Syllabus for Math 103

This syllabus is tentative and will be updated irregularly. The homework page will be updated on the regular basis.

Math 103 will concentrate on Measure Theory and Lebesgue integration with the goal of helping students to prepare for the Analysis certification exam. The exact syllabus will depend on the interests and backgrounds of the students enrolled. We shall start by reviewing Riemann integral and its properties. From there we will cover as much of Chapters 1-7 as time permits. Ideally the students in the class should have had undergraduate classes in abstract analysis.

Lectures	Sections in Text	Brief Description
Day 1:  9/22	1.1-1.3	General discussion, sigma-algebras, measures, completion of measures
Day 2:  9/24	1.4-1.5	Outer measures and briefly inner measures, Borel measure
Day 3:  9/27	Finish 1.5 and 2.1	Lebesgue measure, measurable functions
Day 4:  9/29	2.2 and start 2.3	Integration of nonnegative functions, Monotone Convergence Theorem, start integration of Complex functions, dominated convergence Theorem
Day 5:  10/1	2.3 and 2.4	Lebesgue integral, Egoroff’s Theorem, almost uniform convergence, discuss Lusin’s Theorem if time permits.
Day 6:  10/4	2.5 and start 2.6	Monotone Class Lemma, Fubini-Tonelli Theorem, Lebesgue-measurable sets in R^n
Day 7:  10/6	2.6 and 2.7	Jordan Content, invariance of Lebesgue measure under rotations, integration in polar coordinates (probably briefly)
Day 8:  10/8	3.1 and 3.2	Hahn and Jordan decomposition Theorems, absolutely continuous measures, start Lebesgue-Radon-Nikodym Theorem
Day 9 10/11	3.2 and 3.3	Finish the Lebesgue-Radon-Nikodym Theorem, complex measures, total variation
Day 10:  10/13	3.4	Differentiation on Euclidian spaces, Maximal Theorem, Lebesgue differentiation theorem, density, regular measures, start functions of Bounded variation
Day 11:  10/15	3.5	Functions of bounded variation, absolutely continuous functions, Fundamental Theorem of Calculus for Lebesgue Integrals
Day 12:  10/18	4.1, 4.2	Basic notions of point set topology, Urysohn’s Lemma and Tietze Extension Theorem. Probably without proofs
Day 13:  10/20	4.3, 4.4	Nets and Compacts spaces, continuous image of a compact set, sequentially compact spaces
Day 14:  10/22 Takehome Midterm Exam and the lecture.	4.5	Topology of uniform convergence, Partitions of unity, one-point compactification
Day 15:  10/25	4.6, 4.7	Arzela-Ascoli Theorem and Stone-Weierstrass Theorem
Day 16:  10/27	4.7, 4.8	Complex Stone-Weierstrass Theorem, Stone-Chech compactification, Urysohn Metrization Theorem
Day 17: 10/28 X-hour instead of the class on 10/129	Round up chapter 4	Loose ends
10/29 Homecoming weekend. No class
Day 18:  11/1	5.1, 5.2	Normed vector spaces, Banach spaces, bounded maps, linear functionals
Day 19:  11/3	5.2 and start 5.3	Hanh-Banch Theorem and the Baire Category Theorem
Day 20:  11/5	5.3 start 5.4	Applications of Baire Category Theorem: Uniform boundness principle, Open Mapping and Closed Graph Theorems, topological vector spaces (may be mention topological groups)
Day 21:  11/8	5.4 and 5.5	Strong and weak operator topology, Alaoglu’s Theorem, Hilbert spaces
Day 22:  11/10	5.5	Hilbert spaces, existence of orthonormal bases, Bessel’s Inequality
Day 23:  11/12	6.1	L^p-spaces, Hoelder inequality and Minkowski’s inequality
Day 24:  11/15	6.2 and start 6.3	Dual spaces of L^p-spaces, Chebyshev’s Inequality
Day 25:  11/17	6.3 and 6.4	Minkowski’s inequality, weak L^p-
Day 26:  11/19	Either roundup 6.1-6.4 or do 6.5 interpolations of L^p-spaces	If we go into 6.5 discuss briefly Riesz-Thorin and Marcinkiewics Interpolation Theorems
Day 27:  11/22	7.1 and start 7.2	Regular measures, Radon measures, Riesz representation Theorem, regularity of Radon measures
Thanksgiving recess 11/23-11/28
Day 28:  11/29	7.2 and 7.3	Lusin’s Theorem, semi-continuous functions, dual of C0(X), Riesz Representation Theorem
Day 29:  12/1	Loose ends or 7.4 if we feel heroic	In case we go for 7.4: products of Radon measures and The Fubini-Tonelli Theorem for Radon products