MA598R: Measure Theory

In the summer of 2007, I had the pleasure to help a group of graduate students prepare for their Qualifying exams in Measure Theory.   I taught the course MA598R, which was mainly a thorough review of Torchinsky’s “Real Variables”, together with guided sessions of problem-solving from previous Qualifying exams and lists of problems from Rudin, Torchinsky, Lieb-Loss, and other sources.

Real Variables Analysis (Graduate Studies in Mathematics) (See all Mathematical Analysis Books) Principles of Mathematical Analysis, Third Edition (See all Mathematical Analysis Books)

Lesson Plan and Assignments

Feel free to download the different problem sets below.  In a near future I will also present hints and solutions to some of the harder exercises.

Monday, June 11

Riemann-Stieltjes Integral

  • Functions of Bounded Variation
  • Existence of the Riemann-Stieltjes Integral
  • The Riemann-Stieltjes Integral and Limits

Wednesday, June 13

Abstract Measures. Lebesgue Measure.

  • Algebras and σ-algebras of Sets
  • Additive Set Functions and Measures
  • Properties of Measures
  • Lebesgue Measure on Rn
  • The Cantor Set

Monday, June 18
Second chances: review of Measure Theory

Wednesday, June 20

Measurable Functions

  • Elementary Properties of Measurable Functions
  • Structure of Measurable Functions
  • Sequences of Measurable Functions

Monday, June 25
Second chances: review of Measurable Functions.

Wednesday, June 27


  • The Integral of Nonnegative Functions
  • The Integral of Arbitrary Functions
  • Riemann and Lebesgue Integrals
  • Metric structure of L1
  • The Lebesgue Differentiation Theorem

Monday, July 2
Second chances: review of Integration

Wednesday, July 4
No class

Monday, July 9
Third chances: review of Integration

Wednesday, July 11

Lp Spaces

  • The Lebesque Lp spaces
  • Functionals on Lp
  • Weak convergence

Monday, July 16
Second chances: review of Lp spaces

Wednesday, July 18

Advanced Topics:

  • Covering Lemmas.
  • Absolute Continuity.
  • Approximations to the Identity.
  • The Fourier Transform.

Monday, July 23
Second chances: review of Advanced Topics.

Wednesday, July 25

Qual frenzy

Monday, July 30
Qual frenzy

  1. February 27, 2014 at 12:08 pm

    Hi Francisco,

    I’m a new graduate student in Purdue. I found some links in Prof. Bell’s homepage, including the nice problems you created in the 2007 summer to help students prepare quals. I spent a lot of time thinkng about the ” advanced problems 18″ but still don’t know where to start. I’m sorry to ask you about this now. It has been 7 years’ long since you gave the problems, but I’ve been tortured by it for a long long time.
    I couldn’t get asleep these days because of this problem. I tood a glance of all the books you listed but didn’t find any hint. I think the key to the problem should be the Plancherel Theorem, but I don’t know where to start. Could you please give me some clues whenever you are free?


    • March 4, 2014 at 10:16 pm

      Qinfeng, thanks for reading! I posted a solution to this problem at the following [link]. Let me know if that helps.

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