Before revision: Definitions and first consequences Given a field of sets and a Banach space , a finitely additive vector measure (or measure, for short) is a function such that for any two disjoint sets and in one has A vector measure is called countably additive if for any sequence of disjoint sets in such that their union is in it holds that with the series on the right-hand side convergent in the norm of the Banach space It can be proved that an additive vector measure is countably additive if and only if for any sequence as above one has where is the norm on Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval the set of real numbers, and the set of complex numbers.
After revision: Definitions and first consequences Given a field of sets and a Banach space a finitely additive vector measure (or measure, for short) is a function such that for any two disjoint sets and in one has A vector measure is called countably additive if for any sequence of disjoint sets in such that their union is in it holds that with the series on the right-hand side convergent in the norm of the Banach space It can be proved that an additive vector measure is countably additive if and only if for any sequence as above one has where is the norm on Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval the set of real numbers, and the set of complex numbers.