Lecture log

First week (Week 34)

Wednesday: Introduction to the main topics of the course. After the break we introduced elementary sets and the elementary measure. We talked about some properties of the elementary measure. (Pages 1 to 8 in the book)
Thursday and Friday: We finished our discussion of the elementary measure and moved on to the Jordan measure. We gave several equivalent characterizations, talked about some of its properties and mentioned some examples. (Pages 7 to 14 in the book)

Second week (Week 35)

Wednesday: We showed that unbounded functions are not Riemann integrable. We talked about piecewise constant functions and their basic properties; we defined the Darboux integral. (Pages 14 to 16 in the book)
Thursday: We related the Riemann integral and the Darboux integral. We talked about the connection between Jordan measure and Riemann integral. (Pages 15 to 17 in the book)

Third week (Week 36)

Wednesday: We finished our discussion of the connection between the Riemann integral and the Jordan outer measure. We introduced the Lebesgue outer measure.
Thursday: Properties of the Lebesgue outer measure. We defined Lebesgue measurable sets and their Lebesgue measure.

Fourth week (Week 37)

Wednesday: Finite additivity of Lebesgue outer measure for separated sets, Lebesgue outer measure of elementary sets.
Thursday: We constructed a bounded open subset of the real line which is not Jordan measurable. Lebesgue outer measure of countable unions of almost disjoint boxes.

Fifth week (Week 38)

Wednesday: Properties of Lebesgue measurable sets and Lebesgue measure.
Thursday: Assuming the axiom of choice, we gave an example of a set which is not Lebesgue measurable. Integral of unsigned simple functions.

Sixth week (Week 39)

This week's lectures were cancelled. Next week it will be assumed that everyone has prepared Section 1.3.1 from the book independently.

Seventh week (Week 40)

This week's lectures were cancelled. Next week it will be assumed that everyone has prepared Section 1.3.2 from the book independently.

Eighth week (Week 41)

Wednesday: The unsigned Lebesgue integrals and its properties.
Thursday: Absolute integrability, the Lebesgue integral and its properties, compatibility with the Riemann integral, triangle inequality. In the end we started talking about Littlewood's three principles.

Ninth week (Week 42)

Wednesday: We finished our discussion of Littlewood's three principles. We started studying abstract measure spaces: sigma-algebras, measures, measurable functions, relationship between Borel measurable sets and Lebesgue measurable sets.
Thursday: Properties of measurable functions, simple functions, integral of unsigned simple functions, completion of a measure space.

Tenth week (Week 43)

Wednesday: We discussed the unsigned integral (on abstract measure spaces) and its properties.
Thursday: We discussed absolutely integrable functions (on abstract measure spaces) and their properties. We started talking about the convergence theorems, but so far we only covered the case of uniformly convergent sequences on spaces of finite measure.

Eleventh week (Week 44)

Wednesday: Convergence theorems: examples that motivate the assumptions in the theorems, monotone convergence theorem, Fatou's lemma, dominated convergence theorem.
Thursday: We introduced several new modes of convergence, talked about implications between them and provided some (counter-) examples. In the end of the lecture we began our discussion of the Lebesgue differentiation theorem.

Twelfth week (Week 45)

Wednesday: Proof of the Lebesgue differentiation theorem, assuming the Hardy-Littlewood maximal inequality to be true. We briefly discussed the situation in higher dimension (without proofs). We talked about the rising sun lemma in preparation for the proof of the Hardy-Littlewood maximal inequality.
Thursday: Proof of the Hardy-Littlewood maximal inequality. We started working towards a more general version of the second fundamental theorem of calculus: monotone functions, functions of bounded variation, absolutely continuous functions.

Thirteenth week (Week 46)

Wednesday: Proof of the second fundamental theorem for absolutely continuous functions.
Thursday: Outer measures, pre-measures, and product measures. Caratheodory extension theorem, Hahn-Kolmogorov theorem.

Fourteenth week (Week 47)