46028)]: Let \((f_n)_n\) be an \(s\)-tight sequence of \(E\)-valued Pettis integrable functions that is Pettis uniformly integrable and weakly convergent to a Pettis integrable function \(f:\Omega\to E\). Denote by \(s-Ls\{f_n(\omega)\}\) the Kuratowski limes superior of the sequence \((f_n(\omega))_n\) in the norm topology and assume that for \(\mu\)-almost every \(\omega\) the set \(\overline{\text{conv}}(s-Ls\{f_n(\omega)\})\) is weakly locally compact and does not contain any line. If for almost every \(\omega\) the point \(f(\omega)\) is an extreme point of \(\overline{\text{conv}}(s-Ls\{f_n(\omega)\})\), then \((f_n)_n\) is convergent to \(f\) in the Pettis norm.
scientific article; zbMATH DE number 2101845