Measure and Integration Theory II

5 credits

Syllabus, Master's level, 1MA050

This course has been discontinued.

A revised version of the syllabus is available.

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, 6 November 2007
Responsible department: Department of Mathematics

Entry requirements

120 credit points and Measure and Integration Theory I

Learning outcomes

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

know different convergence concepts for functions such as convergence in measure, almost everywhere and in L^p;

be familiar with L^p as a normed space and with its dual;

know how to use the inequalities of Hölder and Minkowski;

be familiar with extended real-valued and complex measures;

know Riesz's representation theorem;

know the definition and use of regular and complete measures;

be able to give an account of the concepts of bounded variation and absolute continuity, and 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. Riesz's representation theory. Functions of bounded variation. Differentiation of measures and functions. Absolute continuous functions. Complete measures. Regular measures.

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.