Gel'fand integrals and generalized derivatives of vector measures (Q793946)

In this paper, based on the Gelfand integration theory, the authors introduced a notion of \(\Pi\)-generalized (namely, \(\Pi\)-multivalued) derivatives \(\phi_{\nu}\) which can be defined for each pair of a \(\mu\)- continuous \(X^*\)-vaued measure \(\nu\) of finite variation and a family \(\Pi\) of measurable partitions of the base space S. This notion is a generalization of the weak*-derivatives for such measures. They investigated some properties of such derivatives by an argument based on the relative weak*-compactness of the average ranges of vector measures. The fundamental result is the following: Under suitable assumption of \(\Pi\), for every selection f of \(\phi_{\nu}\) there exists a \(\mu\)-null set \(N_ f\) such that \(\| f(s)\| =(d| \nu | /d\mu)(s)\) for \(s\in S\setminus N_ f.\) Making use of this result and the lifting theorem, they also obtained a stronger result in which \(\phi_{\nu}\) is single-valued. Applying this result, they established a generalized Radon-Nikodym theorem for general vector measures, a representation theorem for the dual spaces of \(L_ p(\mu,X)\) and a characterization theorem of Bochner derivatives.