# Measure Theory with Applications (L.7) (850G1)

15 credits, Level 7 (Masters)

Autumn teaching

This module provides a rigorous introduction to abstract measure theory and Lebesgue integration. You will study measures and their properties, learn how to construct measures from outer measures using Caratheodory's method, and use this approach to construct the Lebesgue measure which we will study in detail (and, more generally, Lebesgue-Stiltjes measures). You will then develop the powerful concept of Lebesgue integral, a generalisation of the Riemann integral, and we will make use of it to study various convergence theorems and introduce Lebesgue spaces which are pivotal in applications.

The study of measure theory stems from the need to define an extended notion of length, area, and volume for sets of the respective dimensions that are potentially much more irregular than any set which can be easily imagined, let alone be drawn by hand, such as a circle or a polygon. The collection of sets for which this is possible is the sigma-algebra of Lebesgue measurable sets, and the extended length, area, or volume is the Lebesgue measure of the corresponding dimension. The definition of a measure naturally leads to a theory of integration which is central in many areas of mathematics.

100%: Lecture

### Assessment

20%: Coursework (Portfolio, Problem set)
80%: Examination (Computer-based examination)

### Contact hours and workload

This module is approximately 150 hours of work. This breaks down into about 33 hours of contact time and about 117 hours of independent study. The University may make minor variations to the contact hours for operational reasons, including timetabling requirements.

We regularly review our modules to incorporate student feedback, staff expertise, as well as the latest research and teaching methodology. We’re planning to run these modules in the academic year 2023/24. However, there may be changes to these modules in response to COVID-19, staff availability, student demand or updates to our curriculum. We’ll make sure to let our applicants know of material changes to modules at the earliest opportunity.