## Admission requirements

none

## Description

Main subjects of this course are: measurable sets and function, Lebesgue integral, L^p spaces,

Holder and Minkowski’s inequality, various modes of convergence.

## Course Objectives

The aim of this course is to provide the students with enough background to understand

the concept of Lebesgue measure and be able to operate with Lebesgue integration.

To get a glimpse of the importance of the concepts, consider an interval contained in the

real line or a region in the plane, or simply think of the length of the interval or the area of the region give an idea of the size. We want to extend the notion of size to as large a class of sets as possible. Doing this for subsets of the real line gives rise to Lebesgue measure. In the first part of this course we will discuss classes of sets, the definition of measures, and the construction of measures, of which one example is Lebesgue measure on the line.

After an understanding of the notion of measures, we will proceed to the Lebesgue integral. We talk about measurable functions, define the Lebesgue integral, prove the monotone and dominated convergence theorems, look at some simple properties of the Lebesgue integral, compare it to the Riemann integral, and discuss some of the various ways a sequence of functions can converge.

## Timetable

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

On site lectures

Tutorials

## Assessment method

The final grade consists of homework (30%) and a written (retake) exam (70%). To pass the course, the grade for the (retake) exam should be at least 5 and the (unrounded) weighted average of the two partial grades at least 5.5. No minimum grade is required for the homework in order to take the exam or to pass the course. The homework counts as a practical and there is no retake for it; it consists of 6 assignments, of which the lowest grade can be replaced by A bonus homework.

## Reading list

Lecture notes

http://bass.math.uconn.edu/3rd.pdf opens in new window

Mark Veraar, MEASURE AND INTEGRATION

http://fa.its.tudelft.nl/~veraar/teaching/lecture_notes_KW2.pdfDonald L. Cohn, Measure Theory ISBN: 978-1-4614-6955-1 (Print) 978-1-4614-6956-8 (Online) (available as e-book via Leiden University Library).

P. Billingsley, Probability and Measure, 3d Edition, J. Wiley and Sons.

P. Billingsley, Convergence of Probability Measures, 1999, J. Wiley and Sons.

## Registration

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 both preregister and confirm your participation 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. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

Email: d.e.terhesiu@math.leidenuniv.nl

## Remarks

The course will use Brightspace.