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This text gives a general definition of the (abstract) integral, using the Daniell method. A consequence of this approach is the fact that integration theory on Hausdorff topological spaces appears simply to be a special case of abstract integration theory. The most important tool for the development of the abstract theory is the theory of vector lattices which is presented here in great detail. Its consequent application not only yields insight into integration theory, but also simplifies many proofs. For example, the space of real-valued measures on a d-ring turns out to be an order complete vector lattice, which permits a coherent development of the theory and the elegant derivation of many classical and new results. The exercises occupy an important part of the volume. In addition to their usual role, some of them treat separate topics related to vector lattices and integration theory. This work should be of interest to graduate-level students and researchers with a background in real analysis, whose work involves (abstract) measure and integration, vector lattices, real functions of a real variable, probability theory and integral transforms.
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Since about 1915 integration theory has consisted of two separate branches: the abstract theory required by probabilists and the theory, preferred by analysts, that combines integration and topology. As long as the underlying topological space is reasonably nice (e.g., locally compact with countable basis) the abstract theory and the topological theory yield the same results, but for more compli cated spaces the topological theory gives stronger results than those provided by the abstract theory. The possibility of resolving this split fascinated us, and it was one of the reasons for writing this book. The unification of the abstract theory and the topological theory is achieved by using new definitions in the abstract theory. The integral in this book is de fined in such a way that it coincides in the case of Radon measures on Hausdorff spaces with the usual definition in the literature. As a consequence, our integral can differ in the classical case. Our integral, however, is more inclusive. It was defined in the book "C. Constantinescu and K. Weber (in collaboration with A.