This syllabus is valid: 2007-09-03
and until further notice
Course code: 5MA068
Credit points: 7.5
Education level: Second cycle
Main Field of Study and progress level:
Mathematics: Second cycle, has second-cycle course/s as entry requirements
Grading scale: Three-grade scale
Responsible department: Department of Mathematics and Mathematical Statistics
Contents
The course covers the following topics: Abstract measure- and integration theory, measure- and integration theory on topological spaces and the Lebesgue measure and Lebesgue integral on the n-dimensional space, sigma algebras, measure and outer measure, complete and regular measures, Lebesgue measure in one and several variables, measurable functions and the Lebesgue integral. The second part of the course consists of convergence in measure, almost-everywhere and in Lp-space, absolute continuous and singular measures, Lebesgue decomposition and extended real valued and complex measures. Finally, Hölders and Minkowskis inequalities, Radon-Nikodyms theorem and Riesz representation theorem are treated.
Expected learning outcomes
After the course the students are expected to be able to:
define and understand basic notions in abstract integration theory, integration theory on topological spaces and the n-dimensional space
describe and apply the notion of measurable functions and sets and use Lebesgue monotone and dominated convergence theorems and Fatous Lemma
describe the construction of and apply the Lebesgue integral
describe the construction of product measures and use Fubinis theorem
describe the notion of absolute continuity and singularities of measures and apply Lebesgue decomposition and the Radon-Nikodym theorem
apply Hölders and Minkowskis inequalities and describe Riesz representation theorem
describe the notion of extended real valued and complex measures.
Required Knowledge
Required knowledge to be able to take the course is the course Real analysis (5MA020) or corresponding knowledge.
Form of instruction
Teaching is mainly given by lectures and supervision.
Examination modes
The examination is performed by written exam(s) and/or oral presentations. Possible grades for the course are: failed, passed or passed with distinction.The student has to pass all examinations and compulsory parts of the course in order to have passed the complete course. Those who have received the grade passed at the course cannot be reexaminated in order to achieve a higher grade.
For students that did not pass the first exam, a new opportunity for examination will be arranged. Students that failed exam twice, are entitled to reqest another teacher to grade the course. This request should be posed to the Head of the Department of Mathematics and Mathematical Statistics.
Examination based on this course syllabus is guaranteed at least two years after the student was first registered on the course.
Literature
The literature list is not available through the web.
Please contact the faculty.
