Stability of some types of Radon-Nikodým properties
Let \(X\) be a Banach space, \((\Omega,\Sigma,\mu)\) be a measure space and \(G\) be a compact metrizable Abelian group. Moreover, let \(1\leq p\leq\infty\), \(L^p(\mu,X)\) \((L^p(G,X))\) be the Banach space of all (classes of) \(X\)-valued \(\mu\)-\(p\)-Bochner integrable functions on \(\Omega\) (respectively Bochner integrable functions on \(G\)) with the usual norm, \(M(G,X)\) be the space of all \(X\)-valued countably additive measures of bounded variation and \(E(X)\) be the Banach space of all (classes of) \(X\)-valued measurable functions on \(\Omega\) such that \(\omega\to|f(\omega)|\) belongs to a Köthe function space \(E\) on \((\Omega,\Sigma,\mu)\). The types of Radon-Nikodým properties I-\(\Lambda\)-RNP and II-\(\Lambda\)-RNP of Banach space has been defined by Edgar and Dowling. Conditions that a) the assertions (i) \(X\) has type II-\(\Lambda\)-RNP and (ii) \(E(X)\) has type II-\(\Lambda\)-RNP would be equivalent and b) \(L^p(\mu,X)\) and \(M(G,X)\) would have type II-\(\Lambda\)-RNP whenever \(X\) does are given in the paper under review. Moreover, the conditions that \(L^1(G,X)\) whould have type I-\(\Lambda\)-RNP if and only if it has type II-\(\Lambda\)-RNP and \(E(X)\) has type I-\(\Lambda\)-RNP if \(E\) has type I-\(\Lambda\)-RNP and \(X\) has type II-\(\Lambda\)-RNP are presented.
\(X\)-valued countable additive measures of bounded variation
scientific article; zbMATH DE number 856550