# MATS112 Measure and Integration Theory 2 (4 cr)

**Study level:**

Advanced studies

**Grading scale:**

0-5

**Language:**

English, Finnish

**Responsible organisation:**

Department of Mathematics and Statistics

**Curriculum periods:**

2020-2021, 2021-2022, 2022-2023

## Description

Lp -spaces, general measure spaces, measurable functions and integrals, s-dimensional Hausdorff measure.

## Learning outcomes

After the course the student

- understands the basics of L^p spaces (in particular L^2)
- is able to define general outer measures and use the Caratheodory criterion for measurability
- knows and uses the definitions of general measures and measurable functions in abstract measure spaces.
- knows the basics of Hausdorff measures

## Description of prerequisites

Introduction to mathematical analysis 3, Vector calculus 2, Vector analysis 1 and 2, Measure and integration 1

## Study materials

Luentomoniste

## Literature

- Andrew M. Bruckner, Judith B. Bruckner & Brian S. Thomson: Real Analysis, 2008, www.classicalrealanalysis.com
- Avner Friedman: Foundations of Modern Analysis.
- Terence Tao: An Introduction to Measure Theory
- Elias M. Stein & Rami Shakarchi: Real Analysis.

## Completion methods

### Method 1

**Evaluation criteria:**

Exam after the lectures. Extra benefits from approved homework assignments.

**Time of teaching:**

Period 2

Select all marked parts

### Method 2

**Evaluation criteria:**

Final exam. At least 50% of total points is required for a passing grade.

Select all marked parts

**Parts of the completion methods**

x

### Teaching (4 cr)

**Type:**

Participation in teaching

**Grading scale:**

0-5

**Evaluation criteria:**

Exam after the lectures. Extra benefits from approved homework assignments.

**Language:**

English, Finnish

**Study methods:**

24h lectures, 6 exercise sessions

**Study materials:**

Luentomoniste

**Literature:**

- Andrew M. Bruckner, Judith B. Bruckner & Brian S. Thomson: Real Analysis, 2008, www.classicalrealanalysis.com
- Avner Friedman: Foundations of Modern Analysis.
- Terence Tao: An Introduction to Measure Theory
- Elias M. Stein & Rami Shakarchi: Real Analysis.

#### Teaching

##### 10/28–12/17/2021 Lectures

##### 12/15–12/15/2021 Exam

##### 1/19–1/19/2022 Exam

x

### Exam (4 cr)

**Type:**

Exam

**Grading scale:**

0-5

**Evaluation criteria:**

Final exam. Minimum of 50% of total points is required.

**Language:**

English, Finnish

**Study methods:**

Final exam

**Literature:**

- Andrew M. Bruckner, Judith B. Bruckner & Brian S. Thomson: Real Analysis, 2008, www.classicalrealanalysis.com
- Avner Friedman: Foundations of Modern Analysis.
- Terence Tao: An Introduction to Measure Theory
- Elias M. Stein & Rami Shakarchi: Real Analysis.