Measure and category

2013/2014, sem. 1

Prof. Boris Tsirelson (School of Mathematical Sciences).
Be acquainted with Lebesgue integration and metric spaces. Everything else will be explained from scratch. However, some maturity in analysis is needed.
Grading policy
Written homework and oral exam.

Lecture notes

  1. Basic ideas.
  2. Typical sequences etc.
  3. The Banach-Mazur game.
  4. Choice axioms and Baire category theorem.
  5. Many points of continuity.
  6. Good sets and their equivalence classes.
  7. About Egorov's and Lusin's theorems.
  8. Fubini's theorem and Kuratowski-Ulam theorem.
  9. More on differentiation.
  10. Typical compact sets.
  11. Typical functions via Banach spaces.
  12. Typical functions like to embed.

Additional sources

A. M. Bruckner, J. B. Bruckner, and B. S. Thomson, Real Analysis (2nd Edition), ClassicalRealAnalysis.com (2008). (See Chapter 10.)

John Oxtoby, "Measure and category (a survey of the analogies between topological and measure spaces)", Springer 1971.

A quote:

The Baire category is a profound triviality which condenses the folk wisdom of a generation of ingenious mathematicians into a single statement.

T.W. Körner, "Linear analysis" Sect.6, p.13.