Math 631 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.


Basic multivariable analysis (the equivalent of Math 531 and 532).


  1. Outer measure, measurable sets, sigma-algebras, Borel sets, measurable functions, the Cantor set and function, nonmeasurable sets.
  2. Lebesgue integration, Fatou's Lemma, the Monotone Convergence Theorem, the Lebesgue Dominated Convergence Theorem, convergence in measure.
  3. LΡ spaces, Hoelder and Minkowski inequalities, completeness, dual spaces.
  4. Abstract measure spaces and integration, signed measures, the Hahn Decomposition, the Radon-Nikodym Theorem, the Lebesgue Decomposition
  5. Expectation, variance, and conditional expectation.
  6. Product measures, the Fubini and Tonelli Theorems, Lebesgue measure on real n-space.
  7. Equicontinuous families, the Ascoli-Arzela Theorem.
  8. Hilbert spaces, orthogonal complements, representation of linear functionals, orthonormal bases.
  9. Fourier series and Fourier transform. Parseval’s identity.


  • H.L. Royden, Real Analysis, Chapters 1-7, 11-12.
  • Reed and Simon, Methods of Mathematical Physics I: Functional Analysis, Chapters 1-2.
  • G.B. Folland, Real Analysis, Chapters 0-3, 6.