Author Description

--> This book offers to the reader a self-contained treatment and systematic exposition of the real-valued theory of a nonabsolute integral on measure spaces. It is an introductory textbook to Henstock–Kurzweil type integrals defined on abstract spaces. It contains both classical and original results that are accessible to a large class of readers. It is widely acknowledged that the biggest difficulty in defining a Henstock–Kurzweil integral beyond Euclidean spaces is the definition of a set of measurable sets which will play the role of "intervals" in the abstract setting. In this book the author shows a creative and innovative way of defining "intervals" in measure spaces, and prove many interesting and important results including the well-known Radon–Nikodým theorem. --> Contents: A Nonabsolute Integral on Measure Spaces: Preliminaries Existence of a Division and the H -Integral Simple Properties of the H -Integral The Absolute H -Integral and the McShane-Type Integrals: The Absolute H -Integral and the M -Integral The H -Integral and the Lebesgue Integral The Davies Inetgral and the Davies-McShane Integral Further Results of the H -Integral: A Necessary and Sufficient Condition for H -Integrability Generalised Absolute Continuity and Equiintegrability The Controlled Convergence Theorem The Radon–Nikodým Theorem for the H -integral: The Main Theorem Descriptive Definition of H -integral Henstock Integration in the Euclidean Space Harnack Extension and Convergence Theorems for the H -Integral: The H -Integral on Metric Spaces Harnack Extension for the H -Integral The Category Argument An Improved Version of the Controlled Converge