MA 544 Spring 2021
Instructor: Greg Buzzard Location and time: WTHR 160, MWF 11:30AM12:20PM 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, semicontinuity, sequences of continuous functions and types of convergence, equicontinuity and AscoliArzelà Theorem, StoneWeierstrass theorem.
New topics:
 General σalgebras and measures, construction of the Lebesgue measure: Torchinsky, Chapters IV and V. (Also WheedenZygmund, Chapter III.)
 Properties of measurable functions and sequences of measurable functions: Torchinsky, Chapter VI; Rudin (R & C) Chapter I. (Also WheedenZygmund, Chapter IV.)
 General Integration Theory: Rudin (R & C) Chapter I, Torchinsky, Chapter VII.
 L^{p}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 L^{p}(R^{n})continuous functions of compact support, infinitely differentiable functions of compact support, HardyLittlewood 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.