×

On the control measures of vector measures. (English) Zbl 1076.28011

If \(X\) is a locally convex space such that \(\{0\}\) is a \(G_{\delta}\) set, then every \(X\)-valued \(\sigma\)-additive measure on a \(\sigma\)-algebra possesses a real-valued control measure. If \(\Omega\) is uncountable then there is a measure with values in \(\mathbb R ^{\Omega}\) equipped with the product topology, which does not have a real-valued control measure. If the closed linear span of every bounded sequence in \(X\) possesses a countable total set of continuous functionals, then every \(X\)-valued Borel measure on a Lusin space has a real control measure. The conditions on the measure space and on \(X\) for existence of Rybakov type control measures are studied. Some results for polymeasures are given.

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
PDFBibTeX XMLCite
Full Text: EuDML