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Final Exam

Math 420/507: Notes

Slides from the classes   (section references to [Folland], unless otherwise noted)
  • Class 1 (Sep 5): Introduction to the course

  • Part A: Lebesgue Measure on R, and Abstract Measure Theory (1.1-1.5)

    • Class 2 (Sep 7): Measures and sigma-algebras (1.2,1.3)
    • Class 3 (Sep 10): Properties of measures (1.3)
    • Class 4 (Sep 12): Toward Lebesgue measure: premeasures (1.5)
    • Class 5 (Sep 14): Toward Lebesgue measure: outer measures (1.4, 1.5)
    • Class 6 (Sep 17): Toward Lebesgue measure: Caratheodory's theorem (1.4)
    • Class 7 (Sep 19): Lebesgue measure on R (1.5)
    • Class 8 (Sep 21): Regularity of Lebesgue measure; Cantor sets (1.5)

  • Part B: Integration and Convergence Theorems (2.1-2.3)

    • Class 9 (Sep 24): Measurable functions (2.1)
    • Class 10 (Sep 26): Integration of simple and non-negative functions (2.2)
    • Class 11 (Oct 1): Simple function approximation; monotone convergence (2.2)
    • Class 12 (Oct 3): A.e. convergence; Fatou's lemma (2.2)
    • Class 13 (Oct. 5): Integrating complex functions; dominated convergence (2.3)

  • Part C: Convergence and Approximation of Functions (2.4)

  • Part D: Product Measures and Integration on R^n (2.5-2.6)

    • Class 16 (Oct. 15): Product measures (2.5)
    • Class 17 (Oct. 17): Integration on product spaces (2.5)
    • Class 18 (Oct. 19): Integration on R^n (2.6)

  • Part E: Differentiation of Measures (3.1-3.3)

    • Class 19 (Oct. 22): Signed measures (3.1)
    • Class 20 (Oct. 24): Hahn and Jordan decompositions (3.1)
    • Class 21 (Oct. 26): Absolute continuity of measures (3.2)
    • Class 22 (Oct. 29): The Radon-Nikodym derivative (3.2,3.3)

  • Part F: Differentiation on R^n (3.4)

    • Class 23 (Oct. 31): Differentiation on R^n (3.4)
    • Class 24 (Nov. 2): The maximal theorem (3.4)
    • Class 25 (Nov. 5): Lebesgue differentiation theorem (3.4)
    • Class 26 (Nov. 7): Differentiation of Borel measures (3.4)

  • Part G: Differentiation on R (3.5)

    • Class 27 (Nov. 9): Differentiation of increasing functions (3.5)
    • Class 28 (Nov. 14): BV functions (3.5)
    • Class 29 (Nov. 16): Complex Borel measures on R (3.5)
    • Class 30 (Nov. 19): Absolute continuity (3.5)
    • Class 31 (Nov. 21): Fudamental theorem of calculus (3.5)

  • Part H: Introduction to L^p spaces (6.1)

    • Class 32 (Nov. 23): Holder's inequality (6.1)
    • Class 33 (Nov. 26): Minkowski's inequality; L-infinity (6.1)

  • Part I: Review


    Some supplementary notes (courtesy of Joel Feldman)