Hopefully Interesting

October 29, 2009

A Hands-On Proof of Lusin’s Theorem

Filed under: analysis,math.CA — Pietro @ 3:40 am

Today I’d like to share a cute proof I came up with. Actually this is one of those posts that have been in the oven for the last year and a half, while my wife and I have been busy getting married, travelling across Europe, moving into a new place, defending my MSc in Logic and then moving to Los Angeles. But, since I’m retaking basic Real Analysis as a requirement of the PhD program at UCLA, it seems adequate to revisit this little idea and finally post it.

The proofs I’ve seen of Lusin’s theorem go through Egorov’s theorem; that’s how Stein-Shakarchi, Folland and Wikipedia do it. I don’t feel it is a particularly elegant proof, and it produces a truncated version of Lusin’s theorem, which holds only for sets of finite measure. A simple $\sigma$-finiteness argument takes care of this shortcoming, but one is left with the feeling that Egorov’s theorem is our hammer, and we’re trying to see Lusin’s as a nail. Hence the motivation for a different proof. I hope you will also appreciate how there are no real obstacles in the route below; everything that needs to be done can be done nearly without thinking. (more…)

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 $k$?

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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