Fatou's lemma for Gelfand integrable mappings
The authors give an exact and an approximate version of Fatou's lemma for integrably bounded, Gelfand integrable mappings taking values in the topological dual \(E^*\) of a separable Banach space \(E\) with the weak* topology. An approximate Fatou's lemma is formulated as follows: Let \((\Omega,{\mathcal A},\mu)\) be a complete finite positive measure space and \(E\) a separable Banach space. Let \((f_n)\) be an integrably bounded sequence of Gelfand integrable mappings from \(\Omega\) to \(E^*\) such that the weak* limit \(\lim_n\int_\Omega f_nd\mu\) exists. Then, for every \(w^*\)-neighborhood \(W\) of the origin in \(E^*\), there exists a Gelfand integrable mapping \(f_W : \Omega\to E^*\) such that \(\int_\Omega f_Wd\mu-\lim_n\int_\Omega f_nd\mu\in W\) and \(f_W(\omega)\in\text{Ls}_n\{f_n(\omega)\}\) for a.e.\ \(\omega\in\Omega\), where \(\text{Ls}_n\{x_n\}\) denotes the set of all weak* limit points of a sequence \((x_n)\) in \(E^*\). Their work is directly motivated by an economic model of spatial economies and the above approximate Fatou's lemma forms the central technical tool to prove the existence of general equilibria for such economies.
scientific article; zbMATH DE number 1880387