# MATS111 Measure and Integration Theory 1 (5 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, 2023-2024

## Description

Lebesgue measure and measurable sets, Lebesgue integral and integrable functions, the connection between Lebesgue integral and Riemann integral, convergence theorems, absolutely continuous functions.

## Learning outcomes

After the course one is able

- to define Lebesgue measure and integral
- to study integrability of a function
- to establish and employ basic properties of Lebesgue measure
- to state, prove, and apply the most important convergence theorems
- to explain the relations between Lebesgue and Riemann integrals

## Description of prerequisites

Introduction to mathematical analysis 3, Vector calculus 2, Vector analysis 1

## Study materials

Luentomoniste

## Literature

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

## Completion methods

### Method 1

**Evaluation criteria:**

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

**Time of teaching:**

Period 1

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 (5 cr)

**Type:**

Participation in teaching

**Grading scale:**

0-5

**Evaluation criteria:**

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

**Language:**

Finnish

**Study methods:**

28h lectures, 7 exercise sessions

**Study materials:**

Luentomoniste

**Literature:**

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

#### Teaching

##### 9/1–10/23/2022 Lectures

##### 10/26–10/26/2022 Exam

##### 11/9–11/9/2022 Exam

x

### Exam (5 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

**Study materials:**

Luentomoniste

**Literature:**

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