Homework Assignments:

• Assignment 1
• Assignment 2: part 1, part 2
• Assignment 3: part 1, part 2
• Assignment 4
• Assignment 5: part 1, part 2
• Assignment 6: part 1, part 2

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.

Time and Place:

Lectures: Mo 9:45-11:00, Tu 11:15-12:30 in East Hall 8

Recommended Textbooks:

[1] 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.

[2] D.L. Cohn, Measure theory. Second edition, 2013

A detailed textbook on measure theory and integrals. A lot of material overlaps with [1]. Contains some advanced topics.

[3] G.B. Folland, A Guide to Advanced Real Analysis, 2009

A brief textbook that provides a broad overview of the subject. Essential definitions, major theorems, and key ideas of proofs are included, while technical details are omitted. Very useful as a quick introduction to the subject.

Grading:

The grade is computed as an averaged percent score with the following weights:

• Homework: 20%
• Midterm: 30%
• Final Exam: 50%

Class Schedule

Mo, 05/02/2018: Introduction. Existence of Lebesgue non-measurable sets.

Tu, 06/02/2018: Short discussion of the axiom of choice. Algebras, σ-algebras, measures and their properties. Practice exercises

Mo, 12/02/2018: Further properties of measures; completion of a measure; Some lecture notes

Tu, 13/02/2018: Borel σ-algebra of the real line; premeasure; extension of a premeasure to a measure.

Mo, 19/02/2018: HW 1 is due; Borel measures on the real line. Lebesgue-Stieltjes measures.

Tu, 20/02/2018: Lebesgue measure on the real line; Cantor set, Cantor function

Mo, 26/02/2018: Existence of Lebesgue measurable sets that are not Borel. Outer measures.

Tu, 27/02/2018: Caratheodory's Theorem; extension of a premeasure: proof; Some lecture notes

Mo, 05/03/2018: HW 2 is due; Structure of Lebesgue measurable sets; n-dimensional Lebesgue measure;

Tu, 06/03/2018: Measurable functions and their properties.

Mo, 12/03/2018: Pointwise convergence of measurable functions. The Simple Approximation Lemma and Theorem;

Tu, 13/03/2018: Egoroff's Theorem, Lusin's Theorem. Lebesgue integral of bounded functions on domains of finite measure; Some lecture notes

Mo, 19/03/2018: HW 3 is due; The Bounded Convergence theorem. Lebesgue integral of nonnegative measurable functions;

Tu, 20/03/2018: Fatou's Lemma; the Monotone Convergence Theorem;

---SPRING BREAK---

Mo, 02/04/2018: ---No class---

Tu, 03/04/2018: Midterm; Topics for the Midterm

Mo, 09/04/2018: The General Lebesgue Integral.

Tu, 10/04/2018: The Lebesgue Dominated Convergence Theorem, the (General) Vitali Convergence Theorem.

Mo, 16/04/2018: HW 4 is due; Convergence in measure

Tu, 17/04/2018: Monotone functions. Lebesgue's Theorem

Mo, 23/04/2018: The Vitali Covering Lemma

Tu, 24/04/2018: Functions of bounded variation

Mo, 30/04/2018: HW 5 is due; Absolutely continuous functions

Tu, 01/05/2018: ---No Class, Labor Day---

Mo, 07/05/2018: Indefinite integral. The Fundamental Theorem of Calculus for absolutely continuous functions. Lebegue decomposition of a function of bounded variation.

Tu, 08/05/2018: Normed linear spaces. Examples. Lp-spaces.

Mo, 14/05/2018: HW 6 is due; Hölder's inequality, Minkowski's inequality and corollaries from them.

Tu, 15/05/2018: Topics that were not covered in this course. Concluding remarks.

Final Exam: Fr, 25/05/2018, 16:00 - 18:30 (Research I Lecture Hall) Topics for the Final Exam