MATH 604 • REAL VARIALBES I • Fall 2001/2002

Catalogue Description: Covers algebra of sets, topology of metric spaces, compactness, completeness, function spaces, general theory of measure, measurable functions, integration, convergence theorems, and applications to classical analysis and integration.
Instructor : Dr. Robert Boyer
Office : Korman 292
Telephone : 215-895-1854
e-mail :
Home Page :
Office Hours: Monday 3-5 pm, Wednesday 1-3 pm, Thursday 3-5 pm.
Text : H. L. Royden, Real Analysis , Third Edition, Macmillan, 1987.
Supplementary sources:
W. Rudin, Principles of Mathematical Analysis, Third Editon, McGraw Hill.
A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover Publications.
M. Siegal, Real Variables , Schaum Outline Publications.
P. Halmos, Measure Theory, Springer-Verlag.

Graduate standing, Undergraduates with permission of the instructor.

Core mathematics course. Beginning graduate students in mathematics. Interested students in engineering. Well motivated senior undergraduate mathematics majors.

Schedule of Lectures, Coverage, and Homework

DatePosted NotesTopicsSectionsHomework
9/24/01 Lecture One Review of Advanced Calculus 2.1,2.2,2.3,2.4,2.5 Homework One
10/1/01 Lecture Two Topology of the Reals 2.5,2.6,2.7 Comments on Homework One
10/15/01 Lecture Three Topology of Reals; Lebesgue Measure 2.6, 3.1, 3.2 Homework Two
10/22/01 Lecture Four Measure Theory 3.2,3.3,3.4 Comments on Homework Two
10/29/01 Lecture Five Measurable Functions 3.4,3.5,3.6 Homework Three
11/05/01 Lecture Six Lebesgue Integral 4.1,4.2 Comments on Homework Three
11/12/01 Lecture Seven Convergence Theorems 4.2,4.3,4.4 Homework Four
11/19/01 Lecture Eight Differentiation 5.1,5.2,5.3 Comments on Homework Four
11/26/01 Lecture Nine Hausdorff Measures Hand-Out Homework Five
12/03/01 Lecture Ten Fractal Dimension Hand-Out Comments on Homework Five
12/10/01 Review Material Final    

Role of Course in the Graduate Curriculum:
This course gives an introduction to measure theory and integration. These topics are essential for a deeper understanding of probability theory and stochastic processes as well as approximation theory.

Course Goals:

  1. Development of theoretical understanding of contemporary mathematics.
  2. Development of abstract reasoning and proof writing.
  3. Introduction to measure theory and integration.
Class Format:
The class meets once a week in the evening for three hours. The majority of class time is spent on lecturing on the material. Other activities include question and answer sessions and a review of homework problems.


There will be four or five homework assignments. The questions illustrate the logical structure of the material, theorem proving, and computations.

Homework is worth 80% of the letter grade. The final examination is worth 20% of the grade. The final will have a take-home portion as well as an in-class part.

If a homework cannot be completed because of sickness or business travel, then the student should contact the instructor before the due date. There will be a special assignment available at the end of the term to make up any missed homeworks.

Letter grades will be given on the scale: 85% or better A, 70% -85% B, 55%-70% C, and so on.

Students are expected to attend every lecture. Absences should be arranged with the instructor before lecture either through e-mail or telephone.

Academic Honesty
The university's Academic Honesty policy is in effect for this course. Please read Section 11: "Academic Honesty" in the Drexel University Student Handbook to make sure you are familiar with this policy. An electronic version of the Student Handbook appears on the 1998 Drexel CD. You may visit the university's student handbook.


Chapter Two: The Real Number System
  • 2.1 Axioms for the real numbers
  • 2.2 The natural and rational numbers as subsets of R
  • 2.3 The extended real numbers
  • 2.4 Sequences of real numbers
  • 2.5 Open and closed sets of real numbers
  • 2.6 Continuous functions
  • 2.7 Borel sets

  • Chapter Three: Lebesgue Measure
  • 3.1 Introduction
  • 3.2 Outer measure
  • 3.3 Measurable sets and Lebesgue measure
  • 3.4 A non-measurable set
  • 3.5 Measurable functions
  • 3.6 Littlewood's three principles

  • Chapter Four: The Lebesgue Integral
  • 4.1 The Riemann integral
  • 4.2 The Lebesgue integral of a bounded function over a set of finite measure
  • 4.3 The integral of a non-negative function
  • 4.4 The general Lebesgue integral

  • Chapter Five: Differentiation and Integration
  • 5.1 Differentiation of monotone functions
  • 5.2 Functions of bounded variation
  • 5.3 Differentiation of an integral
  • 5.4 Absolute continuity