Measure Theory (MAE616)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE616
Semester	6
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	Greek
Is the Course Offered to Erasmus Students	Yes (exams in English are provided for foreign students)
Course Website (URL)	-

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.

Syllabus

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

Delivery	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 Eudoxus. Additionally:

• Measure Theory, Donald Cohn, Birkhauser.