Parker Glynn-Adey

Keeping Sets Different

Posted in Math by pgadey on 2011/08/19

Consider \mathcal{B} the Borel \sigma-algebra of Lebesgue measurable subsets of [0,1]. We then define a pseudometric d(A,B) = \mu(A \Delta B) on \mathcal{B} where A \Delta B = A \setminus B \cup B \setminus A is the symmetric difference of sets. Ignoring sets of zero measure, we have that d is indeed a metric. We wish to show that \mathcal{B} is not sequentially compact in its metric.

Write x = d_0.d_1d_2\dots for the binary expansion of x \in [0,1]. Consider E_i = \{x = d_0.d_1d_2\dots\ :\ d_i = 1\}. We compute E_i \Delta E_j = \{x : d_i \neq d_j\}. If we think probabilistically, where the digits d_1 and d_j represent independent coin tosses, we get: \mu(E_i \Delta E_j) = 1/2 for i \neq j. Thus d(E_i,E_j) is constant for i \neq j and hence \{E_i\} can have no convergent subsequences

This came from the September 2005 UoT Analysis comprehensive. The solution is due to Dror Bar-Natan.

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