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School of Mathematics

# MATH30131 - 2006/2007

General Information
• Title: Measure Theory
• Unit code: MATH30131
• Credits: 10
• Prerequisites: MT1101, MT1202
• Co-requisite units:
• School responsible: Mathematics
• Member of staff responsible: Prof Richard Sharp
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## Specification

### Aims

To give an introduction to Lebesgue measure on R. To show how this theory leads to the Lebesgue integral on R, and to introduce the concept of Hausdorff dimension of sets in Rn.

### Brief Description of the unit

Riemann created the integral named after him (familiar to students of MT2222) in 1854. The great achievement of the Riemann integral is that it integrates any continuous function defined on a bounded interval in R. But the Riemann theory has the deficiency that it does not behave well with respect to taking limits.

In 1902, in his thesis, H. Lebesgue introduced the integral that bears his name. His key idea was to extend the notion of length from intervals to more complicated sebsets of R (and Rn). His integral integrates any function which is Riemann integrable, and also has good limit properties.

In this course unit we will give a modern treatment of Lebesgue's theory. Although much of the theory can now be done in much more generality than was the case in Lebesgue's time, this course will be focused on the real line setting. The material is proof oriented and should appeal to students who have successfully taken MATH20222 (Real Analysis). However, MATH20222 is not a prerequisite for the course, and any ideas required that are not in first or second year core course units will be reviewed when needed.

The course unit should be useful to students taking probability course units in years three and four since the ideas of measure and integral have a central role in probability theory.

### Learning Outcomes

On successful completion of the course unit the students will be able to

• understand how Lebesgue measure on R   is constructed.
• understand the notions of measurable functions on R
• know how Lebesgue integration is derived from Lebesque measure
• know how to manipulate integrals and use the basic theorems
• understand the notion of Hausdorff dimension of sets in Rn, and be able to calculate it for simple examples.

### Future topics requiring this course unit

It would be helpful for level 3 and 4 courses in probability.

### Syllabus

• Lebesgue measure on R, measurable sets, Borel sets, non-measurable sets. Informal discussion of extension to Rn, concept of a measure space. [9]
• Measurable functions and their properties, simple functions, measurable functions as limits of simple functions. Integral of (i) simple (ii) non-negative measurable functions, and (iii) general integrable functions. Limit theorems of integration. [9]
• Hausdorff measure and dimension, properties of Hausdorff dimension, calculation of Hausdorff dimension of Cantor set, etc. [6]
Textbooks
M. Capinski and E. Kopp Measure, Integral and Probability Springer (SUMS Series), 1999.
H.S. Bear A Primer of Lebesgue Integration Academic Press, 1995.

### Teaching and learning methods

Two lectures a week and a weekly examples class

Assessment
Two hours end of semester examination; Weighting within unit 100%

## Arrangements

Online course materials are available for this unit.

Last modified: October 05, 2010 5:47:58 PM BST.