## 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

The schedule for the course can be found on MyTimeTable.

## 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

Registration for the course is through Usis.

## Contact

Email: d.e.terhesiu[at]math.leidenuniv.nl

## Remarks

The course will use Brightspace.