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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.
  • The Egorov, Lusin and Fubini 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 measure, rectifiable and purely unrectifiable sets.
  • Sard's theorem
  • Frostman measures
  • 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

Practical information:

The class will meet four times a week online on MS Teams on Mondays 2pm, Tuesdays 1pm, Wednesdays 1pm and Fridays 3pm beginning the 5th October. The aim is to have classes replicate face to face meeting as much as possible and so you are not required to turn off your microphone and camera, though of course you can if you wish. Audience participation is encouraged, which is why this format is chosen.

All sessions will be recorded and made available to the class. If you do not wish to appear in the recordings please use the text chat to ask questions.

The Wednesday classes will be exercise sessions where you can ask questions about the homework. In addition to Teams, these will use Microsoft Whiteboard. This is accessible from within Teams, but a superior experience is provided by the dedicated app (Windows 10 and iPad), so please install these if you can. If you have access to a Surface/stylus/tablet/iPad etc, join the Whiteboard using these in addition to using Teams for video. This will allow you to draw in order to ask and answer questions.