A Fatou lemma for Gelfand integrals by means of Young measure theory
The author shows that a Fatou-type lemma, due to \textit{B. Cornet} and \textit{J.-P. Medecin} [Positivity 6, 297-315 (2002; Zbl 1027.28016)], can be generalized for not necessarily integrably bounded but uniformly integrable, Gelfand integrable functions taking values in the topological dual \(E^*\) of a separable Banach space \(E\) with the weak* topology. For instance, an approximate version of Fatou's lemma is generalized as follows: Let \((\Omega,{\mathcal A},\mu)\) be a complete finite positive measure space and \(E\) a separable Banach space. Let \((f_n)\) be a uniformly integrable sequence of Gelfand integrable functions from \(\Omega\) to \(E^*\). Suppose that the \(a:=w^*\)-\(\lim_n\int_\Omega f_nd\mu\) exists. Then, for every \(w^*\)-neighborhood \(W\) of the origin in \(E^*\), there exists a Gelfand integrable function \(f_W : \Omega\to E^*\) such that \(\int_\Omega f_Wd\mu\in W+a\) and \(f_W(\omega)\in w^*\text{-cl} w^*\text{-Ls}_n\{f_n(\omega)\}\) for a.e. \(\omega\in\Omega\), where \(w^*\text{-Ls}_n\{x_n\}\) denotes the \(w^*\)-limes superior of a sequence \((x_n)\) in \(E^*\). The approach to the proof is completely different from that given by Cornet and Medecin and is based on Young measure theory which was developed by the author himself [\textit{E. J. Balder}, ``New fundamentals of Young measure convergence'', Res. Notes Math. 411, 24-48 (2000; Zbl 0964.49011)].
scientific article; zbMATH DE number 1880388