Measure Theory (AN7)

Graduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN7
Semester	1
Course Title	Measure Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	Course description https://math.uoi.gr Learning Management System http://users.uoi.gr/kmavridi

Learning Outcomes

Learning outcomes

Using the Bloom Taxonomy. All the following sets are considered to be arbitrary subsets of an arbitrary Euclidean normed space of finite dimension. Remembering: The notion of the rectangle and the notion of the volume of a rectangle. The notion of the external measure. The notion of the Lebesgue measure. The notion of the σ-Algebra. The Borel set. The notion of the characteristic function, of the step function, of the simple function and of the measurable function. The “almost everywhere” validity of a property. The notion of the Lebesgue integral. Definition of the L1 space, of the integrable functions. The notion of the absolutely continuous function. The notion of the locally integrable function. The notion of the Lebesgue density. The Lebesgue set of a locally integrable function. Good kernels and approximations to the identity. The notion of the bounded variation function. The notion of abstract measurable spaces. The Caratheodory measurable sets. The metric exterior measures. The notion of the pre-signed measure. Comprehension: The Cantor set. Properties of the exterior measure. Properties of the Lebesgue measure. Translation invariance property of the Lebesgue measure. Conditions under which sets are measurable. Construction of non-measurable sets. Properties of measurable functions. Approximation of measurable functions by simple or step functions. The three principles of Littlewood. Brunn – Minkowskii inequality. Properties of the Lebesgue integral. Relation between the Lebesgue integral and the Riemann integral. Fatou Lemma. Uniform convergence theorem. Riesz – Fischer theorem. Translation invariance property of the Lebesgue integral. Fubini theorem. Relation between integrable and measurable function. The Hardy - Littlewood maximal function. Properties of bounded variation functions. Properties of absolutely continuous and differentiable functions. Properties of abstract measurable spaces. Integration in abstract measurable spaces. Absolute continuity of measures. Applying: Calculating the measure of a set. Finding examples of non-measurable sets. Calculating the Lebesgue integral. Finding the mean value of a function. Evaluating: Teaching undergraduate courses.

General Competences

Production of free, analytic and inductive thinking. Required for the production of new ideas. Working independently. Team work. Decision making.

Syllabus

Measure spaces, Lebesgue measure, measurable functions and Lebesgue integral, Monotone convergence Theorem and Dominated convergence Theorem, relation between Riemann and Lebesgue integral. Product measures, Fubini Theorem. L^p spaces. Signed measures, Hahn decomposition, Radon-Nikodym Theorem. Convergence of sequences of measurable functions.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Teaching is assisted by Learning Management System. Teaching is assisted by the use of online forums where students can participate in order to improve their problem solving skills, as well as their understanding of the theory they are taught. Teaching is assisted by the use of pre-recorded videos.
Use of Information and Communications Technology	Use of Learning Management System, combined with File Sharing Platform as well as Blog Management System for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, and communicating with students. Use of Appointment Scheduling System for organising appointments between students and the teacher. Use of Survey Web Application for submitting anonymous evaluations regarding the teacher. Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly presentations - oral exams, combined with weekly written assignments. In any case, all students can participate in written exams at the end of the semester. The aforementioned information along with all the required details are available through the course’s website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course’s website. Upon request, all the information is provided using email or social networks.

Attached Bibliography

1. R. Bartle - The Elements of Integration and Lebesgue Measure
2. H. Bauer - Measure and Integration Theory
3. H.S. Bear - A Primer of Lebesgue Integration
4. V.I. Bogachev - Measure theory
5. R.M. Dudley - Real Analysis and Probability
6. G. Folland - Real Analysis: Modern Techniques and Their Applications
7. D.H. Fremlin - Measure Theory
8. P. Halmos - Measure theory
9. M.E. Munroe - Introduction to Measure and Integration
10. M.M. Rao - Random and Vector Measures
11. E. Stein and R. Skakarchi - Real Analysis
12. T. Tao - An Introduction to Measure Theory
13. N. Weaver - Measure Theory and Functional Analysis

Also, variety of international, peer-reviewed journals, with related content.