Analysis III/IV

Michaelmas 2017

Time and place:   Lectures: Tue 10:00, Th 15:00, CG85
Problems classes:   Fri 17:00, PH8, Weeks 4,6,8,10
Instructor: Pavel Tumarkin
e-mail: pavel dot tumarkin at durham dot ac dot uk
Office: CM110; Phone: 334-3085
Office hours: Tue 11:00 -- 12:00 and by appointment

Textbooks:

The content of the course can also be found in any standard textbook on Real Analysis.

Preliminary course content (subject to change): Real numbers, countable and uncountable sets, open and closed sets, continuos and uniformly continuos functions, Lebesgue measure, measurable sets, Axiom of Choice and nonmeasurable sets, Cantor set, Lebesgue measurable functions, Lebesgue integral

Schedule:

  • Week 1: Real numbers: ordered fields, completeness; inductive sets, integer and rational numbers (R-F, sections 1.1, 1.2)
  • Week 2: Countable and uncountable sets (1.3)
  • Week 3: Open and closed subsets of real numbers (1.4)
  • Week 4: Sequences of real numbers; continuos functions (1.5, 1.6)
  • Week 5: Continuous functions; outer measure (1.6, 2.2)
  • Week 6: Cantor set, Lebesgue measurable sets (2.7, 2.3)
  • Week 7: Outer and inner approximations of measurable sets, countable additivity (2.4, 2.5)
  • Week 8: Non-measurable sets, Cantor - Lebesgue function (2.6, 2.7)
  • Week 9: Measurable functions: definitions and operations (3.1)
  • Week 10: Limits and simple approximations of measurable functions (3.2)

    Outline: Michaelmas term

    4H Reading material: Section 3.3 of Royden -- Fitzpatrick (pp. 64--67), see the link on duo.

    Homeworks: There will be four homework assignments to be handed in on weeks 3, 5, 7, and 9