Customer Reviews (2)

One of the Greatest TextBooks on Lebesgue Theory
This book is one of the best books ever written on this subject .There are lots of worked out problems (something lacking in most books), and the theorems are fully proved.Lebesgue theory is a very hard subject to learn, and this book makes the learning easier because of the author's desire to reach out.A big danger in trying to learn a subject like Lebesgue measure and integration is that it is easy to learn how to push the symbols on a peice of paper without really understanding what is going on.This book tries to develop the reader's intuition and the ideas behind the symbols. This book is much better than all current textbooks used at American Universities.A great book overall.

Save your money for another book
For starters, this book uses non-standard notations or standard notations with non-standard meaning, which is annoying. For example, the bold capital letter I is used to denote of set of integers (rather than the standard Z). The standard blackboard bold letter N is used for the set of natural numbers, but the author excludes zero from that set (the author actually never defines N but it's apparent that zero is excluded from the given examples.) There are also quite a few typos (to cite just one example on page 20: "In the binary system every natural number x can be uniquely represented by x=sum from n=1 to infinity of x_n / 2^(n-1), where x_n is either 0 or 1."), but I gave up on the book when it claimed that the continuum hypothesis (the question of whether there exists a cardinal number strictly between the cardinality of the integers and the cardinality of the real numbers) was still an unsolved problem! ("This problem is yet unsolved").

For some strange reason, the author also develops the theory of cardinal numbers, which is only very marginally relevant to Lebesgue measure and integration. Sure, it's an interesting topic, but it really distracts from the main topic, and does not really belong in such a book.

The book has some redeeming features. In Appendix III, it has the English translation of Henri Lebesgue's lecture called "The Development of the Notion of the Integral" (reproduced from the book "Lebesgue Integration", by Soo Bong Chae, Marcel Dekker, New York, No. 58 (1980)). This lecture is a masterpiece.

It is also written in an accessible manner with good examples, definitions and theorems clearly stated, and instructive discussions, and a fair number of exercises.

However, these positive qualities are more than offset by the aforementioned flaws.
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Customer Reviews (1)

Opinion on Measure and Integration by Sterling K. Berberian
Once upon a time I was once told, the student of mathematics, in the initial stages of his career is given Abstract Algebra, Mathematical Analysis and Point Set/General Topology from which nothing follows. Well, presenting a student these branches that way is like giving one incomplete jigsaw puzzles and asking him to put them altogether. This was my favorite book on Measure Theory & Integration of which I preferred to Paul Halmos's classic on the similar subject in my undergraduate years. The treatment is very abstract, but if the reader purses for a moment, one starts to see some form of unification and connection running throughout the whole spectrum of Mathematics (Pure/Applied)as far Intergration is concern. The treatment is of course the generalization of the Riemman/Riemman Stieltjes integrals. This then makes one belief that integration is not only done on the real line, but also on more abstract spaces as well(generalization of the real/complex number system). The book explains the how/why as to the justification of the application of integrals to Mathematical Statistics (Probability Theory), Engineering and the other areas of the Mathematical Sciences. The basis of my review is not based on chapter by chapter, but as to the reason why such a book was written in the first place. Usually one takes a course in elementary Calculus thinking it all ends there, but that is not the end of the story. I believe after reading classic text like T.M. Apostol's Mathematical Analysis, Walter Rudin's Principles Of Mathematical Analysis and Royden's Real Analysis, I think Sterling K. Berberian's book should then be read and be placed on every mathematician's bookshelf to explain why we human beings enjoy performing Integration in the first place. All the Theorems and Proofs are nicely written, and the treatment which of course is based on Set Theory is very good. I will recommend this book to every mathematician doing research. ... Read more

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Although of unquestioned power and practical utility, the Lebesgue Theory of measure and integration tends to be avoided by mathematicians, due to the difficulty of obtaining detailed proofs of a few crucial theorems.In this concise and easy-to-read introduction, the author demonstrates that the day-to-day skills gleaned from Legesgue Theory far outweigh the effort needed to master it. This compact account develops the theory as it applies to abstract spaces, describes its importance to differential and integral calculus, and shows how the theory can be applied to geometry, harmonic analysis, and probability. Postgraduates in mathematics and science who need integration and measure theory as a working tool, as well as statisticians and other scientists, will find this practical work invaluable. ... Read more

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This volume is a thorough and comprehensive treatise on vector measures. The functions to be integrated can be either [0,infinity]- or real- or complex-valued and the vector measure can take its values in arbitrary locally convex Hausdorff spaces. Moreover, the domain of the vector measure does not have to be a sigma-algebra: it can also be a delta-ring. The book contains not only a large amount of new material but also corrects various errors in well-known results available in the literature. It will appeal to a wide audience of mathematical analysts.

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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. ... Read more

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Based on courses in measure and integration theory that the author has taught for a number of years in the mathematics department of New Mexico State University. ... Read more

Customer Reviews (1)

Measure and Integration
The book of Swartz goes about several subjects, divides in several chapters. The interesting parts for me were Measure Theory, Integration and the Functional Analysis.
It is not a book to read on the Sunday afternoon, some graduate knowledge of the
mathematics is needed! The defintions, theorems and propostion are nice ordered and
the proofs are good to understand. Lebesgue integration is an important part of
the book. Important are also the Additive Set Functions (finite and countable).
It is interesting to note that in the whole book, there is no picture at all, only
text. In the title of the book is mentioned Functions Spaces, may be some more attention could be spend to it then now is done.
Using the book for a lecture? I should not do, it is more for self-study, with the
help of an expert in the areas as mentioned above. It is good book for on the book-shelf and to search for interesting parts of Measure Theory.
But looking in the book, searching for an answer on a question, will certainly
give rise to other questions. The books of I.K. Rana, An Introduction to Measure
and Integration (sec.ed., GSM vol.45) or V.S. Pugachev, I. N. Sinitsyn: Lectures on Functional Analysis and Applications, are certainly worth to mention and then to use together with the book of Swartz,
Blakske
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By generalizing the concept of point function to that of a function (``soma'' function ) over a Boolean ring, Caratheodory gives in this book an elegant algebraic treatment of measure and integration. ... Read more

