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

Ilmoittaudu

Viestit

Käyttäjän Ilkka Holopainen kuva

Ilkka Holopainen

Julkaistu, 30.8.2017 klo 16:25

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.

Käyttäjän Ilkka Holopainen kuva

Ilkka Holopainen

Julkaistu, 29.8.2017 klo 11:08

Esko Heinonen will be the lecturer on the first week (6th and 7th September).
The first exercise classes will be on 13th and 14th September.

Käyttäjän Ilkka Holopainen kuva

Ilkka Holopainen

Julkaistu, 29.8.2017 klo 10:55

Ensimmäisellä luentoviikolla (6. ja 7.9) luennoitsijana on Esko Heinonen.
Ensimmäiset laskuharjoitukset 13. ja 14.9.

Aikataulu

Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

PäivämääräAikaOpetuspaikka
Ke 6.9.2017
12:15 - 14:00
To 7.9.2017
10:15 - 12:00
Ke 13.9.2017
12:15 - 14:00
To 14.9.2017
10:15 - 12:00
Ke 20.9.2017
12:15 - 14:00
To 21.9.2017
10:15 - 12:00
Ke 27.9.2017
12:15 - 14:00
To 28.9.2017
10:15 - 12:00
Ke 4.10.2017
12:15 - 14:00
To 5.10.2017
10:15 - 12:00
Ke 11.10.2017
12:15 - 14:00
To 12.10.2017
10:15 - 12:00
Ke 18.10.2017
12:15 - 14:00
To 19.10.2017
10:15 - 12:00

Muu opetus

06.09. - 18.10.2017 Ke 14.15-16.00
Riikka Schroderus
Opetuskieli: englanti
07.09. - 19.10.2017 To 14.15-16.00
Ilkka Holopainen
Opetuskieli: englanti

Materiaalit

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):

Tehtävät

Harjoitus 1, Exercise 1 (13-14.9.2017)

Harjoitus 2, Exercise 2 (20-21.9.2017)

Harjoitus 3, Exercise 3 (27-28.9.2017)

Harjoitus 4, Exercise 4 (4-5.10.2017)

Harjoitus 5, Exercise 5 (11-12.10.2017)

Harjoitus 6, Exercise 6 (18-19.10.2017)

Kurssin suorittaminen

Exam and exercise classes. The exam will consists of 5 problems (evaluated as 6 pts each).
The first possible exam is on 1st of November.

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

Kuvaus

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.