# MSc Pure Mathematics and Mathematical Logic / Course details

Year of entry: 2024

## Course unit details:Measure Theory and Ergodic Theory

Unit code MATH61021 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 1 No

### Overview

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.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Real Analysis A MATH20101 Pre-Requisite Recommended
Real Analysis B MATH20111 Pre-Requisite Recommended
Metric Spaces MATH21111 Pre-Requisite Compulsory

### Aims

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.

### Learning outcomes

• 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.

Method Weight
Written exam 50%
Set exercise 50%

### Feedback methods

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

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 117

### Teaching staff

Staff member Role
Donald Robertson Unit coordinator