This is a very hard course, but I recommend taking this with Charles. He is a very clear instructor. Assignments are challenging but the lectures cover everything in great detail and he's very available during office hours.

Sigma-algebras, outer measures, measures, Lebesgue integration, various integral convergence theorems, Radon-Nikodym theory. Decomposition theorems. Covered some topology stuff and Lusin's theorem. We did not get to L^p spaces

Could not ask for a better professor

Begin the assignments right away and work in groups. Go to office hours each week if you can. Make sure you have at least a vague idea of how to prove major theorems - I was unprepared and luckily found out how to cook up the proofs from scratch on the exam but it was not fun!