Another proof that \(L^p\)-bounded pointwise convergence implies weak convergence
Let \(1<p<+\infty\), \(\Omega\subseteq\mathbb{R}^N\) and \((f_n)_{n\in\mathbb{N}}\) be a norm-bounded sequence in \(L^p(\Omega)\) converging to \(f\) pointwise a.e. (or in measure). It is well known that then \(f_n\rightarrow f\) weakly in \(L^p(\Omega)\). The author gives another proof of this fact using a theorem of Banach and Saks.
scientific article; zbMATH DE number 6032466