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 -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…)
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 be a measure space, and a sequence of measurable sets such that . Then the points of that lie in infinitely many 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 to be the set of points lying in infinitely many , and notice that . It follows that is measurable, and contained in each cofinite union . Now, , and the latter goes to zero as , because converges. Thus , as claimed.
Alright, so the set of points lying in “very many” is “very small”. A natural quantitative question to ask at this point is: how small is the set of points lying in “moderately many” ? More precisely, if is the set of points lying in at least of the , and the set of points lying in exactly of the , can we get bounds on and , depending on ?
Today I’d like to talk about a gorgeous theorem in order theory. Since partial orders are such general objects, the theorem potentially has many applications, especially in combinatorics (we shall see a few, in fact); and to top it off, I found a very cute inductive proof.
Here’s the statement:
Theorem 1. Let be a partially ordered set (a poset) in which the largest antichain has size . Then it is possible to decompose into the union of chains.
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…)
(This is a continuation of On Limits I.)
Think of Pelé. When somebody asks “who is Pelé”, you might reply, “the best football player ever”. Notice that you are implicitly defining Pelé in terms of his relationship to other footballers; an observation which is often obscured by the fact that there is also an actual person you can point to, and say “that’s Pelé”.
Now suppose a toy company came out with a football trading card game, where each card represents a famous player, and contains various numerical ratings like precision, speed, stamina etc. You might now explain that Pelé (meaning the Pelé card) is “the card with highest speed and precision”. Even though there is still an actual card you call “Pelé” (and children might think of it this way), it is now clear that the thrust of the definition is quite another: what is important about the Pelé card is its relationship to the other cards, not its particular shape, or whether it’s made of paper or plastic.
Similarly, if you ask a child what is a chess pawn, they will most likely point to the actual, physical chess piece: “that’s a pawn”. Which is fine if we had asked “what is a chess pawn, in the physical world?” However, the deeper content of our question was “what is a chess pawn, in the game of chess?” Since grandmasters are able to play entire games in their minds, a pawn can’t be just a wooden piece. In fact, even novice players experience no confusion if pawns are replaced with (say) beans, as long as they agree to it beforehand. So we see that the main defining feature of chess pawns is not their wooden incarnation, nor their color, but their relation to other pieces; that is, the rules they obey in the game of chess. The physical pawn is just a memory aid. (more…)
What on earth could one usefully say about limits, that hasn’t been printed for centuries on a thousand calculus textbooks?
I don’t know about usefulness, but one can certainly say something different. For all their numerosity, calculus texts are remarkably homogeneous in their treatment of this elementary topic, and I see at least one point that could stand clarification.
Limits — especially expressions involving “dot dot dot”, such as 0.999… — still seem to be a source of confusion for many beginning calculus students. Since all the popular treatises give thorough mathematical definitions of the concept, the problem must lie somewhere other than mathematics. I conjecture that what’s missing is not a more easily understandable definition, but instead a straightforward discussion of how definitions work in mathematics.
To get into definitions, I’m going to start by talking about symbols — in particular, symbols that represent numbers. (more…)
Hello. I am Pietro, a brazilian student of logic and mathematics. Of course, like everybody else, I am much more than just that; but that is what will come through the most in these pages.
In this blog, I hope to share interesting things I come across in logic and math, which may not be widely known. Also, I may occasionally share a different way of thinking about well-known things, if I feel I have developed it enough, and it is uncommonly enough seen in mainstream sources like textbooks and papers. I find it enormously important for people to exchange their “inner ways” of thinking, rather than just the “outer” results they reach; communication becomes much clearer, more interesting, and conductive to progress. I thank Terry Tao, Tim Gowers , the n-Category Café and the Catsters for proving this point so forcefully.
Though I’m brazilian, I will be writing in english, simply because I will be understood by a larger number of people that way. It’s fine to be protective of one’s culture and language, but I feel this is simply not the place. If my experience on orkut is any indication, for every portuguese-speaking reader I lose, I will gain five readers from India.
Finally, the “kc” in this blog’s URL are my other initials.