Book Description
This volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given.Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function.Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas. ... Read more

Customer Reviews (6)

best book to learn measure theory from
this is a great, clearly written book that excels as a book to learn analysis from.the book takes a ground up approach, starting with only the positive real line and generalizing from there.being presented in the most simple context where all the abstraction is stripped away, the essence of the arguments is laid bare and thus the proofs are as understandable as possible.then, once the arguments are made and the intuition is in place, the book proceeds to generalize the results to more abstract circumstances. thus makingthe motivation for using more powerful tools clear.

this style can be compared to that of rudin's classic book, which is largely a disaster to learn out of.for instance, rudin's book oscillates between C and R for whatever gives the most slick, but often least insightful proof.this is fine, and even enjoyable once you understand the subject, but is terrible to learn from for most people.there isn't a better book to learn measure theory and integration out of than this book

One of the best real analysis books
For the past two semesters, our professor used this book as textbook. I didn't buy this book during that time. But when I was preparing for the final exam, I borrowed the book from library, and found it was really well-written. Basic materials are well organized and explained. Some profound and important topics are also covered, like Hardy-Littlewood max function; these materials are also explained in a clear and easy-to-follow way. Thus I decided to buy this book to prepare for my Real Analysis prelim and as future reference.

A (quasi) masterpiece
4 stars, which actually means 4.5. I don't rate it the maximum, because I think it lacks a couple of things to be perfect.

The pros:

1. the theory is built from the very ground up to the "ante-room", so to speak, of further and more advanced developments in abstract measure theory and functional analysis, in a deeply logical and clear way with the highest economy of words and of thought. From this viewpoint, for example, I don't see the fact of setting the theory in the R^n environment as a weakness: on the contrary, since it results from a deliberate choice of the authors, it actually ends up in an element of strength, because the reader/learner can take all the time he/she needs to become familiar with the "exact integration" approach of Lebesgue (which is *completely* different from Riemann's), and to visualize how things are going by using the familiar multivariable environment of R^n.

In other words: the reader can take all the time he/she needs to learn to swim, before he/she actually has to swim on the much longer and more difficult track of abstract measure theory (as a branch of functional analysis). I believe such a gradual approach to be better than a direct one, where from the very first page you are thrown into abstract measure theory, with the risk of being almost completely unable to understand what all that stuff is about.

2. The almost perfect way in which the authors build the theory and logically argument actually makes the book a fantastic school to learn the deep essence of the axiomatic method. This is its greatest strength, in my opinion: that is, the fact that in carefully going through the definitions, lemmas, theorems and corollaries (and in fact *working out* them) you can actually learn what the essence of correct mathematical thinking is. As long as I can remember, there are only a couple of other books, at the same level of this one, which are as good: i.e., Rudin 1 & 2 (the "Principles" and "R&C Analysis") and Einar Hille's "Lectures on Ordinary Differential Equations" (too bad it's definitely out of print. It would be such a great thing to have it reprinted in some economic edition).

The cons:

1. The Theorem of Integration by Substitution isn't demonstrated at all, with the possible exception of a particular case in the problems. Since it is a fundamental result and since its demonstration can be very enlightening from a geometric point of view, I think this is a weakness.

2. The part about Indefinite Integral and Differentiation (Vitali's Covering Lemma, and all the results deriving from it) isn't on the same level of the preceding chapters, and isn't as clear and well built as it is on Royden's "Real Analysis" (another great book): maybe because in the latter it fits naturally into the rest of the book (which is, in the first chapters where the theory is built from the foundations, intrinsically one-dimensional) as a necessary development of what comes before, while in Wheeden-Zygmund it seems to be forced in a book which, until that point, had been developing in an intrinsically multi-dimensional way: and this cannot happen at no cost.

Everything considered, it's worth its price (which, btw, is a little too high for a book of less than 300 pages ;) )

A good textbook for new learners
This book uses both classical and abstract approaches to introduce Lebesgue measure and integral. It starts with the classical approach and bases its presentation on Euclidean space. This makes it easier for new learners like me since it is more intuitive. In later chapters an abstract approach is also used. I find this repetition natural and helpful. This book has almost no typo. Its exercises are reasonably challenging.

It could be improved in page layout if the end of each proof is clearly indicated.

An excellent choice
This is the book we used when I was a grad student. This is indeed quite a nicely written book: logical progression of concepts, a large number of exercises of varying difficulty (hard ones have hints) and no typos (always a big plus with me). All the classical results are included. My only suggestions to make this book better would be to have some longer discussions of the concepts introduced to break the litany of definition-theorem-proofs and to include historical notes. This would make this book a little bit less dry and an even more enjoyable read. Nevertheless, this is one of the best books on the subjects, better than the book by Royden which is also used by some professors. ... Read more

Book Description
Professor Zygmund's Trigonometric Series, first published in Warsaw in 1935, established itself as a classic. It presented a concise account of the main results then known, but on a scale that limited the amount of detailed discussion possible. A greatly enlarged second edition (Cambridge, 1959) published in two volumes took full account of developments in trigonometric series, Fourier series, and related branches of pure mathematics since the publication of the original edition. These two volumes, bound together with a foreword from Robert Fefferman, outline the significance of this text. Volume I, containing the completely re-written material of the original work, deals with trigonometric series and Fourier series. Volume II provides much material previously unpublished in book form. ... Read more

Customer Reviews (1)

aging but still a milestone anyway
I am very surprised to be the first person to write a review on this widely known reference in trigonometric and fourier series; there is nothing much to
say: this is a definite reference although things have certainly changed a bit
since the last time the author did work on it. I found both books in hardback
edition lately at a much cheaper price than as a new paperback and I had an instant use of it when asked a specific question on a math forum by a student...
Definitely belongs to the "Hardy-Littlewood" and "Polya-Szëgo" type but then all the more respectable.
By the way, Zygmund wrote a reference book together with Saks on analytic functions... ... Read more

Book Description
In the theory of convergence and summability whether for ordinary Fourier series or other expansions emphasis is placed on the phenomenon of localization whenever such occurs, and in the present paper a certain aspect of this phenomenon will be studied for the problem of best approximation as well. ... Read more