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# Measure & Integration - MAT00087H

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• Department: Mathematics
• Module co-ordinator: Prof. Victor Beresnevich
• Credit value: 20 credits
• Credit level: H
• Academic year of delivery: 2023-24
• See module specification for other years: 2024-25

## Module summary

This module will introduce measure theory and Lebesgue integration. It will develop powerful tools of the theory of Lebesgue integration and will demonstrate that the Lebesgue integral can be computed by familiar methods whenever they are applicable. Amongst various examples of measures it will construct Lebesgue measure which extends the familiar notions of length, areas and volume, and explore advanced topics such as the Radon-Nikodym derivative and Lebesgue’s density theorem.

## Related modules

### Prohibited combinations

• None

For 2023/24 Metric Spaces MAT00051I is not required as a co-requisite.

## Module will run

Occurrence Teaching period
A Semester 1 2023-24

## Module aims

This module will introduce measure theory and Lebesgue integration. It will develop powerful tools of the theory of Lebesgue integration and will demonstrate that the Lebesgue integral can be computed by familiar methods whenever they are applicable. Amongst various examples of measures it will construct Lebesgue measure which extends the familiar notions of length, areas and volume, and explore advanced topics such as the Radon-Nikodym derivative and Lebesgue’s density theorem.

## Module learning outcomes

At the end of the module students should be able to:

1. Use measure space and apply their properties in examples.

2. Apply Caratheodory's criterion in the context of the construction and properties of Lebesgue measure and measurability of Borel sets

3. Demonstrate knowledge of the construction of the Lebesgue integral and its key properties.

4. Compute Lebesgue integrals using the Fundamental Theorem of Calculus, Monotone and Dominated Convergence Theorems, and the Tonelli and Fubini Theorems.

## Module content

• Measure spaces and measurable sets.

• Construction and properties of Lebesgue measure.

• Measurable functions and their properties.

• Construction and properties of the Lebesgue integral.

• Inequalities for Lebesgue integration.

• Monotone and Dominated Convergence Theorems.

• Comparing Lebesgue and Riemann integrals.

• Product measures and the theorems of Fubini and Tonelli.

• Absolute continuity and the Radon-Nikodym derivative.

• Lebesgue’s differentiation and density theorems.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam: Measure & Integration
3 hours 100

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam: Measure & Integration
3 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

• P. Halmos, Measure theory. 1950.

• W. Rudin, Real and complex analysis. Third Edition, 1987.

• H. L. Royden, P. Fitzpatrick, Real analysis. Third Edition, 1988.

• J. Benedetto, W. Czaja, Integration and Modern Analysis, 2009.

• H. Federer, Geometric measure theory. 1969.

• J. Heinonen, Lectures on Analysis on Metric Spaces. 2000.

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.