Universität Wien

250251 VO Measure and integration theory (2007W)

7.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Details

Language: German

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 03.10. 11:00 - 13:00 Seminarraum
  • Thursday 04.10. 08:00 - 10:00 Seminarraum
  • Wednesday 10.10. 11:00 - 13:00 Seminarraum
  • Thursday 11.10. 08:00 - 10:00 Seminarraum
  • Wednesday 17.10. 11:00 - 13:00 Seminarraum
  • Thursday 18.10. 08:00 - 10:00 Seminarraum
  • Wednesday 24.10. 11:00 - 13:00 Seminarraum
  • Thursday 25.10. 08:00 - 10:00 Seminarraum
  • Wednesday 31.10. 11:00 - 13:00 Seminarraum
  • Wednesday 07.11. 11:00 - 13:00 Seminarraum
  • Thursday 08.11. 08:00 - 10:00 Seminarraum
  • Wednesday 14.11. 11:00 - 13:00 Seminarraum
  • Thursday 15.11. 08:00 - 10:00 Seminarraum
  • Wednesday 21.11. 11:00 - 13:00 Seminarraum
  • Thursday 22.11. 08:00 - 10:00 Seminarraum
  • Wednesday 28.11. 11:00 - 13:00 Seminarraum
  • Thursday 29.11. 08:00 - 10:00 Seminarraum
  • Wednesday 05.12. 11:00 - 13:00 Seminarraum
  • Thursday 06.12. 08:00 - 10:00 Seminarraum
  • Wednesday 12.12. 11:00 - 13:00 Seminarraum
  • Thursday 13.12. 08:00 - 10:00 Seminarraum
  • Wednesday 09.01. 11:00 - 13:00 Seminarraum
  • Thursday 10.01. 08:00 - 10:00 Seminarraum
  • Wednesday 16.01. 11:00 - 13:00 Seminarraum
  • Thursday 17.01. 08:00 - 10:00 Seminarraum
  • Wednesday 23.01. 11:00 - 13:00 Seminarraum
  • Thursday 24.01. 08:00 - 10:00 Seminarraum
  • Wednesday 30.01. 11:00 - 13:00 Seminarraum
  • Thursday 31.01. 08:00 - 10:00 Seminarraum

Information

Aims, contents and method of the course

Although the classical Riemann integral is extremely useful for explicit calculations (due to its close connection with differentiation), it suffers from a number of technical disadvantages (e.g. the difficulty of interchanging integration with limiting processes). Another problem is the fact that relatively few function possess a Riemann integral.

Around 1900 new developments in analysis (especially in the theory of Fourier series) led to a new approach to integration, which was put in final form by Lebesgue. The main difference between Riemann and Lebesgue integration can be described as follows: for the Riemann integral the domain of the function is decomposed into small intervals, on which the function is almost constant, and the integral is approximated by the sum of the areas of the resulting `rectangles'. For the Lebesgue integral the range of the function is divided into small intervals, and the integral is again approximated by sums of areas of rectangles, but the bases of these rectangles are now much more complicated sets than before (in particular they are no longer intervals). In order to pursue this idea one has to develop a concept of `size' or `measure' of quite general sets.

The success of this approach to integration goes well beyond the original problem of integrating functions of one or more real variables: measure theory lies at the root of almost every branch of modern analysis, as well as of probability theory.

Some main topics:

1. Set functions and measures,
2. Measurable functions,
3. The Lebesgue integral,
4. Limit theorems,
5. Fubini's theorem,
6. The Radon Nikodym theorem,
7. Measures on special spaces:
a. Lebesgue measure on Rn: connections with the Riemann Integral, Lebesgue-Stieltjes Integral and differentiation,
b. The integral as a linear functional: the theorem of Riesz about measures and compact spaces,
8. Spaces of integrable functions (Lp-spaces),
9. Probability measures and elementary probability theory.

Assessment and permitted materials

Minimum requirements and assessment criteria

Examination topics

Reading list

1. Taylor, J. C. An introduction to measure and probability. Springer-Verlag, New York, 1997. xviii+299 pp. ISBN: 0-387-94830-9
2. Stroock, Daniel W. A concise introduction to the theory of integration. Second edition. Birkhäuser Boston, Inc., Boston, MA, 1994. viii+184 pp. ISBN: 0-8176-3759-1
3. Bauer, Heinz. Maß- und Integrationstheorie. Second edition. de Gruyter Lehrbuch. [de Gruyter Textbook] Walter de Gruyter & Co., Berlin, 1992. xviii+260 pp. ISBN: 3-11-013626-0
4. Rudin, Walter. Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1

Association in the course directory

MSTM

Last modified: Mo 07.09.2020 15:40