Measure Theory (MAE616)

General

School School of Science Department of Mathematics Undergraduate MAE616 6 Measure Theory Lectures (Weekly Teaching Hours: 3, Credits: 6) Special background 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. Greek Yes (exams in English are provided for foreign students) 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. 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