## Keeping Sets Different

Consider the Borel -algebra of Lebesgue measurable subsets of . We then define a pseudometric on where is the symmetric difference of sets. Ignoring sets of zero measure, we have that is indeed a metric. We wish to show that is not sequentially compact in its metric.

Write for the binary expansion of . Consider . We compute . If we think probabilistically, where the digits and represent independent coin tosses, we get: for . Thus is constant for and hence 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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