general integration and measure

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General Integration And Measure

Author : Alan J. Weir
ISBN : 0521204070
Genre : Mathematics
File Size : 82. 61 MB
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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.

Measure Integration And Functional Analysis

Author : Robert B. Ash
ISBN : 9781483265100
Genre : Mathematics
File Size : 84. 75 MB
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Measure, Integration, and Functional Analysis deals with the mathematical concepts of measure, integration, and functional analysis. The fundamentals of measure and integration theory are discussed, along with the interplay between measure theory and topology. Comprised of four chapters, this book begins with an overview of the basic concepts of the theory of measure and integration as a prelude to the study of probability, harmonic analysis, linear space theory, and other areas of mathematics. The reader is then introduced to a variety of applications of the basic integration theory developed in the previous chapter, with particular reference to the Radon-Nikodym theorem. The third chapter is devoted to functional analysis, with emphasis on various structures that can be defined on vector spaces. The final chapter considers the connection between measure theory and topology and looks at a result that is a companion to the monotone class theorem, together with the Daniell integral and measures on topological spaces. The book concludes with an assessment of measures on uncountably infinite product spaces and the weak convergence of measures. This book is intended for mathematics majors, most likely seniors or beginning graduate students, and students of engineering and physics who use measure theory or functional analysis in their work.

Lebesgue Integration And Measure

Author : Alan J. Weir
ISBN : 0521097517
Genre : Mathematics
File Size : 89. 32 MB
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A textbook for the undergraduate who is meeting the Lebesgue integral for the first time, relating it to the calculus and exploring its properties before deducing the consequent notions of measurable functions and measure.

Measure And Integration

Author : Leonard F. Richardson
ISBN : 0470501146
Genre : Mathematics
File Size : 22. 39 MB
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A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis Measure and Integration: A Concise Introduction to Real Analysis presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis. The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems Lebesgue measure on the line and in Euclidean space Measurable functions, Egoroff's theorem, and Lusin's theorem Convergence theorems for integrals Product measures and Fubini's theorem Differentiation theorems for functions of real variables Decomposition theorems for signed measures Absolute continuity and the Radon-Nikodym theorem Lp spaces, continuous-function spaces, and duality theorems Translation-invariant subspaces of L2 and applications The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, Measure and Integration: A Concise Introduction to Real Analysis is a useful book for courses in real analysis at the graduate level. It is also a valuable reference for practitioners in the mathematical sciences.

An Introduction To Measure And Integration

Author : Inder K. Rana
ISBN : 0821883917
Genre : Lebesgue integral
File Size : 62. 42 MB
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Introduction To Measure And Integration

Author : S. J. Taylor
ISBN : 0521098041
Genre : Mathematics
File Size : 74. 86 MB
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This paperback, gives a self-contained treatment of the theory of finite measures in general spaces at the undergraduate level.

The Theory Of Lebesgue Measure And Integration

Author : S. Hartman
ISBN : 9781483280332
Genre : Mathematics
File Size : 40. 86 MB
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The Theory of Lebesgue Measure and Integration deals with the theory of Lebesgue measure and integration and introduces the reader to the theory of real functions. The subject matter comprises concepts and theorems that are now considered classical, including the Yegorov, Vitali, and Fubini theorems. The Lebesgue measure of linear sets is discussed, along with measurable functions and the definite Lebesgue integral. Comprised of 13 chapters, this volume begins with an overview of basic concepts such as set theory, the denumerability and non-denumerability of sets, and open sets and closed sets on the real line. The discussion then turns to the theory of Lebesgue measure of linear sets based on the method of M. Riesz, together with the fundamental properties of measurable functions. The Lebesgue integral is considered for both bounded functions — upper and lower integrals — and unbounded functions. Later chapters cover such topics as the Yegorov, Vitali, and Fubini theorems; convergence in measure and equi-integrability; integration and differentiation; and absolutely continuous functions. Multiple integrals and the Stieltjes integral are also examined. This book will be of interest to mathematicians and students taking pure and applied mathematics.

A Primer Of Lebesgue Integration

Author : H. S. Bear
ISBN : 0120839717
Genre : Mathematics
File Size : 68. 73 MB
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This successful text offers a reader-friendly approach to Lebesgue integration. It is designed for advanced undergraduates, beginning graduate students, or advanced readers who may have forgotten one or two details from their real analysis courses. "The Lebesgue integral has been around for almost a century. Most authors prefer to blast through the preliminaries and get quickly to the more interesting results. This very efficient approach puts a great burden on the reader; all the words are there, but none of the music." Bear's goal is to proceed more slowly so the reader can develop some intuition about the subject. Many readers of the successful first edition would agree that he achieves this goal. The principal change in this edition is the simplified definition of the integral. The integral is defined either with upper and lower sums as in the calculus, or with Riemann sums, but using countable partitions of the domain into measurable sets. This one-shot approach works for bounded or unbounded functions and for sets of finite or infinite measure. The author's style is graceful and pleasant to read. The explanations are exceptionally clear. Someone looking for an introduction to Lebesgue integration could scarcely do better than this text. -John Erdman Portland State University This is an excellent book. Several features make it unique. The author gets through the standard canon in only 150 pages and then arranges the material into easily digestible units (a proof hardly ever exceeds three-fourths of a page). The author writes with concision, clarity, and focus. -Robert Burckel Kansas State University This text achieves its worthy goals. The author tends to the business at hand. The short chapter on Lebesgue integration is refreshing and easily understood. One can use a semester covering the book, and the students will be well-grounded in the basics and ready for any of a dozen possible second semesters. -Joseph Diestel Kent State University

Real Analysis

Author : J Yeh
ISBN : 9789814578561
Genre : Mathematics
File Size : 51. 55 MB
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This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped. The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians. The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory.

Measure Theory And Integration

Author : Michael Eugene Taylor
ISBN : 9780821841808
Genre : Mathematics
File Size : 27. 78 MB
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This self-contained treatment of measure and integration begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. From there the reader is led to the general notion of measure, to the construction of the Lebesgue integral on a measure space, and to the major limit theorems, such as the Monotone and Dominated Convergence Theorems. The treatment proceeds to $L^p$ spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to $L^2$ spaces as Hilbert spaces, with a useful geometrical structure. Having gotten quickly to the heart of the matter, the text proceeds to broaden its scope. There are further constructions of measures, including Lebesgue measure on $n$-dimensional Euclidean space. There are also discussions of surface measure, and more generally of Riemannian manifolds and the measures they inherit, and an appendix on the integration of differential forms. Further geometric aspects are explored in a chapter on Hausdorff measure. The text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales. This text will prepare graduate students for more advanced studies in functional analysis, harmonic analysis, stochastic analysis, and geometric measure theory.

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