Book Description
Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.
After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.
As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.
Also available, the first two volumes in the Princeton Lectures in Analysis:
Customer Reviews:
great book.......2006-10-19
i found the first three chapters of this book very clear and well written. i'd strongly recommend it for someone looking to learn about analysis on the real line.
Good book for reading and as a graduate student.......2006-07-19
Easy to read. My university is using this book to get the graduate students ready for the real analysis qualifying exam. So go ahead and buy this book if you're planning to work on a PhD in mathematics. If you're not planning to work on a PhD in math, this is still a good book to read if you enjoy studying about the real line.
The book begins with measure theory, integration and differentiation. These are included in Chapters 1 to 3. Then in Chapters 4 and 5, we look into Hilbert spaces. This is similar to studying finite-dimensional inner-product spaces, but here, Hilbert space is infinite-dimensional. However, the analysis is very similar. If you know some linear algebra, it should feel like as if you have already read these two chapters.
Finally in Chapters 6 and 7, we see abstract measure theory, including Hausdorff measure, and we study fractals and self-similar sets. And this concludes the book.
Also recommend Walter Rudin's Real Analysis.
Suffers from all the flaws of a 1st edition.......2005-12-18
This book has a lot of problems. Several sections are poorly written/edited. Several important named theorems are not clearly labeled. Also some of the proofs contain typos or errors. The chapter on differentiation is particularly lacking. The chapter is poorly organized and presented. There is also a glaring TeX error in the chapter.
At Princeton this book is used as part of an undergraduate course, and it shows. This is not the ideal book for a graduate level course in real analysis(though I think it would be very well suited for advanced undergrads). Too much time is spent on Lebesgue measure and integration in the first 2 chapters, and abstract measure theory is not intoduced until chapter 6. Also the Monotone Class theorem is lacking from the chapter on abstract measure theory. Also, the book only touches on functional analysis in the two chapters on Hilbert spaces (where they assume all Hilbert spaces are separable).
On the other hand, the presentations of Lebesgue measure/integration and Hilbert spaces in the book are pretty good. The exercises and problems in teh book (when stated properly) are very good and instructive. Overall this book has a lot of potential to be very good, but seems to be suffering from a lack of revision. Hopefully these issues will be fixed in later editions.
Excellent sourse for graduate analysis.......2005-07-03
This book is the best book on real analysis I have ever studied. It does a wonderful job in bridging undergraduate level with graduate level analysis. I have not seen any book that makes measure and Lebesgue theory so easy to understand.
The books begins by defining what a "measure" is all about. And the description is so intuitive and geometrical that you would wonder why you weren't taught it this way before. The book then goes into Lebesgue theory and all of it suddenly becomes so easy.
The book has plenty of wonderful examples and a good set of over 30 problems per chapter.
Elias Stein (one of the authors) is a very renowned mathematician, and one need not worry about the accuracy of the proofs in the book--they are "bullet-proof", and at the same time succinct.
If you are struggling with W. Rudin's book on Analysis, this book is a MUST for you.
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Measure Theory and Integration (Pure and Applied Mathematics)
M.M. Rao
Manufacturer: Marcel Dekker
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A Modern Theory of Integration (Graduate Studies in Mathematics)
ASIN: 0824754018 |
Book Description
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.
Customer Reviews:
Jon's Review.......2002-04-18
Simply put, M.M. Rao's "Measure Theory and Integration" is an awesome book. It is truly the "Encyclopedia Britannica" of Real Analysis textbooks. This math textbook/reference book contains the most general, yet practical, theorems on the subject known to mankind. I cannot recommend it highly enough.
