The Henstock?Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Perron integral; in particular, it includes the powerful Lebesgue integral. This book presents an introduction of the multiple Henstock?Kurzweil integral. Along with the classical results, this book contains some recent developments connected with measures, multiple integration by parts, and multiple Fourier series. The book can be understood with a prerequisite of advanced calculus.

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

The main topics of this book are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Morever, some relations between integration and differentiation are made clear.The book is self-contained. It is of interest to specialists in the field of real functions, and it can also be read by students, since only the basics of mathematical analysis and vector spaces are required.

This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces.A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc.

"the results of the book are very interesting and profound and can be read successfully without preliminary knowledge. It is written with a great didactical mastery, clearly and precisely It can be recommended not only for specialists on integration theory, but also for a large scale of readers, mainly for postgraduate students".Mathematics Abstracts

This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces. A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc. Contents: Introduction to the Gauge or Henstock-Kurzweil Integral; Basic Properties of the Gauge Integral; Henstock''s Lemma and Improper Integrals; The Gauge Integral over Unbounded Intervals; Convergence Theorems; Integration over More General Sets: Lebesgue Measure; The Space of Gauge Integrable Functions; Multiple Integrals and Fubini''s Theorem; The McShane Integral; McShane Integrability is Equivalent to Absolute Henstock-Kurzweil Integrability. Readership: Upper level undergraduates and mathematicians interested in gauge integrals.