Measure & Ergodic Theory

Assessment

Exam (50%)

An on-campus exam covering all of the intended learning outcomes. The following questions give some idea of the style of the exam.

Practice Exam

Practice Exam Solutions

Coursework (50%)

Each week a take-home test will be assigned. You will submit a solution via Blackboard. There will be ten tests in total. Your top eight test marks will make up your coursework score for the unit.

Weekly Take-Home Tests

Week 1 Coursework Due 1600 on 3rd October

Week 2 Coursework Due 1600 on 10th October

Week 3 Coursework Due 1600 on 17th October

Week 4 Coursework Due 1600 on 24th October

Week 5 Coursework Due 1600 on 31st October

Week 6 is Reading Week. No Coursework

Week 7 Coursework Due 1600 on 14th November

Week 8 Coursework Due 1600 on 21st November

Week 9 Coursework Due 1600 on 28th November

Week 10 Coursework Due 1600 on 5th December

Week 11 Coursework Due 1600 on 12th December

Week 12. No Coursework

Intended Learning Outcomes

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