This text assumes a general background in mathematics and familiarity with the fundamental concepts of analysis. Classical theory of functions, including the classical Banach spaces; General topology and the theory of general Banach spaces; Abstract treatment of measure and integration. For all readers interested in real analysis. I find that using the convexity of the exponential is much clearer.

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Product Description This is the classic introductory graduate text. Heart of the book is measure theory and Lebesque integration. The course was structured as a sort of modified RL Moore method class: there were very few lectures, and we the students could only use theorems and propositions presented in the text if we had gone to the board and presented a valid proof for each.

As such, most of the students learned the fundamentals very well. This in turn made my first graduate course in real variables much easier.

It was disappointing to use Bartle and discover that so many of the problems in Royden, which I had spent countless hours attempting to prove, had been completely worked out in Elements of Integration. In short, Royden makes you work for many most?

Zorach on Aug 22, Many people criticize this book as unclear and unnecessarily abstract, but I think these comments are more appropriately directed at the subject than at this book and its particular presentation. I find this classic to be one of the best books on measure theory and Lebesgue integration, a difficult and very abstract topic. Royden provides strong motivatation for the material, and he helps the reader to develop good intuition.

I find the proofs and equations exceptionally easy to follow; they are concise but they do not omit as many details as some authors i. Royden makes excellent use of notation, choosing to use it when it clarifies and no more--leaving explanations in words when they are clearer. The construction of Lebesgue measure and development of Lebesgue integration is very clear. Exercises are integrated into the text and are rather straightforward and not particularly difficult.

It is necessary to work the problems, however, to get a full understanding of the material. There are not many exercises but they often contain crucial concepts and results. This book contains a lot of background material that most readers will either know already or find in other books, but often the material is presented with an eye towards measure and integration theory. The first two chapters are concise review of set theory and the structure of the real line, but they emphasize different sorts of points from what one would encounter in a basic advanced calculus book.

Similarly, the material on abstract spaces leads naturally into the abstract development of measure and integration theory. This book would be an excellent textbook for a course, and I think it would be suitable for self-study as well. Reading and understanding this book, and working most of the problems is not an unreachable goal as it is with many books at this level.

This book does require a strong background, however. Due to the difficult nature of the material I think it would be unwise to try to learn this stuff without a strong background in analysis or advanced calculus. A student finding all this book too difficult, or wanting a slower approach, might want to examine the book "An Introduction to Measure and Integration" by Inder K.

Rana, but be warned: read my review of that book before getting it. Not perfect, but better than the rest By Mitchel T. Some here have complained about it doing everything twice.

This can be a problem in some cases, such as common texts for a first course in real analysis where topological ideas are covered for Euclidean space first and then again for general metric spaces, but with measure theory, this is the right approach. By on Mar 03, All the negative oppinions say that It repeats allmost identical contents twice. I cannot understand why so many people feel so uneasy about that. Acutally, It does not repeat the "same" thing. In part one, measure theory on the real line is presented and, after you get pretty good understanding and image in the "real world", the abstraction or equivalently, axiomazation of measure In abstract space is given.

I know that many people in this field love the "rudin Style" - Books which contains Definitions and Theorems only. Oh I envy them. I wish I had the ability to understand something from the essense without rumbling the world I can touch for some time. If you agree with the negative oppinions than you can start from part three.

Even those who complain the system of this book could do that cause they already have read the part one and studied the "same thing" Twice. If they have not read them and started from the part three, they would have complained that It was so abstract. Pretty good as a first book, except for chapter 5. A Customer on Sep 17, Royden is pretty good for learning about measure theory for the first time. There are some annoying misprints in the problems which cause headaches for students.

A major wart is that Chapter 5 on differentiation is terrible. He keeps applying the vitali lemma over and over again, confusing the reader because he neglects to even mention Lebesgue points. Not bad for self-study, excellent for reference By Mike Turner on Mar 28, I used Royden 2nd edition as a graduate student over 30 years ago, and have been away from real analysis pretty much ever since not because of the book!

The book also provides a good introduction to metric spaces, including a discussion of Banach and Hilbert spaces. Also included is a very insightful and user friendly introduction to topological spaces. The later portions of the text present a well written development of abstract measure theory including signed measures, the Radon-Nikodym Theorem, Fubini and Tonelli Theorems, and the Riesz Lemma. Overall, the book is an indispensible tool for serious mathematics students.

