My graduate analysis course (I'm in an applied math program) is using the Folland book, which I find quite dense but also pretty clear. I can follow his proofs in the sense that I can read through each line and see mathematically why it's true, but I lack the intuition to see the bigger picture. I'm trying to work on this by going through every proof and seeing if I can understand why the conditions are necessary and find counterexamples.
But one thing I've noticed among my cohort is that the ones who don't seem to be struggling have seen many these problems before. I'll ask them how on earth they knew to do a particular trick on the homework and often the answer is "I've actually seen a variation of this problem before." I've never taken measure theory and I was never the best at analysis, so I feel like I'm behind on two fronts. I'm looking for some sources of exercises in measure theory that I can attempt on my own but check the solutions if I need to. I feel lost on all the homework problems (which are mostly assigned out of Folland's book), and never have any clue how to start any of them. Don't get me wrong, I'm enjoying the class and learning a lot, but it also feels like I've figured out a grand total of maybe two homework problems on my own since the semester began...
I have Bartle's "Elements of Integration and Lebesgue Measure" which is written clearly and helpful if I don't understand something in Folland, but also doesn't have solutions to the exercises as far as I know.