Math 420/507  Real Analysis I  Fall 2016
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
Email: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00AM  10:00AM in Room 202 of Mathematics Building.
Office hours: Wed 1011AM, Fri 121PM 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.

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.
 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.
 Solution of HW 4.
 Homework 5, due on Wednesday November 23
 Section 2.2: 15, 16, 17.
 Section 2.3: 19, 20, 25, 28.
 Solution of HW 5.
 Homework 6, due on Friday December 2
 Section 2.5: 46, 48, 49.
 Section 2.6: 53, 54, 55.
 Solution of HW 6.
Weekbyweek 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 1921)
 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 4952. A good exposition of what we covered is in Chapter 16 of "Real Analysis" by Neal Carothers.
 Week 2: (pages 2127)
 Sigma algebras.
 Measures.
 Week 3: (pages 2631)
 Properties of measures.
 Completion of a measure.
 Outer meaures.
 Week 4: (pages 3137)
 Caratheodory extension theorem.
 Borel measures on the real line.
 LebesgueStieltjes measure.
 Week 5: (pages 3640)
 Approximation of measurable sets from above and below
 Characterization of Lebesgue measurable sets on the real line
 Cantortype constructions
 Disparate notions of size: measure and cardinality
 Week 6:
 Practice problem session
 Midterm review
 Week 7: (pages 4348)
 Measurable functions
 Simple functions
 Week 8: (pages 4852)
 Integration of nonnegative functions
 Monotone convergence theorem
 Fatou's lemma
 Week 9: (pages 5256)
 Integration of complex functions
 Normed vector spaces of integrable functions
 Dominated convergence theorem
 Week 10: (pages 5659, 6465)
 Lebesgue's theorem on Riemann integration
 Product measures
 Towards the ndimensional Lebesgue integral