Faculty of Science AMSI Summer School 2013

Measure theory

Marty Ross


Synopsis

Please contact me if you have any questions about the outline below (martinirossi@gmail.com).

Measure theory is the modern theory of integration, the method of assigning a "size" to subsets of a universal set. It is more general, more powerful and more beautiful (though also more technical) than the classical theory of Riemann integration.

This subject will be a relatively standard introduction to measure theory, with some emphasis on geometric aspects and applications. More generally, the subject will be a revision of (or an introduction to) the results and techniques of real analysis. In broad outline, the topics we intend to cover are:


Contact hours

28 hours spread over the four weeks, with consultation a requested/required.


Prerequisites

We will assume some familiarity with the fundamental concepts of analysis on the real line and in Euclidean Space (infs and sups, convergent sequences, open and closed sets, continuity, completeness and compactness, countability). Corresponding familiarity with these notions in metric spaces would be very helpful but will not be assumed; familiarity with these notions in topological spaces would be dandy.


Assessment

Problems assigned during lectures (50%), plus a take home exam (50%).


Background (pre-summer school) reading

Lecture notes summarising the relevant background on sets and real analysis are available here. Some of this material will be reviewed along the way, however it would definitely be worthwhile taking a good look at the background notes before the start of the Summer School. If it seems appropriate, you may also wish to browse through a real analysis text or two (e.g. Calculus by Spivak, Elementary Analysis by (not Marty) Ross, and Understanding Analysis by (not Tony) Abbott).


Resources

Lecture notes will be provided. There are also many excellent books on analysis and measure theory, which would be good to grab from your library. Some good general texts are: Real Analysis by Royden; Foundations of Real and Abstract Analysis by Bridges; Measure Theory and Integration by de Barra; and An Introduction to Measure and Integration by Rana. For the later, geometric parts of the subject you can take a look at: Geometry of Sets and Measures in Euclidean Space by Mattila; The Geometry of Fractal Sets by Falconer; and Measure Theory and Fine Properties of Functions by Evans and Gariepy. Texts with an emphasis on probability will be less useful, as the language and approach tend to be quite different.


About marty Ross

Marty Ross is a mathematical nomad. He is an American (sort of), who grew up in Australia and received his PhD in minimal surfaces from Stanford University. He has lectured at Melbourne, Monash and La Trobe Universities. When not wandering in the woods, Marty spends a lot of time with his colleagues Burkard Polster and QEDcat, popularizing mathematics and whacking windmills.

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