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 (Unseen examination)