While a great book about measure theory and real analysis in $\mathbb R^n$, the only downside is the lack of exercises. Can anyone provide a good book to supplement it with exercises? I plan to use it for self study.
If you look for a generic collection of problems in measure theory and functional analysis, I would highly recommend:
This is a great collection of problems with complete solutions.
However, if you are looking more for a collection of problems in geometric measure they and Sobolev spaces, as it is represented in the book by Evans and Gariepy, it is difficult to find one. However, the book
W. P. Ziemer, Weakly Differentiable Functions. Sobolev spaces and functions of bounded variation.. Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. (MathSciNet review).
presents material similar to that in the book by Evans and Gariepy and it includes many exercises. Unfortunately the exercises do not have solutions.
Concerning Geometric measure theory and BV functions, I would recommend Functions of Bounded Variation and Free Discontinuity Problems (by Luigi Ambrosio, Nicola Fusco, and Diego Pallara), Oxford Mathematical Monographs. The first two-three chapters may be of interest to you and they contain (non-trivial) exercises.
Concerning Sobolev spaces, I would also recommend the well known book by Brezis, Functional Analysis, Springer, at least for exercises concerning the one-dimensional case.