Customer Reviews

An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Mathematics)

5 star:		(1)
4 star:		(5)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

I am using this book with two advanced undergraduates, after working through a basic introduction to analysis text the previous semester. Each section is about 3 or 4 pages long, giving the main results and most proofs, although some are left as exercises. Each chapter ends with a nice set of exercises, most of which are accessible to students who feel comfortable with epsilon/delta proofs. The book is very short and concise, 145 pages plus an appendix summarizing basic analysis results on sets, countability, functions, and sequences.

Chapter 1 reviews the Riemann integral and some of its drawbacks. Chapter 2 introduces the idea of outer measure and measurable sets, all on the unit interval. The next two chapters discuss properties of measurable sets and measurable functions. Chapters 5 and 6 then cover the Lebesgue integral and convergence theorems. The last three chapers introduce L2 spaces, Fourier series, and proofs of convergence.

All in all this is a good, very cheap way to learn the basics of measure theory and the Lebesgue integral, before moving on to something like Rudin or Royden.

This book was my first contact with measure theory. I read this for self study -- more or less as a leisure book. The material is aimed at undergrads, and probably doesn't assume much past plain ol' college calculus. However, the more you know, the easier it will be to read. Any experience with analysis, and proofs will be helpful. And, in chapter 7, "Function Spaces", a linear algebra course will come in very handy.

Anyway, it's not a hard book to read, but it is very dry. Because the book is so short, there is not much room for anything other than a list of definitions, lemmas and theorems. There isn't really much insight. All the way through chapter 7 I was basically plodding along, simply because I wanted to finish the book. However, I'm glad that I did, because I found chapter 8 really fascinating. I think this chapter (Fourier Series in L^2) really ties the book together because you get to see measure theory and lebesgue integration working in harmony with linear algebra. I never really liked linear algebra that much until I read this chapter.

Unfortunately, chapter 9 was a let down and I actually quit reading a few pages before the end of the book. I had already got what I needed out of it. It's a good intro to measure theory if you just want to see what it is, and not really go into detail with it. A lot of the lemmas, propositions and corollaries are left as exercises. I tried to do a handful of problems from every chapter, especially the ones that fill in the text, and had little or no difficulty with any of them.

I would have given this 3 stars if not for chapter 8. For the price, I would recommend it, especially if you love calculus, but never liked linear algebra, because it will hopefully tie them together for you. Now I can't get enough linear algebra! I know, it's sick ;) Then, with your newfound love of linear algebra, read Hubbard and Hubbard's _Vector_Calculus,_Linear_Algebra,_and_Differential_Forms:_A Unified_Approach_, which is currently blowing my mind :)

This assumes some advanced calculus and then introduces enough Lebesgue integration amd measure theory to explain and prove the basic convergence results for (generalized) Fourier analysis. Second half of second year and above.

Everything is very well motivated and the book is not long, but quite a lot is "left as an exercise for the reader." This really hurts the book for self study in my view. If you have a bigger book on real analysis and want another treatment, or need a refresher this will do nicely. On its own -- you've been warned.

The Fourier series sections leaves something to be desired ( like good integral and graphs or real examples of Fourier series),
but the integration theory , measure and set theory parts are very good as far as they go. The use of a back slash notation for the complement of a set is new to me and as far as I can tell never really defined in the text.
As this book is a cheap Dover text you might want to get one that gives
real examples of Fourier series as well: at least the ones for square and triangular waves used in electrical engineering.

A good basic introduction to Lebesgue integration, but a little sketchy in some places. Each chapter concludes with a good set of exercises. This book is good for getting the "idea" behind Lebesgue integration, but comes up short on many of the details. An excellent companion volume would be Jones' "Lebesgue Integration on Euclidean space".

This is more about Lebesgue integration, covered in the first 112 of 154 pages, than about Fourier Series. But do not be fooled by the low price; this book makes for very good review of the Lebesgue theory, well organized and concise. Or with moderate mathematical maturity, the reader could learn the theory here.

I should mention my own history in approaching this book; last year I took a yearlong graduate analysis class out of Rudin, and now I'm taking a similar analysis class elsewhere, and I read this yesterday for review. In that context it served its purpose very well; in what it presents it's clear and concise, with a good idea of what to detail and what to leave to the reader (many of the proofs are delegated to the exercises). Having said that, "in what it presents" is an important qualifier to use; it's somewhat limited in scope, even given that it's a relatively short book (<150 pages). Most notably, measurable sets are only discussed as subsets of the real line, and the only measure discussed is the Lebesgue measure (on the real line). Furthermore, measurable sets are built up from being subsets of [0,1] to being subsets of arbitrary closed intervals to being subsets of the real line and possibly unbounded -- this seems to me like an unnecessary simplification that largely obfuscates the inherent value of the topics; even sigma algebras are only briefly mentioned in a throwaway section that compares Lebesgue measurable sets to Borel sets. It's due to this overall avoidance of advanced machinery that this book likely isn't very suitable or useful for graduate students who are trying to learn analysis, but I imagine that - aside from its aforementioned use as review - a (reasonably bright) junior or senior undergraduate could find some value in it as a preview of topics to come, as long as its limitations are understood.

I should also mention that I'm not really interested in Fourier analysis and I only skimmed through those chapters (the last two, of nine), so what I say here only applies to the previous chapters. In as much as I can judge such things the Fourier analysis chapters seemed decent. I also didn't do any of the exercises, but from a quick glance they seem a little easy as (ostensibly) graduate-level books go and a little challenging as undergraduate books go. They'd be worth working through for someone unfamiliar with the material.

Since amazon won't let you search inside the book, here's a table of contents:

1. The Riemann Integral
- Definition of the Riemann Integral / Properties of the Riemann Integral / Examples / Drawbacks of the Riemann Integral

2. Measurable Sets
- Introduction / Outer Measure / Measurable Sets

3. Properties of Measurable Sets
- Countable Additivity / Summary / Borel Sets and the Cantor Set / Necessary and Sufficient Conditions for a Set to be Measurable / Lebesgue Measure for Bounded Sets / Lebesgue Measure for Unbounded Sets

4. Measurable Functions
- Definition of Measurable Functions / Preservation of Measurability for Functions / Simple Functions

5. The Lebesgue Integral
- The Lebesgue Integral for Bounded Measurable Functions / Simple Functions / Integrability of Bounded Measurable Functions / Elementary Properties of the Integral for Bounded Functions / The Lebesgue Integral for Unbounded Functions

6. Convergence and Lebesgue Integration
- Examples / Convergence Theorems / A Necessary and Sufficient Condition for Riemann Integrability / Egoroff's and Lusin's Theorems and an Alternative Proof of the Lebesgue Dominated Convergence Theorem

7. Function Spaces and L2
- Linear Spaces / The Space L2

8. The L2 Theory of Fourier Series
- Definition and Examples / Elementary Properties / L2 Convergence of Fourier Series

9. Pointwise Convergence of Fourier Series
- An Application: Vibrating Strings / Some Bad Examples and Good Theorems / More Convergence Theorems