Lecturer: Matthew Hyde

Term(s): Term 1

Commitment: 30 lectures

Assessment: Oral exam

Prerequisites: Familiarity with topics covered in Analysis I, II & III and MA260: Norms, metrics and topologies.

Content:

The first part of this course provides an introduction of measure theory for students of all mathematical backgrounds. We will adopt a more advanced approach than a standard undergraduate module, so there will be new content even for those students who have taken measure theory before. This will cover:

• Measures, Carathéodory's construction, integration and convergence theorems.
• Riesz representation theorem, weak* convergence and Prokhorov's theorem.
• Hardy-Littlewood maximal inequality and Rademacher’s theorem.

The second part provides an introduction to geometric measure theory. Time permitting, we will cover some of the following topics:

• Hausdorff distance.
• Hausdorff measure, rectifiable and purely unrectifiable sets.
• Sard's theorem.
• The Besicovitch projection theorem.

Course notes:

Course notes can be found on the Moodle page.

References:

• Rudin, W.: Real and Complex Analysis
• Federer, H.: Geometric measure theory
• Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces