Physics of Fractal Operators (Institute for Nonlinear Science Series) Book

5/28/2015: So far: A very rigorous text! However, I would refer to other sources (lecture videos, notes, etc.) to grasp a better initial understanding on measure theory itself, and a rather reasonable comparison of Riemann vs. Lebesgue integrals.

Note: MITOpenCourseWare unfortunately doesn't have the lecture videos available, but mathematicalmonk on youtube gives an understandable introduction to measure theory in his Probability Primer's series.

12/8/2015: We went through THE WHOLE BOOK in our measure theory class. I will not comment on how the professor taught the class, but I would have been a lot more pleased if there were more examples and some sort of solution manual/hints for problems in this book. In general, I think having some sort of verification for one's solutions for any problem is essential to her learning process.