Module Catalogue 2024/25

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

To familiarise students with the theory of measure spaces. To understand Lebesgue integration and applications that arise both in pure and applied sciences. To reinforce the ability of students to follow research in Analysis.
Module summary
Measure theory gives the appropriate language for measuring subsets of a space in a systematic way. The common example is Lebesgue measure on the real line which gives the length of an interval. This idea can be used to further produce a notion of integration. Unlike Riemann integration which is based on a partition of the domain of a function, Lebesgue integration relies on partitions of the range. As such it can tackle, in a sense, more functions than usual. Measure theory is a basic tool
for Analysis and Algebra but also has vast applications in Applied Sciences, including Physics, Medicine and Economics. By the end of the course the students will understand Lebesgue integration

in Rn and how it can be used as a language to encode a variety of examples through the notion of Hilbert spaces.

Outline Of Syllabus

Systems of sets and measures.
Measure theory on Rn (Lebesgue integration).
Comparison with Riemann integration

Intended Knowledge Outcomes

Students will learn various techniques for performing algebraic and analytic manipulations on sets,
measures, integrals of functions and their limits. By the end of the course they will be able to
understand the basic terminology and key results that arise in the context of measure theory.
Students will have the opportunity to see applications of these methods in related topics in Analysis.

Intended Skill Outcomes

Students will enhance their ability to understand how measure theory is used in the framework of
Analysis and other fields. They will learn to recognise and apply key tools for Lp-spaces.
Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition):
remember, understand, apply, analyse, evaluate and create

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

General Notes

N/A