Book Description
The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: T. W. Anderson The Statistical Analysis of Time Series T. S. Arthanari & Yadolah Dodge Mathematical Programming in Statistics Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences Robert G. Bartle The Elements of Integration and Lebesgue Measure George E. P. Box & George C. Tiao Bayesian Inference in Statistical Analysis R. W. Carter Simple Groups of Lie Type William G. Cochran & Gertrude M. Cox Experimental Designs, Second Edition Richard Courant Differential and Integral Calculus, Volume I Richard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II D. R. Cox Planning of Experiments Harold M. S. Coxeter Introduction to Modern Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II Bruno de Finetti Theory of Probability, Volume 1 Bruno de Finetti Theory of Probability, Volume 2 W. Edwards Deming Sample Design in Business Research Amos de Shalit & Herman Feshbach Theoretical Nuclear Physics, Volume 1 —Nuclear Structure J. L. Doob Stochastic Processes Nelson Dunford & Jacob T. Schwartz Linear Operators, Part One, General Theory Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Two, Spectral Theory—Self Adjoint Operators in Hilbert Space Nelson Dunford & Jacob T. Schwartz Linear Operators, Part Three, Spectral Operators Herman Feshbach Theoretical Nuclear Physics: Nuclear Reactions Bernard Friedman Lectures on Applications-Oriented Mathematics Phillip Griffiths & Joseph Harris Principles of Algebraic Geometry Gerald J. Hahn & Samuel S. Shapiro Statistical Models in Engineering Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume I—Methods and Applications Morris H. Hansen, William N. Hurwitz & William G. Madow Sample Survey Methods and Theory, Volume II—Theory Peter Henrici Applied and Computational Complex Analysis, Volume 1—Power Series—Integration—Conformal Mapping—Location of Zeros Peter Henrici Applied and Computational Complex Analysis, Volume 2—Special Functions—Integral Transforms—Asymptotics—Continued Fractions Peter Henrici Applied and Computational Complex Analysis, Volume 3—Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions Peter Hilton & Yel-Chiang Wu A Course in Modern Algebra Harry Hochstadt Integral Equations Erwin O. Kreyszig Introductory Functional Analysis with Applications William H. Louisell Quantum Statistical Properties of Radiation Ali Hasan Nayfeh Introduction to Perturbation Techniques Emanuel Parzen Modern Probability Theory and Its Applications P. M. Prenter Splines and Variational Methods Walter Rudin Fourier Analysis on Groups C. L. Siegel Topics in Complex Function Theory, Volume I—Elliptic Functions and Uniformization Theory C. L. Siegel Topics in Complex Function Theory, Volume II—Automorphic and Abelian Integrals C. L. Siegel Topics in Complex Function Theory, Volume III—Abelian Functions & Modular Functions of Several Variables J. J. Stoker Differential Geometry J. J. Stoker Water Waves: The Mathematical Theory with Applications J. J. Stoker Nonlinear Vibrations in Mechanical and Electrical Systems
Customer Reviews:
A good introduction: concise and clear........2007-01-27
The book is concise and easy to follow. The author rarely gives lengthy explanations and analogies, but spends the bulk of the book stating solid facts and proofs. I also like the organization of the book. All definitions and theorems are explicitly stated and indexed, not scattered in paragraphs in the body of the text.
The book misses subjects such as complex measures (they are briefly mentioned), the fundamental theorem of calculus under Lebesgue settings, and probability measures, but its ok since the book is an introduction to the subject. A more comprehensive (and harder to read) book is "Real & Complex Analysis" by Walter Rudin. If you are interested in probability, consider Ptrick Billingsley's book "Probability and Measure".
Good Integration and Measure Into (A Bit Expensive Though).......2005-01-15
The exposition of integration in this book is the clearest I have read. I also found the chapter on modes of convergence, where it laid out the relationship between things such as L^P-convergence and convergence in measure, to be extremely useful. The second half, where it covers topics like Lebesgue measure, repeats some of the same information from the first part which is a bit iritating if you are reading straight throught, but contains a lot of good information. The book is also quite small making it easy to take with you as a quick reference.
Let me warn you though that this is an introduction to integration and measure _not_ an introduction to real analysis. It does not cover important topics like L^P-approximation, differentiation, etc. For a complete treatment of real analysis, I recommend the books "Lebesgue Integration on Euclidean Space" by Frank Jones and the slightly more abstract "Real and Functional Analysis" by Serge Lange.
IF YOU WANT TO UNDERSTAND MEASURE THEORY..........2001-06-04
IF YOU WANT TO UNDERSTAND MEASURE THEORY READ THIS BOOK, MAYBE THE ONLY PROBLEM IS THE LACK OF EXAMPLES BUT THE WAY THAT THE THEORY IS PRESENTED MAKE IT YOUR FIRST CHOICE WHEN YOU TRY TO LEARN MEASURE THEORY.
Excellent as an itroduction and as a reference.......2000-03-31
When I took my first one-semester course on measure and Lebesgue integration my teacher chose Bartle's "The Elements of Integration" as text. After reading many other books on the subject now I'm sure he made a wise decision.
Assuming almost no strong mathematical background, Bartle is able to build up the basic Lebesgue integral theory introducing the fundamental abstract concepts (sigma-algebra, measurable function, measure space, "almost everywhere", step function, etc.) in such an easy way that the student is not only able to handle them but to UNDERSTAND them.
