# Module Information

Module Identifier
MAM7020
Module Title
LEBESGUE INTEGRATION
2010/2011
Co-ordinator
Semester
Semester 2
Pre-Requisite

#### Course Delivery

Delivery Type Delivery length / details
Lecture 19 x one hour lectures
Seminars / Tutorials 3 x 1 hour problem classes
Seminars / Tutorials 3 x 1 hour problem classes

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Assessment Presentation (30 minute presentation on an advanced topic researched by the student)  20%
Semester Exam 2 Hours   80%
Supplementary Assessment Presentation (if failed orginally)  20%
Supplementary Exam 2 Hours   Two hour written examination  80%

### Learning Outcomes

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

### Aims

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 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 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 work on a small project and give a presentation.

### Content

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

Set theory. Sigma algebras, generated signma-algebras. Borel sets. Measures. Examples. Lebesgue measures.

Measurable functions. Simple functions. Fundamental approximation lemma. Lebesgue integrable functions.

Monotone Convergence Theorem. Fatou's inequality. Minkowski's inequality.

L^p spaces. Young's inequality. Holder's inequality. Minkowski's inequaility. Completeness.

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

Product measures. Tonelli-Fubini Theorum.

Topics for projects include:

Fundamental Theorem of Calculus for absolutely continuous functions.

Rearrangements of functions. Hardy-Littlwood 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. Project will help students develop presentational skills.
Improving own Learning and Performance Students are expected to develop their own approach to time-management regarding completion of assignments and projects on time and preparation between lectures.
Information Technology Students will be encouraged to research topics on the internet and can use technology in their presentation.
Personal Development and Career planning Completion of tasks (assignments and presentation) 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 project will make the students independently research a mathematical problem.
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.

### Notes

This module is at CQFW Level 7