Math 720: Measure Theory and Integration

Homework and Solutions


Course Pace

Please let me know about the pace of this course. I will check responses weekly and adjust the pace accordingly.

Logistical Info

Instructor Gautam Iyer. 
Wean Hall #6121
412 268 8419

For MINOR clarifications and logistical queries, my address is:
(replace "NoSPAM" with "cmu").

For hints or longer questions, come by my office hours and speak to me in person.

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Lectures MWF 10:30--11:20 in WEH 8201.
Office Hours Mondays 11:30-12:20.
Mailing list math-720

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Homework Schedule, Exams Dates and Grading Rubric

Homework due Wednesdays, beginning of class.

Late homework will NEVER be accepted. Really.

Midterm Fri, Oct 11. (Closed book, in class.)
Final TBA

The final exam is be scheduled by the registrar at a time that can not be changed by mere mortals. You can find more information here.

Grading Homework: 20%, midterm: 30%, final: 50%.


This is a first graduate course on Measure Theory, and will at least include the following.

  • Outer measures, measures, $\sigma$-algebras, Carathéodory's extension theorem.
  • Borel measures, Lebesgue measures.
  • Measurable functions, Lebesgue integral (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem).
  • Modes of Convergence (Egoroff's Theorem, Lusin's Theorem)
  • Product Measures (Fubini-Tonelli Theorems), $n$-dimensional Lebesgue integral
  • Signed Measures (Hahn Decomposition, Jordan Decomposition, Radon-Nikodym Theorem, change of variables)
  • Differentiation (Lebesgue Differentiation Theorem)
  • $L^p$ Spaces, Hölder's inequality, Minkowskii's inequality, completeness, uniform integrability, Vitali's convergence theorem.

This will be followed by some special topics (e.g. Fourier Analysis).

Course Outline.

  • The course will start by constructing the Lebesgue measure on $\mathbb{R}^n$, roughly following Bartle, chapters 11--16.
  • After this, we will develop integration on abstract measure spaces roughly roughly following Cohn, chapters 1--6 or Folland.
  • If time permits, I will continue with some Fourier Analysis roughly following Folland chapter 8.


Since measure theory is fundamental to modern analysis, there is no dearth of references (translation: I'm not writing lecture notes). Here are a few other nice references I recommend.

  • I recommend buying either Cohn or Folland. Folland has a few nice additional topics (topology, functional analysis, Fourier analysis and probability). Cohn on the other hand is a slightly easier read and more focussed. If you're going to do your Ph.D. in something analysis related, buy Folland. If not, Cohn should suffice for this course.
  • An excellent (but harder) alternative to Folland is Rudin, which I strongly recommend.
  • Alternately, two slightly easier books are Jones or Royden. (If you find this course a little brisk / hard, I recommend reading Royden.)
  • If you prefer learning from lecture notes, here are some by Lenya Ryzhik and Terry Tao. (The last one is available as a PDF, and also as a regular published book.) Alternately, contact Giovanni Leoni for measure theory lecture notes from 2011.
  • An excellent treatment of Fourier Series can be found in Chapter 1 of Wilhelm Schlag's notes. (This has many advanced Harmonic Analysis topics, which I recommend reading later.)

Class policies


  • Late homework will never be accepted.
  • Homework will be assigned every Wednesday, and due the following Wednesday at the beginning of class.
  • Working in groups is encouraged, but solutions must be written up on your own.


  • All exams are closed book, in class exams.
  • Exams from previous years are/will be hosted here, in the Handouts section.

Last Modified: Sun 19 Jul 2015 03:26:32 PM EDT