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Introduction to measure theory


Admission requirements



This course offers an introduction to measure theory, the field of mathematics that is the mathematical foundation of probability theory and mathematical statistics. It also has applications in functional analysis and other areas. The following concepts will be introduced: sigma-algebras, measures, measurable functions and Lebesgue integrals. We shall see how a series can be interpreted as an integral! Additionally we will discuss the most important convergence theorems.

The course will focus on providing intuition about these concepts and the use of the corresponding theory for problem solving. After completing the course you will have sufficient knowledge of measure theory to be able to take the advanced probability courses in the bachelor of mathematics. If the course has peaked your interest in the foundations of measure theory, we can recommend you to take the bachelor-master course Advanced Measure Theory.

Course Objectives

You will learn the basic concepts of measure theory, such as sigma-algebras, measures, measurable functions, Lebesgue integrals and convergence theorems. After taking this course you will be able to use the corresponding theory for problem solving.


You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different

Mode of Instruction

During the semester the lectures and exercise classes will alternate weekly.

Assessment method

The assessment consists of homework exercises and a written test. The final grade is determined by the maximum of the grade for the written test and the weighted average of the homework grade (25%) and the written test (75%). To pass the course the grade for the written test must be at least 5 and the (unrounded) final grade must be at least 5.5. No minimum grade is required for the homework in order to take the written test or to pass the course. The homework counts as a practical and there is no retake for it. In principle there will be 7 homework exercises, the lowest grade of which will not count towards the average grade. The grade of the retake will replace the grade of the written test if it is higher.

Reading list

We will use the book ‘Measures, Integrals and Martingales’ 2nd Edition van Rene Schilling, Cambridge University Press 2017, ISBN 9781316620243. The 1st Edition of this book can be used as well and is available as e-book via Leiden University Library.


From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to register for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course.

Extensive FAQ on MyStudymap can be found here.


Charlene Kalle, Mathematisch Instituut, Snellius office 212, email:


The course will use Brightspace.