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](https://books.google.com/books?id=wB-jCgAAQBAJ).* [Graduate Studies in Mathematics, 166](https://bookstore.ams.org/gsm-166/). American Mathematical Society, Providence, RI, 2015.
(<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=JOUR&pg7=ALLF&pg8=ET&review_format=html&s4=Torchinsky&s5=&s6=&s7=&s8=Books&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=3443231"><FONT FACE="Arial">MathSciNet review</FONT></A><FONT FACE="Arial">).


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* Springer 1989

presents material similar to that in the book by Evans and Gariepy and it includes many exercises. Unfortunately the exercises do not have solutions.