Module Information

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

Course Delivery



Due to Covid-19 students should refer to the module Blackboard pages for assessment details

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

Learning Outcomes

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

demonstrate knowledge of examples of sigma-algebras and measures;

determine the Lebesgue measure of certain Borel sets;

determine whether a function is measurable;

determine whether a function is integrable;

demonstrate that the Riemann and Lebesgue integrals co-incide for a class of functions;

apply convergence theorems to justify the exchange of limiting processes for integrals;

demonstrate knowledge of Lebesgue measure and its properties;

demonstrate knowledge of an advanced development/application of Lebesgue Integration:


Measure theory is one of the main areas of research in mathematical analysis; Lebesgue measure and integration is the most important example. This theory plays a central role in analysis, functional analysis and probability theory. There are applications to the modern theory of partial differential equations and financial modelling. Essential concepts are introduced for any student seeking a deeper understanding of mathematical analysis and its applications.

Brief description

This course is a rigorous 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 powerful theory due to Lebesgue is introduced. The emphasis is on examples and applications of the main theorems rather than proofs of the classical results. Students will be expected to study advanced applications via mini courses


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'r Lemma. Dominated Convergence Theorem.
Lp spaces. Young's inequality. Holder's inequality. Minkowski's inequality. Completeness.
Construction of Lebesgue measure. Translational invariance. Completeness. Characterisation and approximation of Lebesgue measurable sets.
Product measures. Tonelli-Fubini Theorem.
Topics for mini courses include:
Fundamental Theorem of Calculus for absolutely continuous functions.
Rearrangements of functions. Hardy-Littlewood inequality, generalisations and applications.
Weak derivatives and Sobolev spaces in one dimension.
Lebesgue-Stieltjes integral.
Probability spaces, random variables, martingales.
Convolution integrals, smoothing properties.

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.
Information Technology Students will be encouraged to research min course topics on the internet
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.
Research skills The mini coures will make the students research a mathematical topic.
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 7