Uniform integrability and mean convergence for the vector-valued McShane integral
If \(X\) is a Banach space and \(f_k:\mathbb{R} \to X\) is McShane integrable for every \(k \in \mathbb{N}\), \(f_k(x) \to f(x)\) for every \(x\in \mathbb{R}\), the sequence \((f_k)\) is uniformly integrable, then \(f\) is McShane integrable and \(\|f_k-f\|_1 = \sup \{\int _{\mathbb{R}}|x'f|; x' \in X',\|x'\|\leq 1\}\to 0\). By \(X'\) the dual to \(X\) is denoted. Uniform integrability holds in an \(X\)-valued generalization of the B. Levi type monotone convergence theorem.
scientific article; zbMATH DE number 1449425