MA4007: Measure Theory
School | Cardiff School of Mathematics |
Department Code | MATHS |
Module Code | MA4007 |
External Subject Code | G100 |
Number of Credits | 20 |
Level | L7 |
Language of Delivery | English |
Module Leader | Professor Nicolas Dirr |
Semester | Spring Semester |
Academic Year | 2014/5 |
Outline Description of Module
This module provides an introduction to the basic ideas and concepts of measure theory and integration. Measure theory is indispensable for almost all studies in higher mathematics and especially for all subjects based on Hilbert spaces and linear operators (e.g., Quantum mechanics, spectral theory and quantum field theory, representation theory, operator algebras and many more.)
On completion of the module a student should be able to
- Know and understand the concept of a sigma-algebra, a pre-measure and a measure;
- Know and understand the concept of the Lebesgue-Borel measure
- Know and understand the concept of almost everywhere prevailing properties
- Understand the Radon-Nikodym theorem
- Know and understand products measures and Fubini's theorem
How the module will be delivered
30 - 50 minute lectures
Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.
Students are also expected to undertake at least 120 hours private study including preparation of worked solutions for problem classes.
Skills that will be practised and developed
Skills:
The ability to understand courses which require measure theory as a prerequisite, e.g., functional analysis.
Transferable Skills:
Appreciation and understanding of measure theory; ability to understand mathematical concepts based on integration theory.
How the module will be assessed
Formative assessment is carried out in the problem classes. Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.
The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.
The examination paper has a choice of four from five equally weighted questions.
Assessment Breakdown
Type | % | Title | Duration(hrs) |
---|---|---|---|
Exam - Spring Semester | 85 | Measure Theory | 3 |
Written Assessment | 15 | Coursework | N/A |
Syllabus content
- Sigma-algebras, generators,Dynkin systems
- Contents, premeasures, measures
- Lebesgue premeasure
- Extension of a premeasure to a measure
- Lebesgue-Borel measure and measures on the real line
- Measurable mappings and image measures
- Measurable numerical functions
- Integrability
- Almost everywhere prevailing properties
- Convergence theorems
- Radon-Nikodym theorem
- Signed measures and complex valued measures
- Stochastic convergence
- Product measures
- Fubini's theorem
Essential Reading and Resource List
Measure and Integration Theory, Bauer, H., De Gruyter, W.
Measure Theory, Halmos, P., Springer
A Radical Approach to Lebesgue's Theory of Integration, Bressoud, D. M., Mathematical Association of America Textbooks
Background Reading and Resource List
Not applicable.