Decompositions of vector measures in Riesz spaces and Banach lattices

Abstract

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.