Announcement: As it was agreed upon during the class on 17/03/2020, the following changes to the syllabus have been made:

• The Midterm is postponed until further notice. Instead we will continue having weekly homework. The corresponding changes to the class schedule can be seen below.
• If the Midterm is going to happen at some point during one of the classes, then the homework assignments due on the date of the Midterm and the week after will be canceled.
• If we have to continue with online classes until the end of the semester, then there is a high chance that the Midterm will be canceled, and its weight will be distributed between Homework and the Final Exam.
• From 17/03/2020 and until further notice, the classes will be meeting online on MS Teams platform during regular time. Let me know if you have troubles to connect to MS Teams.
• I will keep updating this website by posting homework and topics of the lectures. Some additional materials might be posted on Teams only.

Announcement: There will be an additional lecture on Thursday, February 20th to make up for the canceled lecture on Tuesday, February 18th.

Time: Thursday, February 20th, 9:45-11:00

Location: West Hall 8

Homework Assignments:

• Assignment 1
• Assignment 2
• Assignment 3
• Assignment 4
• Assignment 5
• Assignment 6
• Assignment 7
• Assignment 8
• Assignment 9
• Assignment 10
• Assignment 11
• Assignment 12
• Assignment 13

Summary:

This course continues Analysis II by introducing Lebesgue integration as well as elements of Functional Analysis and Fourier methods in the concrete setting of Lebesgue spaces. The topics include: measurable spaces, Lebesgue measure, Lebesgue integral and its comparison to the Riemann integral, L_p spaces and their properties, functions of bounded variation, absolute continuity, Hilbert spaces, orthonormal systems and Fourier coefficients, Fourier transform.

Main textbook:

H. L. Royden and P. M Fitzpatrick, Real analysis, fourth ed., 2010

Additional (optional) textbook:

G.B. Folland, Real Analysis. Modern Techniques and Their Applications. Second edition, 1999

An advanced textbook that includes a detailed exposition of the theory of measure and integration as well as some elements of general topology, functional analysis, Fourier analysis and some other topics.

Actual Class Schedule (updated regularly)

We are planning to follow the course schedule from last year, however, small deviations are possible.

Tu, 04/02/2020: Introduction. Informal motivation for Lebesgue measure and Lebesgue integral. Existence of Lebesgue non-measurable sets. Discussion of the Axiom of Choice. Construction of the Lebesgue outer measure. (Sections 2.1-2.2)

Tu, 11/02/2020: HW 1 is due; The sigma-algebra of Lebesgue measurable sets. Countable additivity of Lebesgue measure. Caratheodory's Theorem. Abstract definition of measure spaces. (Sections 2.3, 2.5)

Tu, 18/02/2020: HW 2 is due; Continuity of measures. Borel-Cantelli Lemma. Sigma-algebra, generated by a set. Borel sigma-algebra. Borel sets are Lebesgue measurable. Outer and inner approximation of Lebesgue measure. (Sections 2.3-2.5)

Tu, 25/02/2020: HW 3 is due; Cantor set, Cantor function. Existence of Lebesgue measurable sets that are not Borel. (Section 2.7) Measurable functions and their properties. (Section 3.1)

Tu, 03/03/2020: HW 4 is due; More properties of measurable functions. Pointwise (a. e.) convergence of measurable functions. The Simple Approximation Lemma and Theorem. (Sections 3.1, 3.2)

Tu, 10/03/2020: HW 5 is due; Egoroff's Theorem, Lusin's Theorem , definition of the Lebesgue Integral. (Sections 3.2, 3.3, 4.2, 4.3)

Tu, 17/03/2020: HW 6 is due; Definition of the Lebesgue Integral. Its basic properties. The Bounded Convergence theorem. Fatou's Lemma. (Sections 4.2-4.4)

Tu, 24/03/2020: Midterm -- postponed. We will have a regular (online) lecture instead. HW 7 is due; Monotone Convergence Theorem. The Lebesgue Dominated Convergence Theorem. Countable additivity and continuity of integration. Absolute continuity of the integral. The Vitali Convergence Theorem. The General Vitali Convergence Theorem. (Sections 4.3-4.6, 5.1)

Tu, 31/03/2020: HW 8 is due; Convergence in measure. Lebesgue's Theorem. Vitali Covering Lemma. (Sections 5.2, 6.1-6.2)

---SPRING BREAK---

Tu, 14/04/2020: HW 7 is due HW 9 is due; Vitali Covering Lemma. Functions of bounded variation. (Sections 6.2-6.3)

Tu, 21/04/2020: HW 8 is due HW 10 is due; Absolutely continuous functions. The Fundamental Theorem of Calculus for absolutely continuous functions. Lebesgue decomposition of a function of bounded variation. (Sections 6.4-6.5)

Tu, 28/04/2020: HW 9 is due HW 11 is due; Normed linear spaces. Examples. Lp-spaces. Hölder's inequality, Minkowski's inequality and corollaries from them. (Sections 7.1-7.2)

Tu, 05/05/2020: HW 10 is due HW 12 is due; Completeness and separability of Lp-spaces. (Sections 7.3-7.4)

Tu, 12/05/2020: HW 11 is due HW 13 is due;

Final Exam: TBA