RNP and CPCP in Lebesgue-Bochner function spaces
By introducing the notions of a Rademacher tree of measurable sets and a Rademacher sequence of Banach space-valued functions with respect to a \(\sigma\)-algebra \(\Sigma\) of subsets of a set \(\Omega\) and proving some basic results about them, the authors prove the following main result: If \((\Omega, \Sigma, \mu)\) is an atom-free measure space and \(X\) is a Banach space, then every point of sequential continuity of \(B_{L^p(\mu, X)}\) is an extreme point of \(B_{L^p(\mu, X)}\). The above result subsumes the earlier one of M. Smith and B. Turett which states that if \((\Omega, \Sigma, \mu)\) is not purely atomic, then \(L^p(\mu, X)\) with Kadec-Klee property must be strictly convex. Moreover, as an immediate consequence of the main result it is shown that if \((\Omega, \Sigma, \mu)\) is not purely atomic, then RNP (Radon-Nikodym-Property) and CPCP (Convex Point of Continuity Property) (resp. super-RNP and super-CPCP) are equivalent for \(L^p(\mu, X)\) and \(L^p(\mu, X)^*\), \(1< p< \infty\). Also there are obtained many other results, which include a point-version of the Schachermayer's characterization of RNP in terms of strong regularity and the Krein-Milman-property.
Rademacher sequence of a Banach space-valued functions
scientific article; zbMATH DE number 561186
scientific article