Course Syllabus for

# Integration Theory Integrationsteori

## MATM19F, 7.5 credits

Valid from: Autumn 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-08-24

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: MATM19
Teaching language: English

## Aim

For many mathematical investigations the notion of integrability in the sense of Riemann, which is used in the basic courses, is insufficient. Above all, it is difficult to guarantee that the limit of a sequence of Riemann integrable functions is an integrable function. The aim of the course is to acquaint the postgraduate student with the Lebesgue integral, and important theorems valid for it. This theory is indispensable for researchers in, e.g., mathematical analysis, numerical analysis or stochastic processes.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must be able to account for basic concepts and methods within theory of integration.

Competences and Skills

For a passing grade the doctoral student must

• be able to use the theorems in the course to solve mathematical problems, in particular such that are relevant for applications such as if it is permitted to change the order of integration in iterated integrals, and if the integral of the limit of a given sequence of functions equals the limit of the integrals of the functions in the sequence,
• be able to formulate and prove the main theorems in the course.

## Course Contents

Basic theory of Lebesgue integration: basic measure theory, construction of the Lebesgue measure, convergence theorems and Fubini's theorem.

## Course Literature

Cohn, Donald L.: Measure Theory: Second Edition. BirkhĂ¤user, 2013. ISBN 9781461469551.

## Instruction Details

Types of instruction: Lectures, seminars

## Examination Details

Examination formats: Written exam, oral exam, written assignments