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!
Best Answer
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.