NWI-WB088
Measure Theory
Course infoSchedule
Course moduleNWI-WB088
Credits (ECTS)6
CategoryBA (Bachelor)
Language of instructionDutch
Offered byRadboud University; Faculty of Science; Wiskunde, Natuur- en Sterrenkunde;
Lecturer(s)
Coordinator
prof. dr. H.T. Koelink
Other course modules lecturer
Lecturer
prof. dr. H.T. Koelink
Other course modules lecturer
Contactperson for the course
prof. dr. H.T. Koelink
Other course modules lecturer
Examiner
prof. dr. H.T. Koelink
Other course modules lecturer
Academic year2023
Period
KW3-KW4  (29/01/2024 to 31/08/2024)
Starting block
KW3
Course mode
full-time
Remarks-
Registration using OSIRISYes
Course open to students from other facultiesYes
Pre-registrationNo
Waiting listNo
Placement procedure-
Aims
  • The student is familiar with sigma-algebra's, positive and complex measures, the most important constructions of measures and connections between measures such as the Radon-Nikodym theorem, the Hahn and Jordan decomposition.
  • The student understands integrability and related convergence theorems, e.g. monotone convergence theorem, Fatou's Lemma, Lebesgue's dominated convergence theorem..
  • The student understands product measures and Fubini's theorem.
  • The student is familiar with applications in probability theory.
  • The student is able to communicate clearly and in a substantiated way, in particular in setting up proofs of relevant statements.
Content

In the courses Analyse 1 and Analyse 2 you have become familiar with the Riemann integral on R and the Lebesgue integral. The Riemann integral, although natural and very important in applications, is not sufficient for several other areas in mathematics. The reason for this is that the definition is limited to functions on bounded domains, and that extensions to more general domains are described via limit transitions. The Riemann integral relates poorly to limits, e.g. interchanging integrals, differentiating under the integral, interchanging limit and integral for a sequence of functions, etcetera.

At the beginning of the 20th century Lebesgue developed a theory of integration, which was developend into an abstracted theory of integration based on general measure theory. Lebesgue's approach is an extension of the Riemann integral, which overcomes all the disadvantages mentioned above. After an introduction to the underlying structures of measure theory the corresponding integral theory is developed. Important theorems concerning convergence are discussed. This approach is important for other areas in mathematics, such as functional analysis (e.g. completeness of certain spaces of integrable functions), group theory (e.g. Haar measures on Lie groups) and probability theory. In the course we pay attention to applications in the latter area.

The goal of the course is to become familiar with the most important concepts in measure theory and the corresponding integration theory, in particular the convergence theory.

Level

Presumed foreknowledge
Analysis 1, Analysis 2, intro of Topology.
Test information
Written exam
Specifics

Required materials
Book
Donald L. Cohn: Measure theory. Second edition. Birkhäuser/Springer, 2013
ISBN:9781461469551

Instructional modes
Course

Tests
Exam
Test weight1
Test typeExam
OpportunitiesBlock KW4, Block KW4