Real Analysis I

Riemann integration breaks the moment you try to integrate the indicator of the rationals, and this course rebuilds integration from scratch on the firmer ground of measure theory — sigma-algebras, Lebesgue measure, and the convergence theorems that finally let you swap limits and integrals without apologizing. You'll spend the semester proving things on weekly problem sets and two written exams, working through Cohn or Folland line by line until results like Radon-Nikodym and Fubini feel inevitable rather than magical. It's the standard first-year graduate analysis course, the prerequisite for functional analysis, probability theory, and PDE, and the language most of modern mathematics is written in.