MAGIC116: Measure Theory and Ergodic Theory

Course details

A core MAGIC course

Semester

Autumn 2025
Monday, October 6th to Friday, December 12th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
80

Timetable

Mondays
10:05 - 10:55 (UK)
Wednesdays
12:05 - 12:55 (UK)

Description

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.

Prerequisites

Familiarity with undergraduate real analysis and metric spaces

Syllabus

  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

Lecturer

  • DR

    Donald Robertson

    University
    University of Manchester

Bibliography

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 12th January 2026 at 00:00 and is due in before Friday 23rd January 2026 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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