Measure Theory and Integration

Unit aims

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

Unit description

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.

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.

Transferable skills

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

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.

• R. G. Bartle, The Elements of Integration and Lebesque Measure, Wiley Classics Library.
• G. de Barra, Measure Theory and Integration, Ellis Horwood.
• P. Halmos, Measure TheorySpringer-Verlag New York.
• A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional AnalysisDover Publications Inc.

Unit code: MATH30007
Level of study: H/6
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Dr Valeriy Slastikov

Pre-requisites

MATH20006 Metric Spaces

None

Methods of teaching

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

Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

• 100% from a 2 hour 30 minute exam in January.

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.