Aims

To familiarise students with the theory of measure spaces. To understand Lebesgue measure and 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 can be used to further produce a notion of integration. Unlike Reimann 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 students will understand the Lebesgue measure and 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 on Riemann integration.

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 and explanation of methods, 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 for 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.