On this page you will find brief reports from the lectures.
Monday, January 13th: I started by giving an outline of the basic ideas of the course, and then began lecturing from Chapter 1 of the notes. I got to De Morgan's laws, but didn't have time to prove them.
Friday, January 17th: I started by proving one of De Morgan's laws, and continued with families of sets, a notion that will appear over and over again throughout the course. Next I turned to functions, definining injectivity, surjectivity and bijectivity. I proved the results about forward and inverse images of sets (also important for this course), and briefly discussed Cartesian products. I then introduced relations, and showed that the relation defined by a partition is reflexive, symmetric and transitive. Next time I shall start with Proposition 1.5.2. I hope to finish Chapter 1 before or just after the break and then begin Chapter 2.
Monday, January 20th: I finished Chapter 1 and started Chapter where I defined metric spaces and gave some examples. Next time I shall prove the Inverse Triangle Inequality and then start Chapter 2.
Friday, January 24th: I finished Section 2.1 by giving a few more examples of metric spaces (the discrete metric and the Manhattan metric) and proving the Inverse Triangle Inequality, and then covered Section 2.2 more or less as in the notes. At the end, I just had time to discuss interior, exterior and boundary points from Section 2.3.
Monday, January 27th: I covered Section 2.3, following the book quite slowly, and just had time to introduce the notion of a Cauchy sequence from Section 2.4.
Friday, January 31st: I completed Section 2.4 and started Section 2.4. The last thing I proved was that a compact set is always closed and bounded.
Monday, February 3rd: Completed Section 2.5.
Friday, February 7th: I finished Chapter 2. In Section 2.7, I only sketched the main ideas and did not give the details of the proofs.
Monday, February 10th. Covered Sections 3.1 and 3.2 and tried to motivate a little bit for Section 3.3.
Friday, February 14th: I went through Section 3.3 and Section 3.4 up to an including the proof of Lemma 3.4.1.
Monday, February 17th: Completed Section 3.4 and covered 3.5 up to and including 3.5.5.
Wednesday, February 19th: Finished Section 3.5 and sketched the underlying ideas (using Euler's method and a compactness argument to find a sequence converging to a solution) of section 3.6. We shall start the serious work next time.
Monday, February 24th; Completed Section 3.6.
Friday, February 28th: As we had to cancel the problem session this morning, I set aside some time to help with the mandatory assignment, and for that reason we only covered Section 3.7.
Monday, March 3rd: Lectured on Section 3.8. I only had time to sketch the proofs of Lemma 3.8.8 and Proposition 3.8.9, and no knowledge of these will be required for the exam.
Friday, March 7th: Started Chapter 4 by covering 4.1 and most of 4.2.
Monday, March 10th: Finished Section 4.2 and covered Sections 4.3 and 4.4. In Section 4.3, I skipped the proof of Corollary 4.3.2 as I wanted to be sure I had enough time for the proof of Abel's Theorem in Section 4.4, but the result is still part of the syllabus.
Friday, March 14th: Covedered Section 4.5.
Wednesday, March 19th: Covered Section 4.6 up to an including Proposition 4.6.7
Friday, March 21st: Finished Chapter 4.
Monday, March 31st: Started Chapter 5 and covered Section 5.1 up to the formulation of Proposition 5.1.5.
Friday, April 4th: Completeded Section 5.1 and went through Section 5.2 up to and including the proof of Theorem 5.2.5. I left some of the details of the proofs to the audience.
Monday, April 7th: I completed Section 5.2 and covered Section 5.3 up to and including Proposition 5.3.6.
Wednesday, April 9th: Completed Section 5.3 and started 5.4. Next time I shall start with Corollary 5.4.4.
Friday, April 25th: Finished Section 5.4 and continued with Section 5.5 up to and including Corollary 5.5.4.
Monday, April 28th: Finished Section 5.5 and continued with Section 5.6 up to Lebesgue's Dominated Convergence Theorem.
Friday, May 2nd: I started by proving Lebesgues Dominated Convergence Theorem and then skipped the rest of Section 5.6 (the last result there may be regarded as examples of how to use the Dominated Convergence Theorem). I then continued with Section 5.7 where I covered the L1-theory. I introduced the general Lp-spaces and explained that L2 is an inner product space, but I did not cover the material from Lemma 5.7.2 onward. The parts I skipped (Prop 5.6.6 and 5.7.2-5.7.5) will not be part of the curriculum for the exam. Monday I shall start Chapter 6.
Monday, May 5th: Started Chapter 6 where I covered 6.1 and 6.2 up to and including Lemma 6.2.4. I made a mistake in the drawing I made in the proof of Proposition 6.2.3, but the correct drawing should not be hard to make.
Friday, May 9th: Finished Section 6.2, proved Caratheodory's Theoren for algebras (Theorem 6.3.3) an introduced semi-algebras.
Wednesday, May 14th: Completed Section 6.3 and proved Propositions 6.4.2 and 6.4.4. To save time, I shall only sketch the argument for the existence of a nonmeasurable set (Proposition 6.4.6 and Example 1 in Section 6.4). As earlier mentioned, I shall also skip Section 6.5 (as well as 6.6) and proceed to Sections 6.7 and 6.8.
Friday, May16th: I finished Section 6.4 and covered all of 6.7.
Monday, May 19th: Completed the curriculum. Skipped the proof of the Monotone Class Theorem.