Lebesgue Measure & Integration - MAT00013H

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Module will run

Occurrence	Teaching period
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.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	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.

Indicative reading

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