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.
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$\begingroup$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:
- Real Analysis (4th Ed.) by Royden
- Measure, Integration & Real Analysis (Open Access Book) by Sheldon Axler as already mentioned by Axion004
- Basic Analysis: Introduction to Real Analysis (free online textbook) by Jiří Lebl
- Basic Real Analysis (Stony Brook Mathematics)
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):
$\endgroup$ 0 $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 0 $\begingroup$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.
$\endgroup$ $\begingroup$I am doing an independent study in measure theory and integration, and I have found Measure and Integral(3rd edition) by Richard L. Wheeden and Antoni Zygmund to be immensely useful. It covers all topics you mention except for a comprehensive treatment of complex analysis. You can find it here:
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