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.


Copyright Cardiff University. Registered charity no. 1136855