MA 544 Spring 2021
Instructor: Greg Buzzard
Location and time: WTHR 160, MWF 11:30AM-12:20PM
Office hours: TBA
Textbook: S. Axler, Measure, Integration & Real Analysis, Springer, 2020.
Topics: This course is a rigorous introduction to measure theory. We will cover the topics in the Math Graduate Student Handbook as described below, but using the textbook above as a primary source.
Required background: A reference for these background topics is W. Rudin, Principles of Mathematical Analysis – Chapters 2, 3, 4, 7
- Topology of metric spaces (properties of open, closed, compact, and connected sets).
- Continuity, semi-continuity, sequences of continuous functions and types of convergence, equicontinuity and Ascoli-Arzelà Theorem, Stone-Weierstrass theorem.
- General σ-algebras and measures, construction of the Lebesgue measure: Torchinsky, Chapters IV and V. (Also Wheeden-Zygmund, Chapter III.)
- Properties of measurable functions and sequences of measurable functions: Torchinsky, Chapter VI; Rudin (R & C) Chapter I. (Also Wheeden-Zygmund, Chapter IV.)
- General Integration Theory: Rudin (R & C) Chapter I, Torchinsky, Chapter VII.
- Lp-spaces: Torchinsky Chapter XII, Rudin (R & C) Chapter III.
- Product measure, Fubini's Theorem, Rudin (R & C), Chapter VIII.
- Differentiation of monotone functions, functions of bounded variation, and absolutely continuous functions (all on an interval [a; b]), Torchinsky, Chapter III, Section 1, Chapter X; Royden Chapter V; Bañuelos Lecture Notes.
- Convolutions, approximations to the identity, density theorems for Lp(Rn)-continuous functions of compact support, infinitely differentiable functions of compact support, Hardy-Littlewood maximal function. Torchinsky VIII, Section 1 and Torchinsky XIII, Section 2; Bañuelos Lecture Notes.
References for new topics: (books on reserve in the Math Library):
- W. Rudin, Real and Complex Analysis (R&C)
- A. Torchinsky, Real Variables
- H. L. Royden, Real Analysis
- R.L. Wheeden and A. Zygmund, Measure and Integral
- R. Bañuelos, Lecture Notes
Schedule: See Brightspace.
Grading: See Brightspace.
Homework: See Brightspace.