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Syllabus:

# Integration Theory, 7.5 Credits

Swedish name: Integrationsteori

This syllabus is valid: 2017-08-21 and until further notice

Course code: 5MA183

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematics: Second cycle, has only first-cycle course/s as entry requirements
Computational Science and Engineering: Second cycle, has only first-cycle course/s as entry requirements

Responsible department: Department of Mathematics and Mathematical Statistics

Established by: Faculty Board of Science and Technology, 2017-12-12

## Contents

The course covers notions and theorems in measure- and integration theory. The non-negative measures are defined on general sigma algebras. The notion measurability is central in the course. Measurable functions with respect to a given sigma algebra, are studied. Radon-Nikodym's theorem, Fubini's theorem and the convergence theorems are central in the course.

## Expected learning outcomes

For passing a grade, the student must be able to

Knowledge and understanding

• thoroughly account for definitions and theorems cocerning measure- and integration theory
• thoroughly account for the notions measurability and integrability

Skills

• independently construct neasure rooms and functions with given regularity properties
• independently conduct formal proofs of theorems in measure- and integration theory
• independently decide the measurability of sets and functions

## Required Knowledge

The course requires 90 ECTS including 22,5 ECTS in Calculus of which 7,5 ECTS in Multivariable Calculus and Differential Equations and a course in Real Analysis or equivalent. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

The teaching consists of lectures and lessons.

## Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

## Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.

## Literature

### Valid from: 2018 week 3

Cohn Donald L.
Measure theory
2. ed. : New York, NY : Springer New York : 2013 : xxi, 457 s. :
http://www.loc.gov/catdir/enhancements/fy1317/2013934978-d.html
ISBN: 9781461469551 (Print)
Mandatory
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