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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