From the first part of the book I appreciate specially chapters 6, 7, and 10, on L_p spaces, modes of convergence, and product measures, respectively. These chapters contain the most used results of the basic theory, and they are stated exactly in the way one needs them, making the book very useful for future reference.
I like the second part very much also, because it stresses the importance of measure theory by itself and not only as a requisite for integration theory. If you are interested in fractal geometry or geometric measure theory you will find chapters 11 to 17 very helpful.
Since I own this book it has never been lazy in my bookshelf.
A great place to begin.......2000-02-04
Measure and Integration is a daunting subject for mathematical neophytes. Bartle's little volume is the right place to start. I first learned measure theory from it 20 years ago and went on to study functional analysis and stochastic approximation.
I was able to master the material on my own with this book. The problems are at the right level and he begins with the correct level of abstraction. I recommend it over anything else because it is straighforward, clear and focused. Master it then go on to Walter Rudin's Real and Complex Analysis.
Book Description
Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately. Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset. In addition, the text provides students a through treatment of Fourier analysis, while holistically preparing students to become "workers" in real analysis.
Customer Reviews:
great!.......2006-03-30
This is a terrific text for a first course in graduate-level real analysis, and is suitable for self-study. It develops Lebesgue integration theory slowly, in a very clear manner. In addition, the latter part of the book covers the basics of Fourier Analysis and important topics in differentiation. I frequently refer to this book, as the results are easy to find.
Rigor not Rigor Mortis.......2006-02-25
One of the problems with modern mathematics is its obsession with rigor which has been attended, over the last few decades, by a mushrooming of symbols and jargon. Much of it is not clearly related to the ideas they serve to label, as evidenced by such terms as the topological use of "filter" whose etymology is obscure (ascribed by some to H. Cartan). Moreover, the particular subject of Lebesgue integration and its generalizations is made even more confusing by a wide variety of approaches depending on an author's penchants--many of whom are enamored with a purely axiomatic approach and who make little or no appeal to intuition or--God forbid!--pictures. The author of the present work is obviously someone who has actually taught mathematics and taught it lovingly. This book is an excellent read with lots of interesting topics well explained from a student's point of view. There seems to be a nice ramping from the truly elementary to the sophisticated, which means the book will interest experienced mathematicians, scientists and engineers. There are lots of "doable" problems that the reader can solve along the way. For the experienced mathematician these little problems help alot as a refresher (Oh!, now I remember, that's how you do it.). I like the emphasis on Euclidean space. Somehow, I always feel more comfortable there! It gives me things I can actually construct and doodle on paper. And, it allows the author to use a few figures in a meaningful way. Which is another of the book's strong points and if I could recommend a future improvement, it would be to bring on more of those pictures! Tristram Needham has done a nice job along these lines with his book "Visual Complex Analysis." (I ordered several copies as Christmas gifts--just kidding!). Anyone who has taught mathematics and genuinely wished to be understood by his students has, at various times, drawn them pictures. Inside the cover sheets are lists of integration formulae, a fourier transform table, and a table of "assorted facts" on things like the Gamma function; which show that this is not only a book on Lebesgue integration but a calculus book with the Lebesgue integral occupying center stage. Everyone who has been enamored by the notion of the integral--as I was as a freshman calculus student and have been ever since--will want to have this book on their shelf.
an excellent introductory text.......2003-09-24
As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.
The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.
For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.
High Praise for Jones.......2000-08-22
"Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0
< p
< 1.
The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.
Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.
While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!
As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.
treasure trove of mathematical technique.......2000-04-01
This book is a treasure trove of mathematical technique. It covers topics that are relevant to many broad areas of real and functional analysis including signal processing and approximation theory. The author takes the time not only to prove the results, but also to construct the proofs so that the technique is made explicit to the reader. The author also motivates definitions by breaking them into the successively more complicated pieces so as to build intuition in the reader.
I especially recommend this book to anyone who lacks formal training in mathematics or wishes to develop mathematical technique in the areas of real and functional analysis.
Book Description
Measure, Integral and Probability is a gentle introduction that makes measure and integration theory accessible to the average third-year undergraduate student. The ideas are developed at an easy pace in a form that is suitable for self-study, with an emphasis on clear explanations and concrete examples rather than abstract theory. For this second edition, the text has been thoroughly revised and expanded. New features include: · a substantial new chapter, featuring a constructive proof of the Radon-Nikodym theorem, an analysis of the structure of Lebesgue-Stieltjes measures, the Hahn-Jordan decomposition, and a brief introduction to martingales · key aspects of financial modelling, including the Black-Scholes formula, discussed briefly from a measure-theoretical perspective to help the reader understand the underlying mathematical framework. In addition, further exercises and examples are provided to encourage the reader to become directly involved with the material.
