August 23: Covered most of Chapters 1 and 2 in the book except the material on cardinality beyond countability (from bottom of page 9 and out chapter 1). I did, however, prove that the real numbers are not countable (Theorem 2.8).
August 26: Covered most of Chapter 3 (mentioned, but did not prove Theorem 3.7). In Chapter 4 I defined measures and proved Proposition 4.3. We continue with Theorem 4.4 next time,
August 30: I started by proving the most impotant part of theorem 4.4 (that a measure satisfies conditions (iii) and (iii')), and only mentioned the opposite implications (useful, but not fundamental). I completed section 4 by giving examples of measures, and then continued with chapter 5, covering 5.1-5.3 and the first half of the proof of 5.4 (the part showing that D_D is a Dynkin system). On Thursday we only do problems, and the lectures continue on Monday.
September 6: I continued with the proof Theorem 5.4 (even did it twice as it seemed hard to understand) and then turned to Theorem 5.7. I skipped Theorem 5,8 as we shall return to the construction of the Lebesgue measure later, and instead turned to Chapter 6 where I defined semirings, stated Caratheodory's Extension Theorem, and introduced the outer measure. We shall continue with the proof of Caratheodory's theorem next Monday (recall that there is no class on September 9th).
September 13: I continued the proof of Caratheodory's Extension Theorem, but did not finish. On Thursday I start on what the book calls "Step 3".
September 16: I completed the proof of Caratheodory's theorem.
September 20: I first sketched the construction of Lebesgue measure using the Caratheodory Extension Theorem. Then I turned to chapter 7 and proved the main results there. Finally, I started chapter 8 by defining measurable functions and proving Lemma 8.1.
September 27: Finished Chapter 8.
September 30: Covered Chapter 9 up to (but not including) Beppo Levi's Theorem (9.6).
October 4: Completed Chapter 9 and covered Chapter 10 up to an including Theorem 10.4
October 7: I continued with Chapter 10, putting in a proof of Lemma 10.8. Next time I shall (probably) cover Corollary 10.14 and then start Chapter 11.
October 11: Proved Theorem 10.14 and covered Chapter 11 up to and including Theorem 11.5.
October 18: Talked briefly about the relationship between Riemann and Lebesgue integration, and formulated (but did not prove) Theorem 11.8. I skipped the section on "Examples" at the end of Chapter 11, but covered chapter 12 up to and including Minkowski's inequality.
October 21: Proved Riesz-Fischer's Theorem (12.7) and its corollary 12.8
October 25: Discussed the relationship between a.e.-convergence and L^p-convergence a little more thoroughly than the book by exhibiting examples showing that neither implies the other. Covered 12.9 through 12.13. Next time I prove Jensen's inequality, and then we start chapter 13.
November 1: I first proved Jensen's inequality and then covered chapter 13 up to an includíng the Uniqueness Theorem for Product Measures (Theorem 13.4).
November 4: Proved the existence theorem for product measures. To make the proof easier to understand, I first proved the theorem for finite measures and then sketched the extension to the sigma-finite case.
November 8: Proved Tonelli's and Fubini's theorems plus 13.10 and 13.11. I may skip (the proofs of) 13.11-13.13.