Master’s studies

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), 3, 4, 5
  • Established: 2012-03-08
  • Established by: The Faculty Board of Science and Technology
  • Revised: 2015-04-09
  • Revised by: The Faculty Board of Science and Technology
  • Applies from: week 30, 2015
  • Entry requirements: 120 credits including 90 credits in mathematics with Real Analysis.
  • Responsible department: Department of Mathematics

Learning outcomes

In order to pass 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.

Other directives

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

Reading list

Reading list

Applies from: week 30, 2015

  • Adams, Malcolm; Guillemin, Victor Measure theory and probability

    [New ed.]: Boston: Birkhäuser, cop. 1996

    Find in the library

    Mandatory

  • Rudin, Walter Real and complex analysis

    3. ed.: New York: McGraw-Hill, 1986

    Find in the library