**Instructor:** Prof. Robert Haslhofer

**Contact Information:** roberth(at)math(dot)toronto(dot)edu, IC477

**Website:** http://www.math.toronto.edu/roberth/C37.html

**Lectures:** Tuesday 9--11 and Thursday 10--11 in IC326

**Office Hours: ** Tuesday 11--12 and 16--17 in IC477

**Textbook:** E. Stein, R. Shakarchi: *Real Analysis -- Measure theory, Integration and Hilbert Spaces*, Princeton University Press

**Secondary Reference: **H. Royden, P. Fitzpatrick: *Real Analysis*, Pearson

**Topics to be covered:**
Analysis in R^d, Lebesgue measure, Lebesgue integration, Differentiation and Integration, L^p spaces, Hausdorff measure, Fourier analysis

**Grading Scheme:** Homework 30%, Midterm 30%, Final Exam 40%

**Midterm Exam:** Tuesday Feb 13th from 9--11 in IC326

**Final Exam:** Saturday April 14th from 14--17 in IC120

**Remarks:** Please discuss lectures and homework problems among yourselves and with me, and consult other sources. But write up your assignments in your own words, and be ready to defend them! Your work will be judged on the clarity of your presentation as well as correctness and completeness.

**Homework problem sets:**

HW1: Problem Set 1
HW2: Problem Set 2
Practice Problems: Practice Problems
HW3: Problem Set 3
HW4: Problem Set 4
HW5: Problem Set 5

**Weekly schedule:**

**Week 1 (Jan 8-14)**

Tuesday: Why do we need measure and integration theory?

Thursday: Review of basic properties of Euclidean space R^d

**Week 2 (Jan 15-21)**

Tuesday: Review of basic notions from topology

Thursday: Rectangles and cubes

**Week 3 (Jan 22-28)**

Tuesday: The exterior measure

Thursday: The exterior measure (continued)

**Week 4 (Jan 29-Feb 4)**

Tuesday: Measurable sets

Thursday: The Lebesgue measure

**Week 5 (Feb 5-11)**

Tuesday: The Lebesgue measure (continued)

Thursday: The Cantor-Lebesgue function

**Week 6 (Feb 12-18)**

Tuesday: Term Test

Thursday: Measurable functions

**Week 7 (Feb 26-Mar 4)**

Tuesday: Approximation by simple functions, Littlewood's three principles

Thursday: Integration of simple functions

**Week 8 (Mar 5- 11)**

Tuesday: Integration of bounded functions, comparison with the Riemann integral

Thursday: Integration of nonnegative and integrable functions

**Week 9 (Mar 12-18)**

Tuesday: Fatou's lemma, monotone convergence theorem

Thursday: Dominated convergence theorem

**Week 10 (Mar 19-25)**

Tuesday: The space L^1 of integrable functions, completeness

Thursday: dense subsets

**Week 11 (Mar 26 - Apr 1)**

Tuesday: The Hilbert space L^2

Thursday: Fourier series

**Week 12 (Apr 2 - 8)**

Tuesday: Lebesgue differentiation theorem

Thursday: Fundamental theorem of Calculus revisited