Course unit details:
Measure Theory and Ergodic Theory
Course unit fact file
Unit code |
MATH61021 |
Credit rating |
15 |
Unit level |
FHEQ level 7 – master's degree or fourth year of an integrated master's degree |
Teaching period(s) |
Semester 1 |
Available as a free choice unit? |
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.
Assessment methods
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 |
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