# A comprehensive book on graduate real analysis

I'm looking for a comprehensive book/a comprehensive list of books on graduate analysis that covers/cover these topics: Lebesgue measure and integration on $$\mathbb{R}^d$$, the relationships between integrability and differentiability (it must also cover the theory of functions of bounded variation), complex analysis and fourier analysis.

• Stein and Shakarchi's princeton lectures in analysis is excellent
– fwd
Sep 28 at 16:54
• Integration and Modern Analysis by Benedetto/Czaja (2009; 575 pages) is pretty comprehensive (525 references, 14 page subject index, 8 page index of names) and includes a 42 page appendix on Fourier analysis. For complex analysis I strongly recommend getting a separate text, since the subject matter is sufficiently different that you're not going to find a comprehensive book for both (although Rudin is worth looking at). Sep 28 at 18:27

Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland. It presupposes and sparingly applies some complex analysis (as opposed to discussing the fundamentals), but the other mentioned topics are covered extensively from scratch.

The following books might be interesting in your case too:

• Mathematical Analysis (Apostol), since it covers Lebesgue measure and similar topics
• Real Analysis: A Comprehensive Course in Analysis, Part 1 (Berry Simon) is a very good reference too (I really like AMS books)
• A Passage to Modern Analysis (W. J. Terrell) is from AMS too and introduces things more gently (including Lebesgue measures) since it is from the series "Pure and Applied Undergraduate Texts"
• Manifolds and Differential Geometry (J. M. Lee) covers additionally smooth categories

The following books are available as PDF:

Here is a chapter on Lebesgue Integral (the whole book Lectures on Real Analysis might be interesting, since one can see at least the chapter titles including descriptions of its content):

Measure, Integration & Real Analysis by Sheldon Axler. The electronic version is freely available online and the exposition is done in a format similar to Linear Algebra Done Right. The proofs are extremely readable and it has a separate chapter devoted to Fourier Analysis.

You may have used "Baby Rudin" during your undergraduate (aka Principles of Mathematical Analysis by Rudin). He has written a graduate level text, dubbed "Big Rudin", or Real and Complex Analysis. You can find it's official webpage here.