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MAST90012 (620-645)
Measure Theory

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au

Time and Location:
       Lecture Monday 11:00 - 12:00 Richard Berry 215
       Lecture Wednesday 11:00-12:00 Richard Berry 215
       Lecture Thursday 11:00-12:00 Richard Berry 215

Subject Outline

Measure Theory formalises and generalises the notion of integration. It is fundamental to many areas of mathematics and probability and has applications in other fields such as physics and economics. Students will be introduced to Lebesgue measure and integration, signed measures, the Hahn-Jordan decomposition, the Radon-Nikodym derivative, conditional expectation, Borel sets and standard Borel spaces, product measures, and the Riesz representation theorem.

Main Topics

The following is a week by week outline of the course.

  1. Sets, Cardinality, orders, and Zorn's lemma
  2. Topology, compact sets, metric spaces, completeness, uniform continuity
  3. Measures, Caratheodory extension
  4. Integration and measurable functions
  5. Limit theorems for integrals
  6. The Radon-Nikodym theorem
  7. Polish spaces, Borel spaces, Borel sets, measurable sets and measurable functions
  8. Banach spaces
  9. The Reisz Representation theorem
  10. The Stone-Weierstrass theorem
  11. Product measures and Fubini's theorem
  12. Haar measure and ergodic theory

References

  • Notes on the page http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html
  • Wikipedia
  • arXiv
  • MathSciNet
  • Notes on Measure Theory, by Greg Hjorth 2010
  • W. Rudin. Real and Complex Analysis. McGraw – Hill. Third Edition. 1987.
  • P. Halmos. Measure Theory. Springer. 1974.

Week by week topics

  • Week 1, 28 February-4 March 2011: Sets, Cardinality, orders, and Zorn's lemma
  • Week 2, 7-11 March 2011: Topology, compact sets, metric spaces, completeness, uniform continuity
  • Week 3, 14-18 March 2011: Measures, Caratheodory extension
  • Week 4, 21-25 March 2011: Integration and measurable functions
  • Week 5, 28 March - 1 April 2011: Limit theorems for integrals
  • Week 6, 4-8 April 2011: The Radon-Nikodym theorem
  • Week 7, 11-15 April 2011: Polish spaces, Borel spaces, Borel sets, measurable sets and measurable functions
  • Week 8, 18-22 April 2011: Banach spaces
  • Week 9, 2-6 May 2011: The Reisz Representation theorem
  • Week 10, 9-13 May 2011: The Stone-Weierstrass theorem
  • Week 11, 16-20 May 2011: Product measures and Fubini's theorem
  • Week 12, 23-27 May 2011: Haar measure and ergodic theory

Lectures

Use the notes on the page http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html, particularly those under the heading Topological structures.

Also use the Notes on Measure Theory, by Greg Hjorth 2010

Assessment

Up to 40 pages of written assignments (40%: two assignments worth 20% each, due mid and late in semester), a 3 hour written examination (60%, in the examination period).

The plaigiarism declaration is available here.

The first homework consists of the Exercises to Chapters 1,2 and 6 of Rudin, Real and Complex Analysis, McGraw – Hill, Third Edition, 1987. -- These are due 11 April.
The second homework consists of the Exercises to Chapters 3, 4 and 5 of Rudin, Real and Complex Analysis, McGraw – Hill. Third Edition. 1987. -- due 30 May.