Math 720: Measure Theory and Integration

Homework and Solutions


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 2:30--3:20 in WEH 8220.
Office Hours Mondays 10:30-12:20.
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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 10 (closed book).
Either 2:00-3:20 in WEH 8201
or 2:30-3:50 in WEH 8220
Final Fri, Dec 12, 5:30pm--8:30pm in WEH 7201 (and NOT PH A18C). (Closed book.)
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: Fri 22 May 2015 03:12:29 PM EDT