On Some Properties of Vector Measures

Abstract

We study the properties of a parameterized sequence of countably additive vector measures with densities defined on a compact space T with a nonnegative nonatomic Radon measure μ and taking values in a separable Banach space. Each vector measure of this sequence depends continuously on a parameter belonging to a metric space. We assume that a countable locally finite open cover and a partition of unity inscribed into this cover are given in the metric space of parameters. We prove that, for each value of the parameter, there exists a sequence of μ-measurable subsets of the space T which is a partition of T. In addition, this sequence depends uniformly continuously on the parameter and, for each value of the parameter and each element of the initial parameterized sequence of vector measures, the relative value of the measure of the corresponding subset in the partition of T can be uniformly approximated by the corresponding value of the corresponding partition of unity function.