Breadcrumb

Measure Theory and Integration (MATH 34000)

Academic Year:

Contents of this document:

Administrative Information

  1. Unit number and title: MATH 34000 Measure Theory and Integration
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1999 (as Analysis 3)
  6. Unit Organiser: Thomas Jordan
  7. Lecturer: Thomas Jordan
  8. Teaching block: 1
  9. Prerequisites: MATH 20200 Metric Spaces

Unit aims

The aim of the unit is to introduce measure theory and the Lebesgue integral.

General Description of the Unit

The course introduces the Lebesgue integral and develops the elements of measure theory. We will, (i) generalise the notions of "length", "area" and "volume", (ii) find out which functions can be integrated, and (iii) prove the main properties of the Lebesgue integral.

Relation to Other Units

This unit is an element of a sequence of Analysis courses at Levels C/4, I/5, H/6 and 7/M. It is a prerequisite for Advanced Topics in Analysis (not currently running).

Teaching Methods

A standard lecture course of 30 lectures, 3 revision classes and problem classes.

Learning Objectives

At the end of the course the student should know and understand the definitions and theorems (and their proofs), and should be able to use the ideas of the course in unseen situations.

Assessment Methods

The final assessment mark for the unit is calculated from a standard 2 ½-hour written closed-book examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are not permitted.

Award of Credit Points

Credit points are gained by:

  • either passing the examination (pass mark: 40),
  • or getting a final assessment mark of 30 or over and also making satisfactory attempts for at least three of the assigned homeworks. 

Transferable Skills

Assimilation of abstract ideas and reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.

Texts

  • R. G. Bartle, The Elements of Integration and Lebesque Measure, Wiley Classics Library,
  • G. de Barra, Measure Theory and Integration, Ellis Horwood.

 

Syllabus

Extended Real Number Theory, Measureable Functions, Measures, The Integral, Integrable Functions, Lp spaces, Modes of Convergence, Decompostion of Measures, Generation of Measures, Product Measures, Approximation of Measureable Sets, Non-Borel sets.