**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 in IC 326 and Thursday 10--11 in IC320

**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

**PREREQUISITE: MATB43. Note: If you have not taken MATB43 then you can NOT take MATC37** (The only case where I consider waiving this is if you are exceptionally strong, e.g. CGPA above 3.5)

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

**Midterm Exam:** Tuesday Feb 26 from 9--11 in IC326

**Final Exam:** Monday April 15 from 19--22 in IC200

**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 7-13)**

Tuesday: Why do we need measure and integration theory?

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

**Week 2 (Jan 14-20)**

Tuesday: Review of basic notions from topology

Thursday: Rectangles and cubes

**Week 3 (Jan 21-27)**

Tuesday: The exterior measure

Thursday: The exterior measure (continued)

**Week 4 (Jan 28-Feb 3)**

Tuesday: Snowstorm

Thursday: Measurable sets

**Week 5 (Feb 4-10)**

Tuesday: The Lebesgue measure

Thursday: The Lebesgue measure (continued)

**Week 6 (Feb 11-17)**

Tuesday: Snowstorm

Thursday: Measurable functions

**Week 7 (Feb 25-Mar 3)**

Tuesday: Term Test

Thursday: Approximation by simple functions, Integration of simple functions

**Week 8 (Mar 4- 10)**

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

Thursday: Integration of nonnegative and integrable functions

**Week 9 (Mar 11-17)**

Tuesday: Fatou's lemma, monotone convergence theorem

Thursday: Dominated convergence theorem

**Week 10 (Mar 18-24)**

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

Thursday: dense subsets

**Week 11 (Mar 25 - 31)**

Tuesday: The Hilbert space L^2

Thursday: Fourier series

**Week 12 (Apr 1 - 7)**

Tuesday: Lebesgue differentiation theorem

Thursday: Fundamental theorem of Calculus revisited