Measure and Integration Theory II
Syllabus, Master's level, 1MA050
This course has been discontinued.
- Code
- 1MA050
- Education cycle
- Second cycle
- Main field(s) of study and in-depth level
- Mathematics A1F
- Grading system
- Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
- Finalised by
- The Faculty Board of Science and Technology, 29 May 2013
- Responsible department
- Department of Mathematics
Entry requirements
120 credit points including Measure and Integration Theory I
Learning outcomes
In order to pass the course (grade 3) the student should be able to
- describe and use different convergence concepts for functions such as convergence in measure, almost everywhere and in Lp;
- give an account of Lp spaces and their duals;
- use the inequalities of Hölder and Minkowski;
- define and use extended real-valued and complex measures;
- describe Riesz's representation theorem;
- give an account of regular and complete measures;
- define the absolute continuous and singular measures and be able to use Lebesgue decomposition and the Radon-Nikodym theorem;
- give an account of bounded variation and absolute continuity and be able to explain their role in connection with differentiation of functions.
Content
Convergence in measure, almost everywhere, and in Lp. Lp as a normed space. The dual of Lp. Hölder's and Minkowski's inequalities. Real-valued, extended real-valued and complex measures. Complete measures. Regular measures. Riesz's representation theory. Absolutely continuous and singular measures. Lebesgue decomposition and the Radon-Nikodym theorem. The Radon-Nikodym derivative. Functions of bounded variation. Differentiation of measures and functions. Absolute continuous functions.
Instruction
Lectures and problem solving sessions.
Assessment
Written examination at the end of the course combined with assignments given during the course.