# MATS112 Measure and Integration Theory 2 (4 cr)

Study level:
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

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
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.

x

### Exam (4 cr)

Type:
Exam
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.