Measure & Integration - MAT00087H

«Back to module search

  • Department: Mathematics
  • Credit value: 20 credits
  • Credit level: H
  • Academic year of delivery: 2025-26

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

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching period
A Semester 2 2025-26

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. Demonstrate knowledge of the properties of measures and measurable sets by using them in examples.

  2. Demonstrate understanding of Caratheodory's construction of measures including Lebesgue measure.

  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

  • Outer measures, measures and measure spaces.

  • Construction and properties of Lebesgue measure.

  • Construction and properties of the Lebesgue integral.

  • Monotone and Dominated Convergence Theorems.

  • Comparing Lebesgue and Riemann integrals.

  • Theorems of Fubini and Tonelli.

  • Inequalities of Hölder, Cauchy-Schwarz and Minkowski.

  • Absolute continuity and the Radon-Nikodym derivative.

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 student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

[1] H. L. Royden, P. Fitzpatrick, Real analysis. 4th edition, 2010.

[2] P. Halmos, Measure theory. 1950.

[3] W. Rudin, Real and complex analysis. Third Edition, 1987.