Measure Theory

This subject has the following teaching availabilities in 2010:

It is recommended that students have completed a third year subject in metric spaces, measure and integral (equivalent to 620-311 [2008] Metric Spaces and 620-312 [2008] Linear Analysis).

For the purposes of considering requests for Reasonable Adjustments under the Disability Standards for Education (Cwth 2005), and Students Experiencing Academic Disadvantage Policy, academic requirements for this subject are articulated in the Subject Description, Subject Objectives, Generic Skills and Assessment Requirements for this entry.

The University is dedicated to provide support to those with special requirements. Further details on the disability support scheme can be found at the Disability Liaison Unit website: http://www.services.unimelb.edu.au/disability/

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. Hahn-Jordan decomposition. Radon-Nikodym derivative. Conditional expectation. Borel sets and standard Borel spaces. Product measures. The Riesz representation theorem. The Krein-Milman theorem. The Stone-Weierstrass theorem. The measure disintegration theorem. Ergodic theory.

After completing this subject, students will:

• understand the fundamentals of measure theory and have an understanding of how these;
• underpin the use of mathematical concepts such as volume, area, and integration;
• learn an advanced description of the basic notion of integration;
• develop a perspective on the broader impact of measure theory in ergodic theory; and
• have the ability to pursue further studies in this and related areas.

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).

This subject is not available as a breadth subject.

Upon completion of this subject, students should gain the following generic skills:

• the ability to think abstractly;
• problem-solving skills including engaging with unfamiliar problems and identifying relevant strategies;
• analytical skills including the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of an analysis; and
• through interactions with other students, the ability to work in a team.