MATH 232 SF Real Analysis


Schedule: 06:00 PM - 09:00 PM Friday, IB 104
Credit: 3 units
Prerequisite: COI (Consent of Instructor)
Consultation: 10:30 AM - 12:00 NN & 01:00 PM - 02:30 PM Tuesday to Friday, CS Dean's Office, IB Building
Course Outline (Link to Course Syllabus, Link to Lecture Notes):
  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

MATH 132 Q Real Analysis


Schedule: 03:00 PM - 04:30 PM Wednesday & Friday, IB 104
Credit: 3 units
Prerequisite: Math 55 (Elementary Analysis III)
Consultation: 10:30 AM - 12:00 NN & 01:00 PM - 02:30 PM Tuesday to Friday, CS Dean's Office, IB Building
Course Outline (Link to Course Syllabus, Link to Lecture Notes):
  1. Lebesgue Measure
    Lebesgue Outer Measure, σ-Algebras and Borel Sets, Measurable Sets and Lebesgue Measure, Approximation of Measurable Sets, A Non-measurable Set, Measurable Functions, Approximation of Measurable Functions, Littlewood’s Principles
  2. Lebesgue Integration
    The Riemann and Darboux Integrals, Integral of Simple Functions, Integral of Bounded Measurable Functions, Integral of Nonnegative Measurable Functions, The Lebesgue Integral
  3. Differentiation and Integration
    Differentiation of Monotone Functions, Functions of Bounded Variation, Differentiation of an Integral, Absolute Continuity
  4. Lebesgue Spaces
    Convex Functions, Integral Inequalities and Lebesgue Spaces