Can someone please recommend a textbook that gives a substantial treatment of absolute continuity and is accessible (written for students; sticks to the real numbers instead of turning to more abstract generalizations)?
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$\begingroup$ If you have access to interlibrary loan or ProQuest, you may want to look at Walter Eugene Klann, Properties and Applications of Absolutely Continuous Functions, Ed.D. Dissertation (under Donald Dale Elliott), Colorado State College [= University of Northern Colorado], 1968, ix + 190 + 1 pages. I gave a summary of its contents here, where it is bibliographic entry [43]. $\endgroup$– Dave L. RenfroJul 6, 2017 at 19:56
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2 Answers
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A way more substantial (but still accessible) treatment than the one in Royden's book can be found in Leoni's A First Course in Sobolev Spaces.
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The Wikipedia page on absolute continuity frequently uses Royden, H.L. (1988), Real Analysis (third ed.) as a reference. I think newer editions of this book are co-authored by Fitzpatrick, as Google searching the reference brought me to this pdf. See page 119 in the book numbering, page 130 in the pdf. I hope this helps.
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$\begingroup$ If you can find a copy of I. P. Natanson, Theory of functions of a real variable, I think you will be pleased with the treatment. $\endgroup$ Jan 23, 2017 at 17:58