Entry requirements

120 credits including Measure and Integration Theory I. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

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

• describe and use different convergence concepts for functions such as convergence in measure, almost everywhere and in L^p;
• give an account of L^p 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 L^p. L^p as a normed space. The dual of L^p. 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.

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.