Decomposition and Extension of Abstract Measures in Riesz Spaces

Abstract

The aim of these notes is to review some recent developments in the theory of abstract measures taking their values in Riesz space. The terni abstract measure is used were to denote a common abstraction of vector measures and linear operators. The topics considered in this survey are: A common approach to vector measures and linear operators, Jordan and Lebesgue decompositions of abstract measures and their applications to vector measures and linear operators, common extensions of linear operators and of vector measures, and extensions of modular functions. We also propose a number of open problems which may stimulate further research in this area. The material of these notes is based on the monograph by Schmidt [5l], two papers by Schmidt and Waldschaks [55], [56], and the PhD Thesis of Waldschaks [60].