Graduate Students
Graduate Programs     
Graduate Home Ph D Programs Masters Degree Ph D Program Requirements Course Descriptions Current Courses Admissions Current Graduate Students Graduate Student Seminar SIAM Chapter Seminar Recent Graduates Incoming Students

Apply Now
Graduate Courses 21-720
Measure and Integration
12 units

Prerequisites: knowledge of (undergraduate) real analysis


  • Outer measure, measure, $ \sigma$-algebras, Dynkin $ \pi-\lambda$ Theorem, Carathéodory's Extension Theorem
  • Borel measures, Lebesgue measure
  • Measurable functions, Lebesgue integral (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem)
  • Modes of Convergence (Egoroff's Theorem, Lusin's Theorem)
  • Product Measures (Fubini-Tonelli Theorems), n-dimensional Lebesgue integral
  • Signed Measures (Hahn Decomposition, Jordan Decomposition, Radon-Nikodym Theorem, change of variables)
  • Differentiation (Lebesgue Differentiation Theorem)
  • $ L^p$ Spaces (Hölder's inequality, Minkowskii's inequality, completeness, equiintegrability (uniform integrability), Vitali's convergence theorem)