# Syllabus for Integration Theory

Integrationsteori

## Syllabus

• 10 credits
• Course code: 1MA215
• Education cycle: Second cycle
• Main field(s) of study and in-depth level: Mathematics A1N
• Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
• Established: 2012-03-08
• Established by:
• Revised: 2018-08-30
• Revised by: The Faculty Board of Science and Technology
• Applies from: week 24, 2019
• Entry requirements: 120 credits including 90 credits in mathematics with Real Analysis.
English language proficiency that corresponds to English studies at upper secondary (high school) level in Sweden ("English 6").
• Responsible department: Department of Mathematics

## Learning outcomes

On completion of the course, the student should be able to:

• use the concepts of measurable set and measurable function;
• state and explain the construction of the Lebesgue integral and use it;
• apply the theorems of monotone and dominated convergence and Fatou's lemma;
• describe the construction of product measure and to apply Fubini's theorem;
• state and explain properties of Lp spaces;¨
• define absolute continuity and singularity of measures;
• apply Lebesgue decomposition and the Radon-Nikodym theorem.

## Content

Measure and outer measure. Lebesgue measure in one and higher dimensions. Measurable functions. Lebesgue integral and convergence theorems. The connection with the Riemann integral. Product measure and Fubini's theorem. L^1 and L2 theory. Hilbert space. Fourier series and Fourier integrals. Convergence of Fourier series in L2 norm. Convergence in measure, almost everywhere and in Lp. Lp as a normed space. Hölder's and Minkowski's inequalities. Absolute continuity and singularity of measures. Lebesgue decomposition and the Radon-Nikodym theorem. Radon-Nikodym derivative.

## Instruction

Lectures and ptoblem solving sessions.

## Assessment

Written examination at the end of the course in combination with homework assignments according to instructions given at the course start.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

## Other directives

The course cannot be included in higher education qualification together with Measure theory and Integration theory I and II or equivalent courses.