MATH 232 SAT2 Real Analysis


Schedule: 09:00 AM - 12:00 NN, Sat, IB 103
Credit: 3 units
Prerequisite: COI (Consent of Instructor)
Consultation: 02:30 PM - 05:00 PM Mon, Tue, Wed, Thu, and Fri, or by Appointment (College of Science Dean's Office).
Course Outline (Link to Course Syllabus):
  1. Collection of Sets
    Rings, Algebras, Borel σ-algebras, Semirings, Dynkin Systems, Monotone Classes
  2. Measure Theory: Construction, Completion, and Examples
    Contents, Premeasures, Measures, From Premeasures to Measures, Lebesgue Measure, Outer and Inner Measures, Complete Measure Spaces, Borel and Lebesgue-Stieltjes Spaces, Regular Borel Measures on Metric Spaces, Image Measures, Properties of Lebesgue Measure
  3. Lebesgue Integration Theory and Convergence Theorems
    Measurable Real-Valued Functions, Lebesgue Integral of Simple Functions, Lebesgue Integral of Nonnegative Measurable Functions, Lebesgue Integrable Functions, Almost Everywhere Properties, Convergence Theorems, Riemann and Lebesgue Integrals
  4. Product Measures, Iterated Integrals, and Change of Variables Formula
    Initial, Final, and Product σ-algebras, Product Measures, Fubini–Tonelli Theorem, Integration Through Image Measures, Change of Variables for Integration
  5. Boundary Integration, Lebesgue Spaces, and Integral Inequalities
    Boundary Integrals, Gauss Divergence Theorem, Reynolds Transport Theorem, Lebesgue Spaces, Integral Inequalities: Jensen, Hölder, Minkowski, Lyapunov
  6. Decomposition of Measures
    Signed and Complex Measures, Radon–Nikodym Theorem, Lebesgue and Hahn Decompositions
Exercises (deadline of submission in parenthesis):