Information for the final exam

Homework Assignments:

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

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.

Class Schedule

Th, 07/02/2019: Introduction. Informal motivation for Lebesgue measure and Lebesgue integral. Existence of Lebesgue non-measurable sets. (Section 2.1)

Fr, 08/02/2019: Discussion of the Axiom of Choice. Construction of Lebesgue outer measure. (Section 2.2)

Th, 14/02/2019: The sigma-algebra of Lebesgue measurable sets. Caratheodory's Theorem . (Section 2.3)

Fr, 15/02/2019: HW 1 is due; Abstract definition of measure spaces. Continuity of measures. Borel-Cantelli Lemma. Countable additivity of Lebesgue measure (continuation of Caratheodory's Theorem). (Section 2.5)

Th, 21/02/2019: Sigma-algebra, generated by a set. Borel sigma-algebra. Borel sets are Lebesgue measurable. (Section 2.3)

Fr, 22/02/2019: HW 2 is due; Outer and inner approximation of Lebesgue measure. Cantor set. (Section 2.4, 2.7)

Th, 28/02/2019: Cantor function. Existence of Lebesgue measurable sets that are not Borel. (Section 2.7)

Fr, 01/03/2019: HW 3 is due; Measurable functions and their properties. (Section 3.1)

Th, 07/03/2019: More properties of measurable functions. Pointwise (a. e.) convergence of measurable functions. (Sections 3.1, 3.2)

Fr, 08/03/2019: HW 4 is due; The Simple Approximation Lemma and Theorem. Egoroff's Theorem, Lusin's Theorem. (Sections 3.2, 3.3)

Th, 14/03/2019: Lusin's Theorem. Definition of the Lebesgue Integral. (Sections 3.3, 4.2)

Fr, 15/03/2019: HW 5 is due; Definition of the Lebesgue Integral. Its basic properties. (Sections 4.2-4.4)

Th, 21/03/2019: Basic properties of the Lebesgue Integral (continued). (Sections 4.2-4.4)

Fr, 22/03/2019: HW 6 is due; The Bounded Convergence theorem. Fatou's Lemma; the Monotone Convergence Theorem. The Lebesgue Dominated Convergence Theorem. (Sections 4.2-4.4)

Th, 28/03/2019: Review. Countable additivity and continuity of integration. (Section 4.5)

Fr, 29/03/2019: Midterm; Information for the midterm

Th, 04/04/2019: Midterm review. Absolute continuity of the integral. The Vitali Convergence Theorem. (Section 4.6)

Fr, 05/04/2019: HW 7 is due; The General Vitali Convergence Theorem. Convergence in measure. (Sections 5.1-5.2)

Th, 11/04/2019: Lebesgue's Theorem. Vitali Covering Lemma. (Sections 6.1-6.2)

Fr, 12/04/2019: HW 8 is due; Lebesgue's Theorem. Vitali Covering Lemma. (Sections 6.1-6.2)

---SPRING BREAK---

Th, 25/04/2019: Functions of bounded variation. (Section 6.3)

Fr, 26/04/2019: HW 9 is due; Absolutely continuous functions. (Section 6.4)

Th, 02/05/2019: The Fundamental Theorem of Calculus for absolutely continuous functions. Lebegue decomposition of a function of bounded variation. (Section 6.5)

Fr, 03/05/2019: HW 10 is due; Normed linear spaces. Examples. Lp-spaces. (Section 7.1)

Th, 09/05/2019: Hölder's inequality, Minkowski's inequality and corollaries from them. (Section 7.2)

Fr, 10/05/2019: HW 11 is due; Completeness of Lp-spaces. (Section 7.3)

Th, 16/05/2019: Separability of L-p spaces. Linear functionals. The Riesz Representation Theorem for Lp-spaces (without proof). (Sections 7.4, 8.1)

Fr, 17/05/2019: HW 12 is due; Review.

Final Exam: TBA