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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: 2021-22

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## Module will run

Occurrence Teaching cycle
A Spring Term 2021-22

## 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
Online Exam
Lebesgue Measure & Integration
N/A 100

None

### Reassessment

Task Length % of module mark
Online Exam
Lebesgue Measure & Integration
N/A 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.