Module Information

Module Identifier
Module Title
Lebesgue Integration
Academic Year
Semester 2
Other Staff

Course Delivery

Delivery Type Delivery length / details
Lecture 19 x 1 hour lectures
Seminars / Tutorials 3 x 1 hour Problem Classes


Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   Written Examination  100%
Supplementary Exam 2 Hours   Written Examination  100%

Learning Outcomes

On successful completion of this module students should be able to:

1. Demonstrate knowledge of examples of sigma-algebras and measures;
2. Determine the Lebesgue measure of certain Borel sets;
3. Determine whether a function is measurable;
4. Determine whether a function is integrable;
5. Demonstrate that the Riemann and lebesgue integrals co-incide for a class of functions;
6. Apply convergence theorems to justify the exchange of limiting processes for integrals;
7. Demonstrate knowledge of lebesgue measure and its properties.


Riemann integrability. Fundamental Theorem of Calculus. Interchange of limiting processes.

Set theory. Sigma-algebras, generated sigma-algebras. Borel sets. Measures. Examples. Lebesgue measure.

Measurable functions. Simple functions. Fundamental Approximation Lemma. Lebesgue integrable functions.

Monotone Convergence Theorem. Fatou's Lemma. Dominated Convergence Theorem.

Lp spaces. Young's inequality. Holder's inequality. Mikowski's inequality.

Construction of Lebesgue measure. Translational invariance. Completeness.
Characterisation and approximation of Lebesgue measurable sets.

Product measures. Tonelli-Fubini theorem.

Brief description

This course is a rigourous practical guide to the basic technical foundations and main principles which underpin the classical notions of area, volume, and the related idea of an integral. After reviewing the Riemann integral, its properties and limitations, the beautiful and poweful theory due to Lebesgue is introduced. The emphasis is on examples and applications of the main theorems rather than proofs of the classical results.

Module Skills

Skills Type Skills details
Application of Number Required throughout the course
Communication Written answers to exercises must be clear and well structured.
Improving own Learning and Performance Students are expected to develop their own approach to time management regarding completion of assignments on time and preparation between lectures.
Personal Development and Career planning Completion of tasks (assignments) to set deadlines will aid personal development. The course will give indications of whether a student wants to further pursue mathematical analysis and its applications.
Problem solving The Assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Subject Specific Skills Broadens exposure of student to topics in mathematics.
Team work Students will be encouraged to work together on questions during problem classes.


This module is at CQFW Level 6