Kaisa_2012_3_photo by Veikko Somerpuro

The content of the course:
Lp-spaces, Hölder's inequality, Minkowski's inequality, completeness of Lp-spaces
Egorov's theorem, Lusin's theorem
Convolution (approximation of Lp-functions by smooth functions)
Covering theorems
Hardy-Littlewood maximal function
Lebesgue's differentiation theorem
Functions of bounded variation
Absolutely continuous functions



Ilkka Holopainen's picture

Ilkka Holopainen

Published, 28.8.2018 at 13:25

First exercise classes are on Thursday, September 13.
Ensimmäiset laskuharjoitukset ovat torstaina 13 syyskuuta.

Ilkka Holopainen's picture

Ilkka Holopainen

Published, 28.8.2018 at 13:23

To start with: recall the basics of measure theory and Lebesgue's integral (see, for instance, the background material (Lecture notes on Measure and integral).

Study the two Fubini's theorems. These will be used (and referred as Fubini 1 and Fubini 2), although we won't state them during the lectures.


Here is the course’s teaching schedule. Check the description for possible other schedules.

Tue 4.9.2018
12:15 - 14:00
Wed 5.9.2018
14:15 - 16:00
Tue 11.9.2018
12:15 - 14:00
Wed 12.9.2018
14:15 - 16:00
Tue 18.9.2018
12:15 - 14:00
Wed 19.9.2018
14:15 - 16:00
Tue 25.9.2018
12:15 - 14:00
Wed 26.9.2018
14:15 - 16:00
Tue 2.10.2018
12:15 - 14:00
Wed 3.10.2018
14:15 - 16:00
Tue 9.10.2018
12:15 - 14:00
Wed 10.10.2018
14:15 - 16:00
Tue 16.10.2018
12:15 - 14:00
Wed 17.10.2018
14:15 - 16:00

Other teaching

06.09. - 18.10.2018 Thu 14.15-16.00
Ilkka Holopainen
Teaching language: English


Lecture notes: Reaalianalyysi I (Ilkka Holopainen).
Lecture notes: Real Analysis I (Ilkka Holopainen).

Background material: Lecture notes on Measure and integral (both in Finnish and in English):


Conduct of the course

Exam and exercise classes. The exam will consists of 5 problems (evaluated as 6 pts each).
The first possible exam is on 31st of October.

Remember to register for the exam in WebOodi at least 10 days before the exam!

You will get extra credit points by solving the home work assignments:
25% = +1p, 35% = +2p, 45% = +3p, 55% = +4p, 65% = +5p ja 75% = +6p


Compulsory course.

Master's Programme in Mathematics and Statistics is responsible for the course.

Belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Measure and integral

Barchelor studies

The course gives the basic knowlegde on real analysis that is of fundamental importance on analysis.

Recommended time/stage of studies for completion: 1. year

Term/teaching period when the course will be offered: varying

Basics on real analysis, like L^p spaces, convolution, covering theorems, Lebesgue's differentiation theorem, BV- and absolutely continuous functions.

Required: Reaalianalyysi I, luentomoniste.

Recommended: R. Gariepy, W. Ziemer: Modern real analysis. F. Jones: Lebesgue integration on Euclidean space.

Lectures and exercise classes.

Exam and excercises, Course will be graded with grades 1-5.

Exam, other methods will be described later.