Customer Reviews:
absolutely useless.......2007-07-05
It starts out okay, good overview of measurable sets and the like. However, it does not even have the essential core theorem to the discipline stating when it is possible to integrate a function! one of the great thing about Lebesgue integration is that a function is integrebale in this sense IF AND ONLY IF the function is measurable. thats the whole point of having measurable functions. there is no if and only if theorem for RS integration. Plus other things, like it talks vaguely about 'randomly choosing a point' but with no precise definition. Things like that.
You are better off buying a classic by Halsey Royden or Walter Rudin, or something like that. This book is useless.
Excellent Book.......2007-06-27
The text is written at a level which is suitable for the classroom or self-teaching by an advanced student. The authors spare few details. I am very satisfied with my purchase.
Very good introduction to measure theory.......2007-04-13
Very good intro for first encounters with measure theory. Throughout the application in probability theory is emphasized. The necessity of each concept introduced is motivated with clear examples. Interesting problem sets are provided after each section; their solutions are given in the appendix.
good book.......2007-02-06
This a good book but will be a bit difficult for engineering graduates like me.. Should know real analysis and set theory to venture into this one.
clear introduction to measury theory.......2006-11-09
very clear measury theory introduction....with many detail solution to exercise....not bad !!
Book Description
This monograph presents a complete and rigorous study of modern functional analysis. It is intended for the student or researcher who could benefit from functional analytic methods, but does not have an extensive background and does not plan to make a career as a functional analyst. It develops the topological structures in connection with measure theory, convexity, Banach lattices, integration, correspondences (multifunctions), and the analytic approach to Markov processes. Many of the results were previously available only in works scattered throughout the literature. The choice of material was motivated from problems in control theory and economics, although the material is more applicable than applied.
Customer Reviews:
An excellent treatment of mathematical methods for economist.......1998-10-20
The monograph covers advanced mathematical methods for economists. It includes chapters on general topology, topological vector spaces, Riesz spaces and Banach lattices, measure and integration, etc. While the book does not contain (hardly) any economics, the mathematics covered is selected under the aspect of later applications to economics. The book contains for example a long chapter on correspondences, a topic which is hardly covered by any standard math book. The presentation of the mathematics is throughout clear and precise. The advantage of the book is that it covers a wide range of mathematical topics, which could not be found together in a book before. Graduate students in economic theory can use it as a text book, but it can also be used as a reference book. The only lacks of the book are that there are no exercises and that not all math areas important to economics (e.g. differential topology) are covered. Overall, this is an excellent book and should become part of the library of everybody interested in mathematical economics.
Book Description
An accessible, clearly organized survey of the basic topics of measure theory for students and researchers in mathematics, statistics, and physics
In order to fully understand and appreciate advanced probability, analysis, and advanced mathematical statistics, a rudimentary knowledge of measure theory and like subjects must first be obtained. The Theory of Measures and Integration illuminates the fundamental ideas of the subject-fascinating in their own right-for both students and researchers, providing a useful theoretical background as well as a solid foundation for further inquiry.
Eric Vestrup's patient and measured text presents the major results of classical measure and integration theory in a clear and rigorous fashion. Besides offering the mainstream fare, the author also offers detailed discussions of extensions, the structure of Borel and Lebesgue sets, set-theoretic considerations, the Riesz representation theorem, and the Hardy-Littlewood theorem, among other topics, employing a clear presentation style that is both evenly paced and user-friendly. Chapters include:
* Measurable Functions
* The Lp Spaces
* The Radon-Nikodym Theorem
* Products of Two Measure Spaces
* Arbitrary Products of Measure Spaces
Sections conclude with exercises that range in difficulty between easy "finger exercises"and substantial and independent points of interest. These more difficult exercises are accompanied by detailed hints and outlines. They demonstrate optional side paths in the subject as well as alternative ways of presenting the mainstream topics.
In writing his proofs and notation, Vestrup targets the person who wants all of the details shown up front. Ideal for graduate students in mathematics, statistics, and physics, as well as strong undergraduates in these disciplines and practicing researchers, The Theory of Measures and Integration proves both an able primary text for a real analysis sequence with a focus on measure theory and a helpful background text for advanced courses in probability and statistics.
