### Measure Theory (MATH4405)

The University of Queensland (Semester 2, 2010)

### random chords applet

This applet illustrates a result by Martin Bridgeman and David Dumas; measure theory and ergodic theory are at its heart!
Press Add Chords many times and watch the show.

### materials

I have the generous permission by Greg Hjorth to use his notes on Measure Theory.

You can also watch all of Marty Ross's AMSI lectures in 2006 by following the link on this page.

We participate in the CARMA-AMSI short course on Haar measure given by Joe Diestel. Lecture notes and other materials can be found here.

Measure Theory on Eric Weisstein's World of Mathematics
Haar Measure on Eric Weisstein's World of Mathematics
Ergodic Theory on Eric Weisstein's World of Mathematics

### course description

I intend to follow Greg's notes, augmenting and subtracting material from time to time.

Ideally, I'd like to cover the following notions, topics and examples:

Notions: Inner and outer measures; Vitali covering theorem, Lebesgue measure, Hausdorff dimension. Change of variables. Proofs by approximation. Convergence theorems. Radon-Nikodym theorem and conditional approximation. Product measures and Fubini-Tonelli theorem.

Selected topics from ergodic theory (such as invariant measures, Haar measure, Poincaré recurrence theorem) and from geometric measure theory (such as the area and co-area formulas).

Examples: Hamiltonian dynamics, Bernoulli shifts, rotations of the circle, horocycle flow on a hyperbolic surface, Hausdorff dimension of a fractal set.

This is, of course, more material than can be covered in 13 weeks, but it gives us a menu to choose from.

### learning activities

Mo 09:00-09:50 67-641 (lecture)
Tu 11:00-11:50 67-641 (lecture/tutorial)
Th 14:00-15:50 67-442 (lecture/tutorial))

### consultation hours

WE 11:00-11:50 67-710
FR 10:00-10:50 67-710

Problem Set 1
Problem Set 2
Problem Set 3
Problem Set 4
Problem Set 5
Problem Set 6

### assignments (pdf)

Assignment 1 (due Thursday, 12 August)
Assignment 2 (due Monday, 6 September)
Assignment 3 (due Monday, 4 October)
Assignment 4 (due Monday, 25 October)

### final exam

TBA
(10 min perusal; 120 min duration)

### general references

"Real Analysis" by Halsey L. Royden (Wiley, 1989)
"Real and Complex Analysis" by Walter Rudin (McGraw-Hill, 1986)
"Counterexamples in analysis" by Bernard R. Gelbaum and John M.H. Olmsted (Holden-Day, 1964)

### specialised references

"Measure Theory and Fine Properties of Functions" by Lawrence C. Evans and Ronald F. Gariepy (CRC Press, 1992)
"Geometric measure theory: a beginner's guide" by Frank Morgan (Academic Press, 1988)
"Ergodic theory" by Karl Petersen (Cambridge University Press, 1983)
"The geometry of fractal sets" by K.J. Falconer (Cambridge University Press, 1984)