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Intégration is the sixth and last of the Books that form the core of the Bourbaki series; it draws abundantly on the preceding five Books, especially General Topology andTopological Vector Spaces, making it a culmination of the core six. The power of the tool thus fashioned is strikingly displayed in Chapter II of the author's Théories Spectrales [MR35 #4725], an exposition, in a mere 38 pages, of abstract harmonic analysis and the structure of locally compact abelian groups.

The present volume comprises Chapters 1-6 in English translation (a second volume will contain the remaining Chapters 7-9). The individual fascicles of the original French edition have been extensively reviewed [Chs. 1--4, MR 14, 960, 2e édn. MR 36 #2763; Ch. 5, MR 18, 881, 2e edn. MR 35 #322; Ch.6, MR 23 #A2033; Chs. 7-8, MR31 #3539; Ch. 9, MR 43 #2183].

Chapters 1-5 received very substantial revisions in a second edition, including changes to some fundamental definitions. Chapters 6-8 are based on the first editions of Chs. 1-5. The English edition has given the author the opportunity to correct misprints, update references, clarify the concordance of Chapter 6 with the second editions of Chapters 1-5, and revise the definition of a key concept in Chapter 6 (measurable equivalence relations). ... Read more

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Designed for the full-time analyst, physicists, engineer, or economist, this book attempts to provide its readers with most of the measure theory they will ever need. Given the choice, the author has consistently optedto develop the concrete rather than the abstract aspects of topics treated.

The major new feature of this third edition is the inclusion of a new chapter in which the author introduces the Fourier transform. In that Hermite functions play a central role in his treatment of Parseval's identity andthe inversion formula, Stroock's approach bears greater resemblance to that adopted by Norbert Wiener than that used in most modern introductory texts. A second feature is that solutions to all problems are provided.

As a self-contained text, this book is excellent for both self-study and the classroom. ... Read more

Customer Reviews (2)

Hard to understand!
I bought this book for my class, but honestly I felt hard to understand the concepts presented in this book.

A perfect place to start Real Analysis
This is a perfect textbook for a one-semester course in measure and integration at a graduate level. It's concise but complete and detailed, rigour but motivated and inspiring. It's more technical than Bartle's little book on integration. It also has solution manual in the end to check on. I hope more readers enjoy such a wonderful germ written by a star form MIT. ... Read more

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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. ... Read more

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This book gives a general definition of the (abstract)integral, using the Daniell method. A most welcome consequence of thisapproach is the fact that integration theory on Hausdorff topologicalspaces appears simply to be a special case of abstract integrationtheory. The most important tool for the development of the abstracttheory is the theory of vector lattices which is presented here ingreat detail. Its consequent application not only yields new insightinto integration theory, but also simplifies many proofs. For example,the space of real-valued measures on a d-ring turns out to be anorder complete vector lattice, which permits a coherent development ofthe theory and the elegant derivation of many classical and newresults. The exercises occupy an important part of the volume. In addition totheir usual role, some of them treat separate topics related to vectorlattices and integration theory. Audience: This work will be of interest to graduate-levelstudents and researchers with a background in real analysis, whosework involves (abstract) measure and integration, vector lattices,real functions of a real variable, probability theory and integraltransforms. ... Read more

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This book is concerned with three main themes. The first dealswith ordering structures such as Riesz spaces and lattice orderedgroups and their relation to measure and integration theory. Thesecond is the idea of fuzzy sets, which is quite new, particularly inmeasure theory. The third subject is the construction of models ofquantum mechanical systems, mainly based on fuzzy sets. In this waysome recent results are systematically presented.
Audience: This volume is suitable not only for specialists inmeasure and integration theory, ordered spaces, probability theory andergodic theory, but also for students of theoretical and appliedmathematics. ... Read more

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Measure theory is a classical area of mathematics born more than two thousand years ago. Nowadays it continues intensive development and has fruitful connections with most other fields of mathematics as well as important applications in physics.

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Non-Additive Measure and Integral is the firstsystematic approach to the subject. Much of the additive theory(convergence theorems, Lebesgue spaces, representation theorems) isgeneralized, at least for submodular measures which are characterizedby having a subadditive integral. The theory is of interest forapplications to economic decision theory (decisions under risk anduncertainty), to statistics (including belief functions, fuzzymeasures) to cooperative game theory, artificial intelligence,insurance, etc.
Non-Additive Measure and Integral collects the results ofscattered and often isolated approaches to non-additive measures andtheir integrals which originate in pure mathematics, potential theory,statistics, game theory, economic decision theory and other fields ofapplication. It unifies, simplifies and generalizes known results andsupplements the theory with new results, thus providing a sound basisfor applications and further research in this growing field ofincreasing interest. It also contains fundamental results ofsigma-additive and finitely additive measure and integration theoryand sheds new light on additive theory.Non-Additive Measureand Integral employs distribution functions and quantilefunctions as basis tools, thus remaining close to the familiarlanguage of probability theory.
In addition to serving as an important reference, the book can be usedas a mathematics textbook for graduate courses or seminars, containingmany exercises to support or supplement the text.
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