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.

$\begingroup$ Have you given a look to Functions of Bounded Variation and Free Discontinuity Problems (Luigi Ambrosio, Nicola Fusco, and Diego Pallara), Oxford Mathematical Monographs? The first twothree chapters may be of interest to you and they contain (nontrivial) exercises. $\endgroup$ – Y.B. Jan 15 '19 at 12:52

$\begingroup$ Concerning Sobolev spaces, I would also recommend the well known book by Brezis, Functional Analysis, Springer. $\endgroup$ – Y.B. Jan 15 '19 at 12:54

$\begingroup$ I will check out the first one, seems interesting! $\endgroup$ – James Baxter Jan 15 '19 at 12:55

1$\begingroup$ This is a very good question since it is really difficult to find a reasonable collection of problems in elementary geometric measure theory and Sobolev spaces. $\endgroup$ – Piotr Hajlasz Jan 15 '19 at 13:05
If you look for a generic collection of problems in measure theory and functional analysis, I would highly recommend:
A. Torchinsky, Problems in real and functional analysis. Graduate Studies in Mathematics, 166. American Mathematical Society, Providence, RI, 2015. (MathSciNet review).
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. SpringerVerlag, 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.

2$\begingroup$ What a nice book the one by Torchinksy, I did not know it! It is a pity it does not have exercises on GMT. It reminds me somehow of the CostaraPopa (which is btw more on the functional analytic point of view). Thanks in any case for pointing it out! $\endgroup$ – Y.B. Jan 15 '19 at 13:14
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 twothree chapters may be of interest to you and they contain (nontrivial) exercises.
Concerning Sobolev spaces, I would also recommend the well known book by Brezis, Functional Analysis, Springer, at least for exercises concerning the onedimensional case.

$\begingroup$ Brezis does not have exercises in Sobolev spaces except Sobolev spaces in dimension one. $\endgroup$ – Piotr Hajlasz Jan 15 '19 at 13:02

$\begingroup$ @PiotrHajlasz Uh you are right, I thought it had problems also for multid case, but I got confused, you are right. Thanks! $\endgroup$ – Y.B. Jan 15 '19 at 13:06