Customer Reviews:
The New Standard for Measure Theory Books.......2004-07-14
This is a fantastic book on measure theory. The focus is on measure theory on its own right and not on probability. I was lucky to come across this book while canvassing the measure theory books at our library. I looked at the books by Billingsley, Halmos, Chung, Resnick, Rao, Rudin, Pollard, Dudley, Nielson, Stroock, Williams, Pitt, and many others. Hand-down, Vestrup is the best.
I believe after scrutinizing so many books, I have a very good baseline to judge Vestrup's work. Here are a few specific reasons:
(1) If you don't like detail and revel in banging your head against the walls to figure out the skipped details in Billingsley, this is not the book for you. But If you are a first timer to measure theory, this is as good as it will get; All the major results of measure theory are presented in detailed and clear manner with few skipped details and few not-so-obvious "it is obvious" remarks.
(2) Vestrup has a lot of exercises with lots of helpful hints. Some problems at first appear to be long and intimidating till you look closely and discover that Vestrup leads you through the problems with his hints.
(3) Certain topics central to understanding of measure theory were given cursory coverage by most of the books mentioned above. Not Vestrup. For example, Vestrup devotes a whole chapter to extensions. This is just one example of many central ideas Vestrup develops meticulously and painstakingly.
This book is fairly new and I think its popularity will grow as more students and professionals discover it. I suppose the only criticism I have is that the typesetting can be improved (second edition maybe?)
There are a few other good books (Ash, Bartle, and Royden) that are out there that you may consider but again Vestrup trumps them all. Whatever you decide on, I strongly warn against using Billingsley.
Book Description
Lebesgue integration is a technique of great power and elegance which can be applied in situations where other methods of integration fail. It is now one of the standard tools of modern mathematics, and forms part of many undergraduate courses in pure mathematics.
Dr Weir's book is aimed at the student who is meeting the Lebesgue integral for the first time. Defining the integral in terms of step functions provides an immediate link to elementary integration theory as taught in calculus courses. The more abstract concept of Lebesgue measure, which generalises the primitive notions of length, area and volume, is deduced later.
The explanations are simple and detailed with particular stress on motivation. Over 250 exercises accompany the text and are grouped at the ends of the sections to which they relate: notes on the solutions are given.
Customer Reviews:
Great book.......2004-05-05
Its a very good text for a first meeting on Lebesgue integration, measure and functional analysis. Rigorous, elegant and simple. A quality book for pure and applied mathematics.
However I found two little mistakes:
In page 151, in the proof of the integral of a transformation, he makes use of the Dominated Convergence Theorem two times (one first time, at the begining of page 151, is right). Thats wrong because we can't "dominate" the function "g" when K -> inf. The correct proof involves divide the function in positive and negative parts and then aplicate Monotone Convergence Theorem. The same in the end of the proof when he generalizes to infinite measure sets.
In page 157, equation (7) should be verified when ||h||
<2*delta, not ||h||
Anyway, a brilliant text. Purchase it.
Good introduction to the theory of Lebesgue integration.......2004-02-26
I picked up this book on a trip to London. I've known some complex analysis and real analysis, and I decided to learn some Lebesgue on my own; ergo the purchase of this book. The style of writing is very lucid: quite informal at times, and the math part is really well-presented (the explanation on 'measure zero' set, for example, is clear, and mathematically rigorous). The topics chosen are not in-depth (I learnt much more on the topic during an actual course in college), but the book definitely works well as a supplement reading, when you are taking real analysis course.
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Lebesgue Measure and Integration: An Introduction (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)
Frank Burk
Manufacturer: Wiley-Interscience
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ASIN: 0471179787 |
Book Description
A superb text on the fundamentals of Lebesgue measure and integration.
This book is designed to give the reader a solid understanding of Lebesgue measure and integration. It focuses on only the most fundamental concepts, namely Lebesgue measure for R and Lebesgue integration for extended real-valued functions on R. Starting with a thorough presentation of the preliminary concepts of undergraduate analysis, this book covers all the important topics, including measure theory, measurable functions, and integration. It offers an abundance of support materials, including helpful illustrations, examples, and problems. To further enhance the learning experience, the author provides a historical context that traces the struggle to define "area" and "area under a curve" that led eventually to Lebesgue measure and integration.
Lebesgue Measure and Integration is the ideal text for an advanced undergraduate analysis course or for a first-year graduate course in mathematics, statistics, probability, and other applied areas. It will also serve well as a supplement to courses in advanced measure theory and integration and as an invaluable reference long after course work has been completed.
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