# MATS111 Measure and Integration Theory 1 (5 cr)

Study level:
0-5
Language:
English, Finnish
Responsible organisation:
Department of Mathematics and Statistics
Curriculum periods:
2020-2021, 2021-2022, 2022-2023

## 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

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
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.

x

### Exam (5 cr)

Type:
Exam
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.