Theory of Tensor Integration

This book is intended as an introduction to the theory of tensor integration. The main object of this book is to study the integration theory of vector valued functions with respect to a countably additive vector measures on a ¿- algebra or on a ¿- ring. A systematic description of the development of tensor integration is given here. In this book we study the Banach space of all tensor integrable functions and discuss many properties such as order continuity, weakly compactly generated, weak sequential completeness and separability of this space. In chapter III we developed the same integration theory for the spaces of p-tensor integrable functions and weakly p-tensor integrable functions and discuss several basic properties of these spaces. In chapter IV we introduce the idea of tensor integration theory and investigated many properties of the space of all tensor integrable functions when the vector measure is defined on ¿-rings.