Geometry and integration for operator valued measures
The average range of a measure, valued in a Banach space \(E\) with respect to an operator-valued measure \(m\) with values in \(L(E,E)\), the \(m\)-convex hull of a subset of \(E\) and the core with respect to \(m\) of an \(E\)-valued function are defined and studied assuming that \(m\) verifies the Price's axiom (some of the results hold without this assumption). Several interesting relations are studied between the average range of a measure defined by an \(m\)-integrable \(E\)-valued function \(f\), the core and the essential range of \(f\), and also between their closed \(m\)-convex hulls. As an application, a mean-value theorem and a criterion of measurability are stated for the Dobrakov integral. Also a Radon-Nikodým theorem about derivation of \(E\)-valued measures with respect to measures valued in \(L(E,E)\) is proved for the Dobrakov integral by means of the generalized concept of the average range of a measure (instead of the usual techniques of localization and control).
scientific article; zbMATH DE number 62063
scientific article