Measure and Integration Theory I

5 credits

Syllabus, Master's level, 1MA049

This course has been discontinued.

A revised version of the syllabus is available.

Code: 1MA049
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)
Finalised by: The Faculty Board of Science and Technology, 6 November 2007
Responsible department: Department of Mathematics

Entry requirements

120 credit points and 90 credit points Mathematics

Learning outcomes

In order to pass the course (grade 3) the student should

be able to use the measurability concept for functions and sets;

be able to use the "almost everywhere" concept;

know the construction of the Lebesgue integral and how to use it;

know how to use the theorems about monotone and dominated convergence, and Fatou's lemma;

be familiar with the construction of product measures;

know how to use Fubini's theorem;

be familiar with the properties of absolute continuity and singularity for measures and know how to use Lebesgue decomposition and Radon–Nikodym's theorem.

Content

Sigma algebras. Measure and exterior measure. Lebesgue measure in one and several dimensions. Measurability of functions. The Lebesgue integral and its relation to the Riemann integral. Absolute continuous and singular measures. Lebesgue decomposition and Radon–Nikodym's theorem, Radon–Nikodym derivative.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.