MATH 4280
Measure Theory
Fall 2017

This is the homepage for MATH 4280 (Measure Theory). This page will be updated throughout the term with information for our course, including homework assignments, review materials, and more.


  • A practice final exam has been posted below.
  • Solutions for Assignment 7 have been posted below.
Course Information

Course meets every MW from 2:00 p.m. - 3:50 p.m. in Knudson Hall, room 201.

Instructor: Ronnie Pavlov
Office: Knudson Hall 204
Phone: (303)-871-4001
Office hours: Monday 4-5, Tuesday 9-10, or by appointment (try to give 24 hours notice for appointments outside scheduled office hours!)


Text: Real Analysis: Modern Techniques and Their Applications, 2nd edition, by Gerald Folland.

This book is available at the DU Bookstore.

Course summary

We will cover introductory aspects of measure theory. In particular, we'll discuss what a measure is, what it can be used for, and how one can construct them. We'll define measurability and integrability of functions from this new viewpoint, and discuss how it yields a more general type of integration for functions from the reals to the reals than classical Riemann integration (namely, Lebesgue integration). This will allow us to revisit various topics you've seen from real analysis (such as convergence theorems, pointwise vs. uniform convergence, Fundamental Theorem of Calculus) in a greater and more useful generality. Depending on how much time remains, we may cover some more advanced topics such as L^p spaces, Fourier analysis, or Haar measures on topological groups.

A running theme throughout the course will be comparing measure-theoretic and topological properties. For instance, we'll discuss and prove Littlewood's so-called three principles of real analysis: any (Lebesgue) measurable set is "almost" a union of intervals, any (Lebesgue) measurable function is "almost" a continuous function, and any series of functions which converge pointwise converge "almost" uniformly.

The prerequisites are some course in topology/metric spaces (MATH 3110/4110 or MATH 3260/4110) and real analysis (MATH 3161). If you are not sure whether you are a good candidate for the class, please feel free to come talk to me.

Grading scheme

Your term grade will consist of homework assignments, one midterm exam, and one final exam, broken down in the following way:

40% Homework
25% Midterm Exam
35% Final Exam


Homework assignments must be turned in at the BEGINNING of class on the due date. Assignments turned in after the first 10 minutes of class will be counted as late, and subject to the usual late homework penalty scheme (described below.)

Late assignments (without documented illness/emergency) will have a percentage subtracted according to the following policy:

3-5 days late: -50%
>5 days late: not accepted


You will have one midterm exam on Monday, October 16th and a final exam on Monday, November 20th. The final exam will be in our classroom during classtime (2:00 p.m. - 3:50 p.m.) Review materials will be posted here later in the term.

Course Policies

Students in this course are expected to abide by the University of DenverÓ³ Honor Code and the procedures put forth by the Office of Citizenship and Community Standards. Academic dishonesty - including, but not limited to, plagiarism and cheating - is in violation of the code and will result in a failing grade for the assignment or for the course. As student members of a community committed to academic integrity and honesty, it is your responsibility to become familiar with the DU Honor Code and its procedures: see

Students with Disabilities: If you qualify for academic accommodations because of a disability or medical issue, please submit a faculty letter to me from Disability Services Program (DSP) in a timely manner so that your needs may be addressed. DSP determines accommodations based on documented disabilities/medical issues. DSP is located on the 4th floor of Ruffatto Hall, 1999 E Evans Ave, 303-871-2278. Information is also available online at; see the Handbook for Students with Disabilities.

Religious Accommodations: University policy grants students excused absences from class or other organized activities for observance of religious holy days, unless the accommodation would create an undue hardship. Faculty are asked to be responsive to requests when students contact them in advance to request such an excused absence. Students are responsible for completing assignments given during their absence, but should be given an opportunity to make up work missed because of religious observance. Once a student has registered for a class, the student is expected to examine the course syllabus for potential conflicts with holy days and to notify the instructor by the end of the first week of classes of any conflicts that may require an absence (including any required additional preparation/travel time). The student is also expected to remind the faculty member in advance of the missed class, and to make arrangements in advance (with the faculty member) to make up any missed work or in-class material within a reasonable amount of time.