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.

Source, CC BY-SA 3.0

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

1. Topology of metric spaces (properties of open, closed, compact, and connected sets).
2. Continuity, semi-continuity, sequences of continuous functions and types of convergence, equicontinuity and Ascoli-Arzelà Theorem, Stone-Weierstrass theorem.

New topics:

1. General σ-algebras and measures, construction of the Lebesgue measure: Torchinsky, Chapters IV and V. (Also Wheeden-Zygmund, Chapter III.)
2. Properties of measurable functions and sequences of measurable functions: Torchinsky, Chapter VI; Rudin (R & C) Chapter I. (Also Wheeden-Zygmund, Chapter IV.)
3. General Integration Theory: Rudin (R & C) Chapter I, Torchinsky, Chapter VII.
4. L^p-spaces: Torchinsky Chapter XII, Rudin (R & C) Chapter III.
5. Product measure, Fubini's Theorem, Rudin (R & C), Chapter VIII.
6. 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.
7. Convolutions, approximations to the identity, density theorems for L^p(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.