Lebesgue Integration by Riesz Method textbook

Question

As an undergraduate student I am recommended to use Soo Bong Chae's Lebesgue Integration as a textbook for a course of Lebesgue Integral. The book is far from satisfying my personal needs as it gets quite complicated in proofs and quite a lot of arguments in proofs are not explained (simply left to reader). But this book approaches Lebesgue Integration by Riesz Method, meaning that Lebesgue integration is approximated by sequences of step functions and also Lebesgue measure is considered to be the consequence of the integration theory. Nonetheless, in my search through Google, I did not come across a textbook which consists of mainly Lebesgue Integration and Measure by Riesz approach and written for undergraduates. So, I need your help to suggest me another book on the subject.

Answer 1

This answer is obviously a little bit dated, but for anybody looking in the future: The Lebesgue Integral for Undergraduates by William presents Lebesgue Integration using the Riesz approach. In the book, he calls it the "Daniell-Riesz" approach (after Percy J. Daniell). I read the book with a professor through an independent study course.

It's not a full treatise on the subject (the author is upfront about its limitations in the introduction). The book definitely helped me when I later encountered measure theory in a real analysis course.

If anyone is interested, you can find PDFs of the Preface and Table of Contents on the MAA's website:

Preface: https://www.maa.org/sites/default/files/pdf/ebooks/TLI_Preface.pdf

Table of Contents: https://bookstore.ams.org/cdn-1646297439663/text-27/~~FreeAttachments/text-27-toc.pdf

** I'm not affiliated with the author/MAA/etc