Measure theory

Owner	Bachelor Wiskunde
Coordinator	dr. E. Musta
Part of	Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Bachelor Wiskunde, year 3Dubbele bachelor Wis- en Natuurkunde, year 3Bachelor Bèta-gamma, major Wiskunde, year 3Dubbele bachelor Wiskunde en Informatica, year 3

Course manual 2021/2022

Course content

The course is an introduction to abstract measure and integration theory, which goes beyond the classical theory of Riemann’s integration. This new perspective on integration can be applied to abstract spaces, providing a foundation for modern probability theory and functional analysis.

We start by introducing the notion of a measure, which evolves from the idea of assigning a weight or a size to a set in an abstract space, generalizing the common ideas of length, area or probability. Even though not all sets can be measurable, it suffices to restrict to certain sets. This leads to the definition of a sigma-algebra. Afterwards, we introduce the measurable functions and define integration of such functions with respect to a measure. We will see  how the new Lebesgue integral relates to the Riemann integral and learn about its main properties such as: the monotone and dominated convergence theorems, the substitution rule and absolute continuity.  The notions of measure, measurability and Lebesgue integral will be extended to product spaces allowing for multi-dimensional integration.

Study materials

Literature

• Measures, Integrals and Martingales, 

  Schilling, René L,  Cambridge: Cambridge University Press, 2005

Objectives

• • The student is able to understand basic concepts of measure and integration theory as demonstrated by the ability to provide definitions of such concept and to show whether, for example, a collection of sets is a sigma-algebra, a given function is measurable or integrable with respect to a measure.
• • The student is able to compare the Lebesgue integral with the Riemann integral and reflect on the advantages and necessity for this new approach.
• • The student is able to write elementary proofs of statements regarding measure and integration theory, as well as more advanced proofs under guidance.
• • The student is able to apply the theory for solving abstract and concrete problems, in particular concerning applications of the convergence theorems for integrals and Fubini’s theorem.
• • The student is able to perform computations with the Lebesgue measure and use its special properties.

Teaching methods

• Lecture
• Seminar
• Self-study

Learning activities

Attendance

Programme's requirements concerning attendance (OER-B):

• Each student is expected to actively participate in the course for which he/she is registered.
• If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
• It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
• In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
• In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article A-6.4, an other arrangement will be proposed in consultation with the study advisor.

Assessment

Item and weight	Details
Final grade
0.25 (25%) Midterm
0.6 (60%) Final exam	Must be ≥ 5
0.15 (15%) Homework assignments

Written final exam, and a written midterm exam after 7 weeks. The final exam will be 60% of the final grade, and the midterm exam will be 25%. The remaining 15% will be the average of the grades for the homework assignments.  The grade of the final exam must be at least 5. The homework and the midterm do not count towards the resit. 

Inspection of assessed work

The inspection moments of the midterm and final exam will be communicated via Canvas. 

Assignments

There will be 4 homework assignments with deadlines  24 September, 15 October, 19 November and 10 December. Each assignment will be published two weeks in advance on Canvas. The homework assignments are individual and should be uploaded on Canvas. Both solutions written by hand or in Latex are accepted but make sure that it is readable and uploaded as a single pdf file. The assignments are graded on a scale 1-10 and their average will count for 15% of the final grade. 

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Processed course evaluations

Hieronder vind je de aanpassingen in de opzet van het vak naar aanleiding van de vakevaluaties.

Contact information

Coordinator

• dr. E. Musta