Hopefully Interesting

March 14, 2008

Quantitative Borel-Cantelli Lemma

Filed under: math.CA,motivated theorems — Pietro @ 12:00 pm
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Everyone knows and loves the Borel-Cantelli lemma: it is widely used, intuitive and has a cute proof. Let’s recall it.

Borel-Cantelli Lemma. Let (X,\mathcal{M},\mu) be a measure space, and (E_n)_{n\in\mathbb{N}} a sequence of measurable sets such that \sum_{n\in\mathbb{N}} \mu(E_n) < \infty. Then the points of X that lie in infinitely many E_n form a set of measure zero.

Proof. We present the usual slick proof of the lemma, and postpone intuition until the development of a quantitative statement. Define F_\infty to be the set of points lying in infinitely many E_n, and notice that F_\infty = \bigcap_{n\in\mathbb{N}} \bigcup_{k\geq n} E_k. It follows that F_\infty is measurable, and contained in each cofinite union \bigcup_{k\geq n} E_k. Now, \mu(\bigcup_{k\geq n}E_k) \leq \sum_{k\geq n} \mu(E_k), and the latter goes to zero as n\to\infty, because \sum_{n\in\mathbb{N}} \mu(E_n) converges. Thus \mu(F_\infty)=0, as claimed. \Box

Alright, so the set of points lying in “very many” E_n is “very small”. A natural quantitative question to ask at this point is: how small is the set of points lying in “moderately many” E_n? More precisely, if F_k is the set of points lying in at least k of the E_n, and G_k = F_k \backslash F_{k+1} the set of points lying in exactly k of the E_n, can we get bounds on \mu(F_k) and \mu(G_k), depending on k?

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December 28, 2007

Length and Area I

Filed under: math.CA — Pietro @ 6:54 am
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This is a topic which has fascinated me for years, and on which I’ve spent many (fortunately unsuccessful) hours trying to prove theorems that are actually false. I find it to be a good introduction to the weirdness of analysis: you know, the gradual realization that much of our intuition about lines, surfaces and continuity is simply not applicable to the analytical formalization of lines, surfaces and continuity. Some mathematicians, like Weyl and Brouwer, blamed this weirdness on the fact that the intuitive notion of a continuum (the line) is not very well captured by something as disconnected as a set of points (the real numbers).

In any case, we have to develop some new intuitions if we’re going to expect, rather than be surprised by, the Koch snowflake, nonmeasurable sets, and families of curves that have positive area but only at the endpoints. (more…)

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