Lectures log

First week

  • Monday: Introduction to the main topics of the course.
  • Tuesday: Some naïve set theoretical topics (which included proving that Q is countable and that R is uncountable). The real numbers system, including the density of Q in R. Review of basic topological concepts on R: convergent sequences, monotone sequences, open and closed sets.

This first week we have very roughly reviewed the content of sections 1.1, 1.2, 1.3, 2.1, 2.2 and 2.3. You are responsible for thoroughly reading these sections from the textbook.

Second week

  • Monday: We reviewed a few more topological concepts (leftover from the first week), namely: compact sets and Heine-Borel theorem. We then defined and studied the Cantor set and the devil's staircase function. Here are the slides on these topics.
  • Tuesday: The Jordan measure and its connection with the Riemann integral. I am following Tao's book (check the course information) and plan to cover roughly section 1.1. Prologue: The problem of measure. So far we have discussed the elementary measure, the definition and some basic facts about the Jordan measure.

Third week

  • Monday: We finished talking about the Jordan measure and began discussing its connection with the Riemann integral, following section 1.1. in Tao's book. Here are the slides shwon in class on Jordan measure.
  • Tuesday: The Darboux integral, its equivalence with the Riemann integral and their basic properties (finishing section 1.1. in Tao's book). Continuituity (everywhere) implies Riemann integrability.

Fourth week

  • Monday: We wrapped up our discussion of the Riemann-Darboux integral: more on continuity and integrability (the goal is to show that if a bounded function is continuous except for a set of Jordan measure zero, then it is Riemann integrable); convergence (pointwise and uniform) of functions and Riemann integrability. We reviewed the definition of the Jordan measure and discussed its shortcomings.
  • Tuesday: We defined the Lebesgue outer measure, gave several examples, learned and played with the \(\frac{\varepsilon}{2^n}\) trick. Then we defined the concept of Lebesgue measurable sets (via Littlewood's first principle, following Tao's book).

Fifth week

  • Monday: We studied the properties of the Lebesgue outer measure (section 1.2.1 in Tao's book).
  • Tuesday: We finished up section 1.2.1 and continued on with the properties of Lebesgue measurable sets (section 1.2.2 in Tao's book).

Sixth week

  • Monday: Lebesgue measurability (Section 1.2.2 in Tao's book).
  • Tuesday: More on Lebesgue measurability.

Seventh week

  • Monday: Carathéodory criterion for measurability, uniqueness of Lebesgue measure, plus a preview of the Lebesgue integral.
  • Tuesday: An example of a set which is not Lebesgue measurable (section 1.2.3 in Tao's book), then began the study of the Lebesgue integral with the integral of unsigned simple functions (section 1.3.1 in Tao's book).

Eighth week

  • Monday: Basic properties of the integral of simple functions, measurable functions (section 1.3.2).
  • Tuesday: Unsigned Lebesgue integral and absolute integrability (sections 1.3.3 and 1.3.4).

Ninth week

  • Monday and Tuesday: continuation of last week - the unsigned Lebesgue integral, its basic properties, including: horizontal and vertical truncation, linearity, area under the graph, uniqueness, compatibility with the Riemann integral.

Tenth week

  • Monday and Tuesday: Signed measurable functions, absolutely integrable functions, the normed space \(L^1 (\mathbb{R}^d)\), Markov's inequality and its corollaries, approximation of \(L^1\) functions, Lusin's theorem.

Eleventh week

  • Monday and Tuesday: We covered Littlewood's three principle (proving Lusin's theorem and describing Egorov's). We then started the study of abstract measure spaces. So far we have managed to learn about \(\sigma\)-algebras, (countably additive) measures and the concept of measurable functions. Next up: finish discussing measurable functions, define the integral …

Twelfth week

  • Monday and Tuesday: Measurable functions and their basic properties, the integral of a measurable function on an abstract measure space. The convergence theorems (monotone convergence and Tonelli so far).

Thirteenth week

  • Monday and Tuesday: The convergence theorems and their proofs: the monotone convergence theorem, Borel-Cantelli's lemma, Fatou's lemma, the dominated convergence theorem. The definitions of different modes of convergence: pointwise, uniform, pointwise a.e., in \(L^\infty\)-norm, almost uniform, in \(L^1\)-norm and in measure. Some of the basic relationships between them, including the case of a finite measure space. We then began discussing the differentiation theorems: a quick review of calculus, followed by the formulation of the differentiation theorem for continuous functions with compact support.

Fourteenth week

  • Monday: The proof of the Lebesgue differentiation theorem. We did this by using the density of the compactly supported continuous functions in \(L^1 (\textbf{R})\) and the Hardy-Littlewood maximal inequality (for absolutely integrable functions on R). We then presented the rising sun lemma and used it to prove the Hardy-Littlewood maximal inequality.
  • Tuesday: Absolutely continuous functions and the fundamental theorem of calculus II (for Lebesgue integrals). Abstract construction of measures (Carathéodory's extension theorem and Kolmogorov's extension theorem). Product measures.
2015-11-16, silviusk