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 Rn 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 Lp-spaces, Hoelder inequality and Minkowski’s inequality Day 24:  11/15 6.2 and start 6.3 Dual spaces of Lp-spaces, Chebyshev’s Inequality Day 25:  11/17 6.3 and 6.4 Minkowski’s inequality, weak Lp- Day 26:  11/19 Either roundup 6.1-6.4 or do 6.5 interpolations of Lp-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