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MA947 - Graduate Real Analysis

Lecturer: David Bate

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.

References:

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