**Instructor:** Prof. Robert Haslhofer

**Contact Information:** roberth(at)math(dot)toronto(dot)edu, BA6208, 8-4632

**Website:** http://www.math.toronto.edu/roberth/RealAnalysis1.html

**Lectures:** Monday 11--12 and Wednesday 10--12 in BA6183

**Teaching assistant:** Justin Ko, justinp.ko(at)mail(dot)utoronto(dot)ca

**Office Hours:** Monday 2--4 in BA6208 (Robert Haslhofer) and Tuesday 11--1 in the math lounge / tutorial room (Justin Ko)

**Textbook:** G. Folland, *Real Analysis -- Modern Techniques and Their Applications*, Wiley (either edition)

**Secondary References:**

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

E. Lieb, M. Loss: *Analysis*, American Mathematical Society

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

**Topics to be covered:**

*Measure theory:* abstract measure spaces, construction of measures, regularity properties, Lebesgue measure, Hausdorff measure, Wiener measure

*Integration theory:* construction of integrals, basic properties, convergence theorems, integration on product spaces

*Functional Analysis:* Riesz representation theorem, L^p spaces, H^1, Sobolev inequality, compactness theorems

*Lebesgue Differentiation:* signed measures and differentiation, Lebesgue-Radon-Nikodym theorem, Lebesgue-differentiation on R^n, BV, FTC

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

**Midterm Exam:** Wednesday, Nov 2 from 10am--12am.

**Final Exam:** Monday, Dec 12 from 9am--12am.

**Remarks:** Please discuss lectures and homework problems among yourselves and with me and the TA, 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.

**WEEKLY SCHEDULE**

**Week 1 (Sep 12-18)**

Introduction and basic notions (Intro of Stein-Shakarchi, Folland 1.1--1.4)

M: Why do we need measure and integration theory?

W: sigma-algebras, measures, outer measures, Caratheodory's theorem

Problem Set 1 (due Sep 21)

**Week 2 (Sep 19-25)**

Construction of measures (Stein-Shakarchi 6.1, Royden-Fitzpatrick 20.2)

M: premeasures, Caratheodory-Hahn extension theorem

W: Lebesgue measure

Problem Set 2 (due Sep 28)

**Week 3 (Sep 26-Oct 2)**

Construction of measures (Stein-Shakarchi 6.1, 7.1--7.2)

M: Regularity properties of measures

W: Hausdorff measure, Wiener measure

Problem Set 3 (due Oct 5)

**Week 4 (Oct 3-9)**

Construction of integrals and basic properties (Folland 2.1--2.2)

M: measurable functions

W: simple functions, definition of integral

Problem Set 4 (due Oct 12)

**Week 5 (Oct 10-16)**

Convergence theorems and integrable functions (Folland 2.2--2.3)

(M: no lecture -- Thanksgiving)

W: monotone convergence, Fatou's lemma, dominated convergence, L^1

Problem Set 5 (due Oct 19)

**Week 6 (Oct 17-23)**

L^p spaces (Folland 6.1--6.2, Lieb-Loss Chap. 2)

M: Hölder inequality, Minkowski inequality, completeness

W: interpolation, weak convergence, dual of L^p

Problem Set 6 (due Oct 26)

**Week 7 (Oct 24-30)**

Radon measures (Folland 7.1--7.2, 4.5)

M: LCH spaces, Radon measures, density of C_c in L^p

W: Riesz representation theorem and applications

Old Midterm (for practice)

**Week 8 (Oct 31-Nov 6)**

Littlewood's three principles (Folland 7.2, 2.4)

M: Regularity of Borel measures, Egoroff's theorem, Lusin's theorem

W: Midterm test

Problem Set 7 (due Nov 9)

**Week 9 (Nov 7-13)**

Integration on product spaces (Folland 2.5)

(M: no lecture -- fall break) - but I do have office hours!

W: product measures, monotone class lemma, Fubini-Tonelli theorem

Problem Set 8 (due Nov 16)

**Week 10 (Nov 14-20)**

Signed measures and Lebesgue-differentiation (Folland 3.1, 3.2, 7.3)

M: signed measures, Hahn/Jordan-decomposition, total variation

W: absolute continuity, Lebesgue-Radon-Nikodym theorem, conditional expectation, dual of C_0

Problem Set 9 (due Nov 23)

**Week 11 (Nov 21-27)**

Lebesgue-differentiation on Euclidean space (Folland 3.4, 3.5, 1.5, Stein-Shakarchi 3.1, 3.3)

M: Hardy-Littlewood maximal inequality, Lebesgue-differentiation theorem

W: distribution function, BV and signed measures, absolute continuity, FTC

Problem Set 10 (due Nov 30)

**Week 12 (Nov 28-Dec 4)**

The Sobolev space H^1 (Lieb-Loss Chap. 7, 8)

M: Definition and basic properties, approximation by smooth functions, weak compactness

W: Gagliardo-Nirenberg-Sobolev inequality, Rellich-Kondrachov theorem

Problem Set 11 (due Dec 7)

**Week 13 (Dec 5)**

Application: The ground state of the Schrödinger equation (Lieb-Loss Chap. 11)