Introduction

Integration theory was originally developed for real-valued functions with respect to real-valued measures. There are a great number of expositions devoted to this, most notably the classical treatises by Lebesgue [116], Carathéodory [30], Radon [158] and Daniell [36]. More recent treatments in the works of Bourbaki [25], Hahn and Rosenthal [80], Halmos [83] and Saks [182] contain excellent historical notes and references on the subject. Vector integration was introduced in the first half of the last century, and exhaustive discussions of the field can for example be found in the works of Dunford and Schwartz [54], [55], [56], Diestel and Uhl [43] or Graves [75].

The aim of this book is to develop a general theory of integration which simultaneously deals with extended real-valued, vector-valued, operator-valued and cone-valued measures and functions. All except the last of these topics have been explored extensively, and integration theory as presented in the available standard texts uses different approaches in each of these cases. Both finitely and countably additive measures have been considered. However, finitely additive measures yield only limited results and are therefore not widely used in analysis. As a consequence only countably additive measures shall be considered in this book.