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

Howard J. Wilcox (Author), David L. Myers (Author), Mathematics (Author)

Book Description

Publication Date: January 4, 1995 | ISBN-10: 0486682935 | ISBN-13: 978-0486682938 | Edition: Dover ed

Undergraduate-level introduction to Riemann integral, measurable sets, measurable functions, Lebesgue integral, other topics. Numerous examples and exercises.

Product Details

• Paperback: 159 pages
• Publisher: Dover Publications; Dover ed edition (January 4, 1995)
• Language: English
• ISBN-10: 0486682935
• ISBN-13: 978-0486682938
• Product Dimensions: 9.2 x 6.2 x 0.4 inches
• Shipping Weight: 9.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #276,443 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

15 of 15 people found the following review helpful:

4.0 out of 5 stars Gentle Intro to Measure Theory, April 4, 2002

By 

Douglas Anderson (Moorhead, MN USA) - See all my reviews
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This review is from: An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Mathematics) (Paperback)

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.

8 of 9 people found the following review helpful:

4.0 out of 5 stars so-so, December 20, 2004

By 

Daniel the Ripper - See all my reviews

This review is from: An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Mathematics) (Paperback)

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 :)

6 of 7 people found the following review helpful:

4.0 out of 5 stars direct intuitive treatment but much "left as an exercise", November 8, 2000

By 

Ken Braithwaite (inkster, MI USA) - See all my reviews
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This review is from: An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Mathematics) (Paperback)

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.