I have good knowledge of Elementary Real analysis. Now I'd like to study measure theory by myself (self-study). So please give me direction for where to start? Which book is good for starting? I have Principles of Mathematical Analysis by W. Rudin and Measure Theory and Integration by G. de Barra. Which book is rich in examples and exercises? Please suggest to me. Thanks in advance.
I wouldn't recommend Baby Rudin (Principles of Mathematical Analysis) as your first text in measure theory. The book is amazing and has great examples, but is terse, formal, and hard for some students to follow. To an extent this depends on where you are in your education, but when I learned measure theory first year, my prof's lectures made far more sense than Rudin. That said, Rudin is full of wonderful example problems, and if you have both books already you should use both books.
I have discovered Yeh's Real Analysis: Theory Of Measure And Integration recently, and I recommend it warmheartedly! Easy to read I think, especially for self study. It has a problem & proof supplement by the way.
Rudin (not PMA but RCA) is very good on the long term, but hard for a first encounter with measure theory. PMA is not mainly about measure theory IIRW.
I recommend "An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Mathematics)". It's extremely concise and well-written. This is more about Lebesgue integration, covered in the first 112 of 154 pages, than about Fourier Series. But do not be fooled by the low price; this book makes for very good and gentle introduction to the Lebesgue theory, well organized and concise. There are MANY exercises with hints after each section.