It is a very readable introduction to ideas central to mathematics as well as an invaluable reference. First, this book have numerous misprints, especially in exercises, so you might often find yourself trying to prove something which is false and, in this case, it is really a waste of time. Second, the 1st section of a book, develops measure theory for a real line, while the 3rd -- for general measure spaces. Now, this is a real waste, as the general theory, basically consists of the same theorems, so there is not point in proving everything twice.

It is absolutely excellent and I vastly prefer it to Rudin. Covers some basics of metric spaces, then some fairly nice amounts of topology, measure theory, topological groups, etc. Very pleased : Maybe good as a supplement, or a first time looking at the material By C. Of the three, Royden is the only one to fully develop the Lebesgue measure and the associated integral before developing a more general theory of measure and integration.

Furthermore, he does not develop Hilbert and Banach space theory, the very basics of functional analysis, to anywhere near the extent that Folland and Rudin do. There is some debate as to whether it is better to start with the Lebesgue integral, and then talk about abstract integration, or the other way around. For the most part, the exercises are fairly trivial, and if they are difficult, or require a bit of creativity, Royden often gives you lots and lots of hand-holding, sometimes even in the form of sketching out the proof for you.

In spite of the relatively low difficulty level, most of the exercises are fairly instructive, in so far as they highlight, elucidate, and expand upon the material. For the most part, this book is not bad. It makes a good supplement to a book like Rudin or Folland, as it is less abstract, and does a better job motivating the material. But if you are a serious student of mathematics, particularly the pure variety, this is really not the book you should be using.

It is just too easy. Readable, very well written By Professor Joseph L. Mccauley on May 10, With basic knowledge of point set theory, a mathematically-oriented physics student can use this book for self-study.

I used it as advanced grad student to learn measure theory and Lesbesgue integration. I certainly remained a beginner surely could not have passed a typical math exam in analysis but was nevertheless able to apply the basic ideas of meassure theory some decades later to resolve a subtle question about fractals.

A classic in real analysis. Rogelio on Oct 17, This book, as far as I consider it, is a classic. Although it has some errata and stuff, it is one of the best books ever written in Real Analysis.

Then, about the book, it takes to know some modern algebra, set theory and basic analysis to study from this fine book. The book is rich on references where one could further study material related to real analysis to the higher level. I think this book will be very useful to serious mathematicians and to those self-studying guys who love the rigour.

Excellent introduction By Alex P. Keaton on Mar 05, I appreciate the approach Royden chooses in beginning with Lebesgue measure first then addressing general measure at the end of the book. For those of you frustrated by this approach, you can jump right to chapters 11 and 12 for a general treatment first.

Very nicely done. Classic text, but a poor reference. It has, however, become a bit dated. First off, the method of developing the Lebesgue integral before general measure theory is out of style. It is now generally accepted that learning the relatively easy concept of general measure theory first, and then the Lebesgue measure as an example, is a superior pedagogical approach. Complementary in the sense that the book motivates the material and gives explanations without leaving the reader to any important developments, critical in that the book is more or less useless as a reference.

All of this said, I would still recommend this book for study. It explains well and would be a good read for self study. As for the criticisms that label the book as either "too difficult" or "too dense," disregard them. Those who make these claims are probably just not very good with Analysis. For a book that is truly awful, see M. It was a very good introduction. I enjoyed reading it By Waseem on Oct 23, I am a researcher and a scientist trained in engineering and computer science.

As I progressed in my career, I felt the need of deeper meaning of everyday math I encountered. Somehow, I became aware of Hilbert spaces. To understand the concept, I started reading about them. Quickly, I realized the concept of Analysis. I enjoyed reading it. It provided me the foundation, I was looking for. Then I struggled through five or six different graduate level texts. I have only read the first two chapters, and everything suddenly started making sense and gel together.

I am awestruck, how Royden has figured out a way to organize all the concepts in some kind of natural sequence, and the amount of details on each concept.

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This is the classic introductory graduate text. Most helpful customer reviews 0 of 0 people found the following review helpful. A classic in real analysis. Rogelio This book, as far as I consider it, is a classic. Although it has some errata and stuff, it is one of the best books ever written in Real Analysis. Then, about the book, it takes to know some modern algebra, set theory and basic analysis to study from this fine book. The book is rich on references where one could further study material related to real analysis to the higher level.

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