## 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.

## BPM 1.4.2 (Sp 84)

Let for . We show that is monotonically increasing. First we rewrite . Taking derivatives, and doing some algebra, we get . We wish to show that everywhere. This would follow from . However, this follows from the convexity of . Which says that for all .

We compute . Taking we see that . We then have that . Since is monotonically increasing, and positive, we have that as . We compute . Applying concavity, we have for all . We then check that , where is by applying . It follows that .

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