MA947-15 Graduate Real Analysis
Introductory description
Introduce the fundamental concepts and results of measure theory and to present more advanced topics not usually covered in the undergraduate courses.
Module aims
Measures, integration, basic properties and convergence theorems. Lebesgue measure.
Egorov, Lusin and Fubini theorems.
Riesz representation theorem and weak* convergence.
Lebesgue density theorem, almost everywhere differentiation of monotone functions, Rademacher’s theorem.
Hausdorff measure, rectifiable sets.
Selected topics in Geometric Measure Theory, for example, Sard's theorem, Frostman measures, Besicovitch projection theorem
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Measures, integration, basic properties and convergence theorems. Lebesgue measure.
Egorov, Lusin and Fubini theorems.
- Riesz representation theorem and weak* convergence.
- Lebesgue density theorem,
- almost everywhere differentiation of monotone functions,
- Rademacher’s theorem.
- Hausdorff measure, rectifiable sets.
- Selected topics in Geometric Measure Theory, for example, Sard's theorem, Frostman measures, Besicovitch projection theorem
Learning outcomes
By the end of the module, students should be able to:
- Measures, integration, basic properties and convergence theorems. Lebesgue measure. Egorov, Lusin and Fubini theorems. Riesz representation theorem and weak* convergence. Lebesgue density theorem, almost everywhere differentiation of monotone functions, Rademacher’s theorem. Hausdorff measure, rectifiable sets. Selected topics in Geometric Measure Theory, for example, Sard's theorem, Frostman measures, Besicovitch projection theorem
Indicative reading list
Rudin, W.: Real and Complex Analysis
Loeb, P.A: Real Analysis
Halmos, P. R.: Measure Theory
Subject specific skills
Develop a deep understanding and applicability of the following topics:
- Measures,
- integration,
- basic properties and convergence theorems.
- Lebesgue measure.
- Egorov, Lusin and Fubini theorems.
- Riesz representation theorem and weak* convergence.
- Lebesgue density theorem,
- almost everywhere differentiation of monotone functions,
- Rademacher’s theorem.
- Hausdorff measure, rectifiable sets.
- Sard's theorem, Frostman measures, Besicovitch projection theorem
Transferable skills
- sourcing research material
- prioritising and summarising relevant information
- absorbing and organizing information
- presentation skills (both oral and written)
Study time
| Type | Required |
|---|---|
| Lectures | 30 sessions of 1 hour (100%) |
| Total | 30 hours |
Private study description
Review lectured material.
Work on suplementary reading material.
Source, organise and prioritise material for additional reading.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group A
| Weighting | Study time | |
|---|---|---|
| Oral Exam | 100% | |
|
An oral exam involving a presentation by the student, followed by questions from the panel (2 members of the department) |
||
Feedback on assessment
Students will receive feedback from the course instructor after the oral exam, to cover also areas like presentation skills and use of technologies (or blackboard)
There is currently no information about the courses for which this module is core or optional.