Customer Reviews

Real Analysis (3rd Edition)

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1 star:		(4)

 

 

I used this text as senior undergraduate in an introductory course to real variables. 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.

The biggest downside however is that most graduate students don't have the time needed to dedicate to the various problems in this text, which is why Royden is probably not the best choice for a first year graduate text. Instead I would recommend Bartle's Elements of Integration and Lebesgue Measure as a first year grad text on the subject. 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?) important results, and in the long run this makes for a much stronger understanding of the material- if you have the time to devote to it.

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.e. Rudin). 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 index and table of notation are excellent and contribute to this book's usefulness as a reference.

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.

I'm a PhD student in mathematics at Georgia Tech. I used this when I first took graduate real analysis at North Dakota State, and then used Wheeden and Zygmund's Measure and Integral here at Georgia Tech as well as Folland's Real Analysis when studying for comps. Time and time again, I found myself going back to Royden for his well-written expositions that left enough out to keep you paying attention but wasn't so sparse that you couldn't figure out what was going on. 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. I saw it first hand last fall, as my colleagues in another section treated Lebesgue measure on the real line as a special case and did things in generality, while my section dealt with R^n first and then moved on to general measures. In the end, I'm quite sure the section that I was in had a firmer grasp on the material.

Royden's classic work has withstood the test of time, and deserves to remain a standard text for years to come.

There are three books that are usually used for a first graduate course in analysis, including measure theory, namely Rudin's Real and Complex Analysis, Folland's Real Analysis, and Royden's book. Of the three, I would say Royden's book is the easiest, both in terms of the exposition, material, and exercises. 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. Personally, I found the development of the Lebesgue integral a bit tedious; the whole thing works a bit better when you first talk about abstract integration, which really isn't a terribly difficult concept, prove the basic integration theorems, then show how to construct an outer measure, and suddenly, the Lebesgue measure and integral just falls into place. I'm not sure anything is lost in the process.

The biggest shortcoming in this book would have to be the exercises: for the most part, they are not very difficult, particularly when you compare them to say, Rudin's text. 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. The exercises here can work well if you want some extra practice that won't take up too much time. If you're a student of econ, or physics, or you just feel like learning graduate-level real analysis, then this book is probably adequate (although I should qualify that statement by saying that I know nothing of econ and little of physics). 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.

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 think It's the best way of explaining something. 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. but It's pretty hard thing to do for me, and I'm sure that most others also are. If you agree with the negative oppinions than you can start from part three. you can get all the contents of Part One. and hopefully, You would be able to Understand all the materials. But I don't think that you will get an clearer image than those who have startde from part one.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.

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.

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(!), but because of being in computers). I've taken a renewed interest in the subject (I'm a pretty random person) and have been surprised at how the material has come back to me, I think because of the readability of the text. It's true, Royden challenges the reader at every turn, but if one has acquired the level of mathematical maturity commensurate with strong interest in analysis, the challenges are appropriate, in my opinion

Halsey Royden's Real Analysis has become the de facto standard for teaching a graduate course on real analysis and integration. 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.

That said, Royden is very good at explaining things in more detail; it is both a complement and a criticism that the book manages to cover a good deal less than Folland's text in almost twice the length. 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. What's more is the fact that in the book's 444 pages it only manages to cover about half of what Gerald Folland goes through in his shorter book; this makes the ridiculous price of the book even less justifiable.

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.A. Armstrong's Basic Topology.

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.

Royden's text begins with a careful development of Lesbesgue measure and integration, with a discussion of differentiation and the L^p spaces. 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.