Math 420/507 - Real Analysis I - Fall 2016
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00AM - 10:00AM in Room 202 of Mathematics Building.
Office hours: Wed 10-11AM, Fri 12-1PM or by appointment.
Course information handout containing syllabus and grading policies.
The final exam for the course has been scheduled on December 9, 7pm. Location TBA.
All the homework problems are from the textbook. Homework assignments will be collected at the beginning of class on the indicated day. Late homework assignments will not be accepted.
- Homework 1, due on Monday September 19
- Section 1.2: Exercises 3, 4, 5.
- Section 1.3: 11, 13.
Solution of HW 1.
- Homework 2, due on Monday October 3
- Section 1.3: 15, 16.
- Section 1.4: 19, 23.
- Section 1.5: 29, 30.
- Solution of HW 2.
- Homework 3, due on Monday October 17
- Section 1.4: 18, 21, 22.
- Section 1.5: 31, 33.
- Solution of HW 3.
- Homework 4, due on Monday November 7
- Section 2.1: 3, 4, 8, 9, 10.
- Section 2.2: 13, 14.
Week-by-week course outline
Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.
Week 3: (pages 26-31)
- Week 1: (pages 19-21)
- Some shortcomings of Riemann integration
- Lebesgue integral - an informal overview
- The problem of measure and existence of nonmeasurable sets
- We spent some time this week motivating the point of Lebesgue integration and the concept of measure. The textbook takes a slightly different route where this material appears in Chapter 2, pages 49-52. A good exposition of what we covered is in Chapter 16 of "Real Analysis" by Neal Carothers.
- Week 2: (pages 21-27)
- Sigma algebras.
Week 4: (pages 31-37)
- Properties of measures.
- Completion of a measure.
- Outer meaures.
Week 5: (pages 36-40)
- Caratheodory extension theorem.
- Borel measures on the real line.
- Lebesgue-Stieltjes measure.
- Approximation of measurable sets from above and below
- Characterization of Lebesgue measurable sets on the real line
- Cantor-type constructions
- Disparate notions of size: measure and cardinality
Week 7: (pages 43-48)
- Practice problem session
- Midterm review
Week 8: (pages 48-55)
- Measurable functions
- Simple functions
- Integration of nonnegative functions
- Monotone convergence theorem