## 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.

### 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