MAGIC116: Measure Theory and Ergodic Theory

An introduction to measure theory and ergodic theory.

Measure theory – an abstraction of what it means to assign size to objects – underpins most of modern analysis and probability, serving as a foundation for such diverse topics as harmonic analysis, stochastic differential equations, fractal geometry and ergodic theory. It came to prominence at the beginning of the 20th century when Lebesgue used it to develop an entirely new approach to integration that overcame the deficiencies of older integrals.

Ergodic theory lies at the confluence of dynamical systems and measure theory. The abstract nature of measure theory has led to breakthrough applications of ergodic theory in many different branches of mathematics. For example, ergodic theory played a central role in the proof of the Green-Tao theorem on arithmetic progressions in primes, and in Margulis’s resolution of the Oppenheim conjecture.

In this course we will learn about the abstract theory of measures and the theory of integration that sits on top of it. Then we will cover examples and properties of measure-preserving transformations and the pointwise ergodic theorem and applyi ergodic theory to other parts of mathematics.

Familiarity with undergraduate real analysis and metric spaces

1. σ-algebras and measures
2. Constructing measures
3. Measurable functions and integration
4. Lebesgue spaces
5. Fubini and disintegration
6. Examples of ergodic theorems
7. Measure-preserving transformations and ergodicity
8. The pointwise ergodic theorem
9. The Koopman operator
10. Entropy and Markov chain