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
Course information handout containing syllabus and grading policies.
The final exam for the course has been scheduled on December 9, 7pm. Location BUCH B208.
Homework
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.
- Homework 2, due on Monday October 3
- Section 1.3: 15, 16.
- Section 1.4: 19, 23.
- Section 1.5: 29, 30.
- Homework 3, due on Monday October 17
- Section 1.4: 18, 21, 22.
- Section 1.5: 31, 33.
- Midterm held on Friday October 21
- Homework 4, due on Monday November 7
- Section 2.1: 3, 4, 8, 9, 10.
- Section 2.2: 13, 14.
- Homework 5, due on Wednesday November 23
- Section 2.2: 15, 16, 17.
- Section 2.3: 19, 20, 25, 28.
- Homework 6, due on Friday December 2
- Section 2.5: 46, 48, 49.
- Section 2.6: 53, 54, 55.
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 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.
- Measures.
- Week 3: (pages 26-31)
- Properties of measures.
- Completion of a measure.
- Outer meaures.
- Week 4: (pages 31-37)
- Caratheodory extension theorem.
- Borel measures on the real line.
- Lebesgue-Stieltjes measure.
- Week 5: (pages 36-40)
- 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 6:
- Practice problem session
- Midterm review
- Week 7: (pages 43-48)
- Measurable functions
- Simple functions
- Week 8: (pages 48-52)
- Integration of nonnegative functions
- Monotone convergence theorem
- Fatou's lemma
- Week 9: (pages 52-56)
- Integration of complex functions
- Normed vector spaces of integrable functions
- Dominated convergence theorem
- Week 10: (pages 56-59, 64-65)
- Lebesgue's theorem on Riemann integration
- Product measures
- Towards the n-dimensional Lebesgue integral