Measure Theory (MAE616)

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School of Science

Academic Unit

Department of Mathematics

Level of Studies


Course Code




Course Title

Measure Theory

Independent Teaching Activities

Lectures (Weekly Teaching Hours: 3, Credits: 6)

Course Type

Special background

Prerequisite Courses

None (from the typical point of view). In order to be able to follow this course, the knowledge from the following courses are required: Infinetisimal Calculus I, Introduction to Topology.

Language of Instruction and Examinations


Is the Course Offered to Erasmus Students

Yes (exams in English are provided for foreign students)

Course Website (URL) See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

After completing this course the students will

  • Have knowledge of the basic properties of σ-algebras, of measures and especially of Lebesgue measure on the set R of real number and on the Euclidean space R^k.
  • Know the basic properties of measurable functions, the definition of Lebesgue integral in a random measure space.
  • Be able to apply the basic theorems concerning Lebesgue intergral (Monotone Convergernce Theorem, Dominated Convergence Theorem).
  • Understand the difference between Riemann integral and Lebesgue integral on R.
General Competences

The course promotes inductive and creative thinking and aims to provide the student with the theoretical background and skills to use measure theory and integration.


Algebras, σ-algebras, measures, outer measures, Caratheodory's Theorem (concerning the construction of a measure from an outer measure). Lebesgue measure, definition and properties. Measurable functions. Lebesgue integral, Lebesgue's Monotone Convergernce Theorem, Lebesgue's Dominated Convergence Theorem. Comparison between Riemann integral and Lebesgue integral for functions defined on closed bounded integrals of the set of reals.

Teaching and Learning Methods - Evaluation


Teaching on the blackboard by the teacher.

Use of Information and Communications Technology

Communication with the teacher by electronic means (i.e. e-mail).

Teaching Methods
Activity Semester Workload
Lectures 39
Personal study 78
Solving exercises 33
Course total 150
Student Performance Evaluation

Exams in the end of the semester (mandatory), potential intermediate exams (optional), assignments of exercises during the semester (optional).

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

  • Measure Theory, Donald Cohn, Birkhauser.