MA2224 : Lebesgue Integral

Hilary term, 2013

Lecturer: Masha Vlasenko

Lebesgue integral extends Riemann-Darboux itegral in a sense that one can integrate a broader domain of functions. This course is intended to introduce students to the theory of measure and integration. We will construct Lebesgue integral on the real line. The only prerequisites are basic knowlegde of analysis and ability to understand and write proofs.

The course will be accompanied by weekly tutorials. Tutorials take place every Friday at 10 am in SNIAM lecture hall.


Literature
[B] H.S.Bear, A Primer on Lebesgue Integration
[RT] Richard Timoney's lecture notes


Grading
3 hour examination in May (joint with metric spaces). The final mark will be 80% of the exam mark + 20% of the continuous assessment mark. The continuous assessment consists of two kinds of work:
In case you've got more then 20% in the continuous assessment by the end of the course, your mark will be cut down to 20%. One person can be a reporter only once.The questions on the final exam will be close to the ones in the assignments, but also there will be some theoretical questions.

Lecture notes and assignments
  1. What is the course about? <pdf>
  2. Countable and uncountable sets, inverse images, characteristic functions and Venn diagrams. <pdf>
  3. Lebesgue outer measure on (0,1).<pdf>
  4. Measurable sets. <pdf> Carathéodory's criterion and set-theoretic operations on measurable sets. <pdf>
  5. Sigma-algebra of measurable sets. <pdf>
  6. Some examples: a non-measurable set, an uncountable set of measure zero. Borel sets. <pdf>
  7. The Lebesgue integral for bounded functions. <pdf> Revision. <pdf>
  8. Properties of the Lebesgue integral. <pdf> Proof of the Bounded Convergence Theorem. <pdf>
  9. The integral of unbounded functions, the non-negative case. <pdf>
  10. The integral of unbounded functions, the general case. The Lebesgue Dominated Convergence Theorem. <pdf>
  11. An overview of further results: differentiation and integration, absolutely continuous and singular measures, the Lebesgue Decomposition Theorem. <pdf>