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# Lebesgue Measure & Integration - MAT00013H

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• Department: Mathematics
• Module co-ordinator: Prof. Victor Beresnevich
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2022-23

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## Module will run

Occurrence Teaching cycle
A Spring Term 2022-23

## Module aims

This module aims to introduce Lebesgue's theory of measure and integration, which extends the familiar notions of volume and "area under a graph" associated with the Riemann integral. It will be demonstrated that the Lebesgue integral can be computed by familiar methods whenever they are applicable (anti-differentiation in dimension one, repeated one-dimensional integrals in higher dimensions), and that it is sufficiently wide in scope to give the powerful convergence theorems needed for more advanced applications.

## Module learning outcomes

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

• Understand the construction and properties of Lebesgue measure, including the notion and properties of null set;

• Understand the construction of the Lebesgue integral and know its key properties;

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

## Module content

Syllabus

• Construction and properties of Lebesgue measure.

• Lebesgue measurable sets, countable additivity of Lebesgue measure.

• Measurable functions and their properties.

• Construction and properties of Lebesgue integral.

• The use of the Fundamental Theorem of Calculus.

• Monotone and Dominated Convergence Theorems.

• Theorems of Fubini and Tonelli.

## Assessment

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

None

### Reassessment

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

## Module feedback

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

John J. Benedetto, Wojciech Czaja, Integration and Modern Analysis. Birkh?er. 2009.

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.