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: PreliminariesExistence of a Division and the H-IntegralSimple Properties of the H-IntegralThe Absolute H-Integral and the McShane-Type Integrals: The Absolute H-Integral and the M-IntegralThe H-Integral and the Lebesgue IntegralThe Davies Inetgral and the Davies-McShane IntegralFurther Results of the H-Integral: A Necessary and Sufficient Condition for H-IntegrabilityGeneralised Absolute Continuity and EquiintegrabilityThe Controlled Convergence TheoremThe Radon–Nikodým Theorem for the H-integral: The Main TheoremDescriptive Definition of H-integralHenstock Integration in the Euclidean SpaceHarnack Extension and Convergence Theorems for the H-Integral: The H-Integral on Metric SpacesHarnack Extension for the H-IntegralThe Category ArgumentAn Improved Version of the Controlled Convergence Theorem Readership: Graduate students and researchers interested in analysis. Keywords: Henstock-Kurzweil Type Integral;Generalized Intervals;Nonabsolute Integration;Measure Spaces;Locally Compact Hausdorff Space;Radon-Nikodym Theorem;Controlled Convergence Theorem;Harnack ExtensionReview: Key Features: To our knowledge there is no book on integration theory whose setting is measure spaces with a topological structureThe theory is developed in a progressive and elementary manner in that the fundamental properties are first established before further results are proved. That way, even though the setting is abstract, this book is accessible to any undergraduate who has done an advanced calculus courseThe key idea behind each original concept is always explained in an intuitive manner before the formal definitions and results are presented in detail

Significantly revised and expanded, this authoritative reference/text comprehensively describes concepts in measure theory, classical integration, and generalized Riemann integration of both scalar and vector types-providing a complete and detailed review of every aspect of measure and integration theory using valuable examples, exercises, and applications. With more than 170 references for further investigation of the subject, this Second Edition provides more than 60 pages of new information, as well as a new chapter on nonabsolute integrals contains extended discussions on the four basic results of Banach spaces presents an in-depth analysis of the classical integrations with many applications, including integration of nonmeasurable functions, Lebesgue spaces, and their properties details the basic properties and extensions of the Lebesgue-Carathéodory measure theory, as well as the structure and convergence of real measurable functions covers the Stone isomorphism theorem, the lifting theorem, the Daniell method of integration, and capacity theory Measure Theory and Integration, Second Edition is a valuable reference for all pure and applied mathematicians, statisticians, and mathematical analysts, and an outstanding text for all graduate students in these disciplines.

This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.

This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.

This is a sequel to Dr Weir's undergraduate textbook on Lebesgue Integration and Measure (CUP. 1973) in which he provided a concrete approach to the Lebesgue integral in terms of step functions and went on from there to deduce the abstract concept of Lebesgue measure. In this second volume, the treatment of the Lebesgue integral is generalised to give the Daniell integral and the related general theory of measure. This approach via integration of elementary functions is particularly well adapted to the proof of Riesz's famous theorems about linear functionals on the classical spaces C (X) and LP and also to the study of topological notions such as Borel measure. This book will be used for final year honours courses in pure mathematics and for graduate courses in functional analysis and measure theory.

This book describes integration and measure theory for readers interested in analysis, engineering, and economics. It gives a systematic account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes measure from the Lebesgue-Stieltjes integral.

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.