Introduction to Real Analysis

Course information

Instructor

Cătălin I. Cârstea

Email

carstea@nycu.edu.tw

Class meetings

Tuesdays, periods 7–8 (lectures); Fridays, period 5 (exercises)

Classroom

SA211[GF]

Credits / course code

3 credits · SCMA20018

Office hours

Thursdays 13:00–14:00, SA338[GF], and by appointment

Textbook

Terence Tao, An Introduction to Measure Theory, AMS, 2011

No Friday class: September 25, October 9, December 25.

Homework and course work

• Homework 1 (PDF) — Elementary and Jordan measure
• Homework 2 (PDF) — Lebesgue outer measure
• Homework 3 (PDF) — Lebesgue measurability and approximation
• Homework 4 (PDF) — Measurable functions and simple integrals
• Homework 5 (PDF) — The unsigned Lebesgue integral
• Homework 6 (PDF) — Signed integrals and the Riemann integral
• Homework 7 (PDF) — Borel sets and measurable functions

Homework is ungraded. We will work through selected homework problems in class on Fridays. Attempt the problems before class and be ready to present and discuss solutions.

Use the AMS edition for exercise numbering. The earlier free edition is also acceptable; some numbering differs.

Course description

Measure theory and Lebesgue integration, from Euclidean spaces to abstract measure spaces. Emphasis on proofs, examples, and applications.

Prerequisites. Introductory analysis or advanced calculus; Riemann integration, basic metric spaces, and proof writing.

Assessment

Class participation and coursework

30 points

Midterm exam

40 points

Final exam

40 points

Tentative exams: midterm October 27; final December 22.

Planned course outline

Tao §1.1

Elementary and Jordan measure (selected topics)

Tao §1.2

Lebesgue measure

Tao §1.3

Measurable functions and the Lebesgue integral

Tao §1.4

Abstract measure spaces and convergence theorems

Tao §1.5

Modes of convergence (selected topics)

Tao §1.7

Measure extension; product measures; Fubini–Tonelli

If time permits: differentiation theory (§1.6), Riesz representation, or an introduction to L^p spaces.