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Module Number:
| 14277
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| Module Title: | Measure and Integration Theory |
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Maß- und Integrationstheorie
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| Department: |
Faculty 1 - Mathematics, Computer Science, Physics, Electrical Engineering and Information Technology
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| Responsible Staff Member: | -
Prof. Dr. rer. nat. habil. Wachsmuth, Gerd
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| Language of Teaching / Examination: | English |
| Duration: | 1 semester |
| Frequency of Offer: |
On special announcement
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| Credits: |
8
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| Learning Outcome: | After successfully completing the module, students know the basic concepts of measure and integration theory and are enabled to deal with them confidently. They possess an enhanced capacity for abstraction. |
| Contents: | - Algebras of sets and measures: σ-algebras and their generators, outer measure, pre-measures, measures, Caratheodory condition, extension of pre-measures to measures, Lebesgue measure on R, uniqueness theorems for measures
- Measurable functions and integration: measurable maps and image measures, integration of measurable functions, theorems of Beppo-Levi, Fatou and Lebesgue, measures with densities, theorem of Radon-Nikodym
- Product measures: products of measurable spaces, product measures, Fubini's theorem, Lebesgue measure in n-dimensional space, transformation rule
- Absolute continuity and Lebesgue's differentiation theorem
- Hausdorff measure
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| Recommended Prerequisites: | Knowledge of the content of the modules
- 11103 : Analysis I
- 11104 : Analysis II
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| Mandatory Prerequisites: | None |
| Forms of Teaching and Proportion: | -
Lecture
/ 4 Hours per Week per Semester
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Exercise
/ 2 Hours per Week per Semester
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Self organised studies
/ 150 Hours
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| Teaching Materials and Literature: | - M. Brokate, G. Kersting: Maß und Integral, Birkhäuser, 2011
- H. Bauer: Maß- und Integrationstheorie, de Gruyter, 1990
- J.L. Doob: Measure Theory, Springer, 1994
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| Module Examination: | Prerequisite + Final Module Examination (MAP) |
| Assessment Mode for Module Examination: | Prerequisite:
- Successful completion of homework
Final module examination:
- Written examination, 90 min. OR
- Oral examination, 30 min.
It will be announced in the first course whether the examination will organized in written or oral form. |
| Evaluation of Module Examination: | Performance Verification – graded |
| Limited Number of Participants: | None |
| Part of the Study Programme: | -
Master (research-oriented) /
Mathematics /
PO 2025
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| Remarks: | - Study programme Mathematics M.Sc.: Compulsory elective module in the complex „Stochastics“
- Study programme Mathematik B.Sc.: Compulsory elective module in the complex „Vertiefung“ (to a limited extend)
- Study programme Wirtschaftsmathematik B.Sc.: Compulsory elective module in the complex „Vertiefung“ (to a limited extend)
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| Module Components: | - Lecture: Measure and Integration Theory
- Accompanying exercise
- Related examination
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| Components to be offered in the Current Semester: | |
