MSc Pure Mathematics / Course details

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 before applying ergodic theory to other parts of mathematics.

The unit aims to: Introduce the abstract theory of integration with respect to a measure, introduce measure-preserving transformations, and apply ergodic theory to other parts of mathematics.
 

• Recognise, deduce and apply properties of sigma-algebras and measures.
• Construct measures using Caratheodory’s extension theorem and the Riesz representation theorem. 
• Compute integrals of measurable functions.
• Define Lebesgue spaces and deduce whether a given function belongs to a specific Lebesgue space.
• Determine whether transformations are measure-preserving or ergodic.
• Interpret applications of the pointwise ergodic theorem to measure-preserving transformations.
• Distinguish measure-preserving transformations via their dynamical properties.
• Describe applications of ergodic theory to other areas of mathematics.

For weekly problem sheets, feedback will be returned scripts, within a week of submission