Measures with finite semi-variation (Q1280220)

For Banach spaces \(E\) and \(F\), let \({\mathcal L}(E,F)\) denote the Banach space of all continuous linear mappings \(E\to F\). The author shows that for a Banach space \(E\) the following are equivalent: (a) For any ring \({\mathcal A}\) and any Banach space \(F\), every bounded additive set function \({\mathcal A}\to{\mathcal L}(E,F)\) has finite semivariation. (b) For any \(\sigma\)-ring \({\mathcal A}\) and any Banach space \(F\), every countably additive set function \({\mathcal A}\to{\mathcal L}(E,F)\) has finite semivariation. (c) \(E\) has finite dimension. He also extends his result to the case of locally convex spaces instead of Banach spaces.