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!