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".

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 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.