MAT 1000 / MAT 457 Real Analysis I

Fall 2015

Web page:

Class Location & Time: Mon, 12:00 - 1:00 PM and Wed, 11:00 AM - 1:00 PM; BA6183

Instructor: Ilia Binder (, BA6118.
Office Hours: Mon 10:00 - 11:30 AM.

Teaching Assistant: Abhishek Oswal (
Office Hours:  Fri 2:00 - 3:00 PM.

Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley 2nd edition, 1999

Other references:

  1. Elias Stein and Rami Shakarchi, Measure Theory, Integration, and Hilbert Spaces
  2. Eliott H. Lieb and Michael Loss, Analysis AMS Graduate Texts in Mathematics, 14 (either edition)
  3. H.L. Royden, Real Analysis, Macmillan, 1988.
  4. A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, 1975.

Online material:

Terry Tao's blog:

Prerequisite:  MAT357H1.


Measure Theory: abstract measures and σ-algebras, Monotone Class Theorem, outer measures and Caratheorodry's theorem, Borel sets and Lebesgue measure, signed measures
Integration: convergence theorems, Fubini's theorem, convolutions
Lebesgue Differentiation: Hardy-Littlewood maximal function, density points, Radon-Nikodym theorem
Functional Analysis: Banach and Hilbert spaces, dual spaces, Lp-spaces.

Topics covered in class.

September 14: The introduction. Existence of Lebesgue non-measurable set. Folland, section 1.1.
September 16: σ-algebras. Monotone Class Theorem. Measures: formal definition and basic properties. Completion of a measure. Folland, sections 1.2, 1.3; Lieb-Loss, section 1.3.
September 21: Outer measures. Caratheodory's Theorem. Premeasures. Folland, section 1.4.
September 23: Extension of measure from an algebra to σ-algebra. Construction and basic properties of Lebesgue-Stilties measures. Folland, sections 1.4, 1.5.
September 28: Measurable functions. Simple functions. Folland, section 2.1.
September 30: Lebesgue integral: simple functions, non-negative functions, general functions. The space L1. Dominated and Monotone convergence Theorems. Fatou's lemma. Folland, sections 2.2, 2.3.
October 5: Comparison of Lebesgue and Riemann Integration. Littlewood's three principles. Egorov's Theorem. Folland, section 2.3; Stein-Shakarachi, section 1.4.3.
October 7: Lusin's Theorem. Convergence in measure. Product of measures. Fubini-Tonelli Theorem. Folland, sections 2.4, 2.5.
October 14: Lebesgue measure in Rn. Signed and Complex measures. Hahn and Jordan decompositions. Total variation. Mutual singularity and absolute continuity. Folland, sections 2.6, 3.1, 3.3.
October 19: Midterm review.
October 21: Midterm at Exam Centre, room 300 (255 McCaul St).
October 26: Lebesgue-Radon-Nykodim Theorem. Functions of Bounded Variation. Folland, sections 3.2, 3.5.
October 28: Functions of Bounded Variation. Absolutely continous functions. Hardy-Littlewood Maximal Function. A Vitali-type Lemma. Folland, sections 3.4, 3.5.
November 2: Hardy-Littlewood Theorem. Lebesgue Differentiation Theorem. Lebesgue points. Folland, section 3.4.
November 4: Radon-Nykodim derivative of regular measures. Applications to the functions of bounded variation. Metric and normed vector spaces, Banach and Frechet spaces. Linear operators. Folland, sections 3.4, 3.5, 5.1.
November 9: No class: Fall break.
November 11: Dual spaces, Hahn-Banch Theorem and its applications. Baire Cathegory Theorem. Open mapping theorem. Folland, sections 5.2, 5.3.
November 16: Closed Graph Theorem. Uniform Boundedness principle. Weak and weak* convergence. Alaoglu's Theorem. Folland, sections 5.3, 5.4.
November 18: Hilbert spaces: definition and examples, orthonoraml systems. Dual space. Folland, section 5.5.
November 23: Lp-spaces. Duality. Folland, sections 6.1, 6.2.
November 25: Dual to Lp. Hausdorff measure and dimension. Folland, section 6.2; Stein-Shakarchi, sections 7.1, 7.2.
November 30: Hausdorff dimension of self-similar sets. Stein-Shakarchi, section 7.2.
December 2: Hausdorff dimension of self-similar sets. Riesz representation Theorem. Stein-Shakarchi, section 7.2; Folland, section 6.1.
December 7: Riesz representation Theorem. Folland, sections 6.1, 6.2.
December 9: Office hours in BA6183, from 11.10 to about 1 pm.


Assignment #1, due September 30.

Assignment #2, due October 7.

Assignment #3, due October 14.

Assignment #4, due November 4.

Assignment #5, due November 11.

Assignment #6, due November 23.

Assignment #7, due November 25.

Assignment #8, due December 2. A misprint in the first problem is corrected on November 30.

Midterm Test. There will be an in-class midterm test on Wednesday, October 21. It will be held at Exam Centre, room 300 (255 McCaul St). The midterm will cover the first two chapters of Folland. No aides are allowed for this test.
Recommended preparation (do not turn in): all homework problems and the following problems from Folland: 1.1, 1.8, 1.10, 1.17, 1.20, 1.23, 1.29, 1.30, 1.31, 1.32, 1.33, 2.2, 2.9, 2.14, 2.16, 2.25, 2.29, 2.37, 2.48, 2.56.

Final exam. The final exam will be held on December 11, 10am - 1pm, at SF1105. You will be allowed to use one one-sided letter-sized page of notes. Textbooks or calculators are not allowed for this exam.
Recommended preparation (do not turn in): all homework and midterm preparation problems, the following problems from Folland: 3.5, 3.15, 3.20, 3.23, 3.26, 3.35, 3.42, 5.4, 5.12, 5.18, 5.24, 5.26, 5.37, 5.42, 5.51, 5.66, 6.15, 6.19, 6.21; and the following exercises from Stein-Shakarchi: 7.5, 7.8, 7.10.

Grading. Grades will be based on eight homework assignments (5% each), Midterm test (20%), and Final exam (40%). I will also occasionally assign bonus problems.

Late work. Extensions for homework deadlines will be considered only for medical reasons. Late assignments will lose 20% per day. Submission on the day the homework is due but after the tutorial is considered to be one day late. Special consideration for late assignments or missed exams must be submitted via e-mail within a week of the original due date. There will be no make-up midterm tests or final. Justifiable absences must be declared on ROSI, undocumented absences will result in zero credit.

Academic Integrity.
Honesty and fairness are fundamental to the University of Toronto’s mission. Plagiarism is a form of academic fraud and is treated
very seriously. The work that you submit must be your own and cannot contain anyone elses work or ideas without proper
attribution. You are expected to read the handout How not to plagiarize ( to-plagiarize) and to be familiar with the Code of behaviour on academic matters, which is linked from the UTM calendar under
the link Codes and policies.