# Any suggestions about good Analysis Textbooks that covers the following topics?

I am an undergraduate math major student. I took two courses in Advanced Calculus (Real Analysis): one in Single variable Analysis, and the second in Multivariable Analysis. We basically used Rudin's Book "Principles of Mathematical Analysis" as a textbook, and we covered the first 9 chapters of the book(Namely: 1. Real and Complex Number Systems. 2- Basic Topology. 3-Numerical Sequences and Series. 4- Continuity. 5- Differentiation. 6- Riemann-Stieltjes Integral. 7- Sequences and Series of Functions. 8- Some special Functions. 9-Functions of several variables).

I am looking for a good (and easy to read) textbook, preferably with many examples (or solved problems) that covers the following topics:

• algebras and measures;
• the measure theoretic integral (in particular, the N-dimensional Lebesgue integral);
• convergence theorems;
• product measures;
• absolute continuity;
• signed measures;
• the Lebesgue-Stieltjes integral.

This is also another description of the topics covered that I found on the syllabus of the course: "Brief review of set operations and countable sets. Measure theory, integration theory, Lebesgue measure and integrals on $\mathbb R^n$, product measure, Tonelli-Fubini theorem. Functions of bounded variation, absolutely continuous functions"

I appreciate any kind of suggestion about a good textbook that I can use to learn the topics above by self-study. I prefer if you can tell me about the easy-to-read ones with examples and solved problems, because it's very hard for me to understand analysis without solving examples and problems. Thanks in advance for the help!

Royden, "Real Analysis" Part I + first few chapters of Part III.

Folland, "Real Analysis: Modern Techniques and Their Applications"

Cohn, Measure Theory

Since you mentioned baby Rudin, first half of big Rudin ("Real and Complex Analysis") is actually not bad, but is perhaps a bit unmotivated when you first approach the subject.

I think you can try to read Folland's book carefully. I think it is not a hard book to read and the exercises are reasonable. It is also one of the standard reference books one used in graduate school.

Another thing which you must know is to use the OCW MIT website. There are plenty of course notes on Lebesgue integration and measure theory if I recall correctly from my memory. And there are problem sets and (sometimes) solutions as well.

The other colleague mentioned Tao. I think his measure theory book is excellent and really serves well for the basic learner. His second book covered more in depth and included slightly more advanced topics like Riesz-Thorin interpolation theorem, though if I recall correctly this is "covered" in Folland as well. Similarly you may read Elias Stein's Real Analysis in his analysis series.

I have read Halmos's measure theory book, and my impression is this book is now out of date. I mean the terminologies are not used nowadays in research anymore. Also his book is very "algebraic" as it has few graphs. I think this is a draw back for a first time learner.

• It's worth noting that Folland and Stein have different approaches to presenting the material. Folland likes abstraction and begins with abstract measure spaces, omitting the discussion of $L^p$ and applications until Ch. 6. Stein likes concreteness and begins with constructing Lebesgue measure, then goes on to a complete discussion of integration and $L^p$. Abstract measure spaces are put aside until Ch. 6. Also, Stein likes to walk you through problems with a number of hints, whereas with Folland you're mostly on your own. I prefer Stein's approach, but I think Folland is a good resource. Oct 28, 2013 at 23:43

T. Tao, Analysis II covers the topic that you need. But for the measure theory, I think "Paul R. Halmos,measure theory" is good.

• It depends. The book by Halmos, while undoubtedly "good", is hardly the right book to study for a basic course in measure theory. Actually I think that most analysts can safely ignore the most of it. Oct 28, 2013 at 22:38