Talk:Vitali covering lemma

Constant should depend on the dimension...

I'm fairly certain the 3 should be raised to the power of the dimension. —Preceding unsigned comment added by 128.208.116.167 (talk) 03:53, 12 January 2010 (UTC)

The unsigned comment is incorrect. When making such an assertion one should give a reason (first of all, what is wrong with the provided proof?). This lemma applies to arbitrary metric spaces irrespective of any notion of dimension. 68.121.170.218 (talk) 07:33, 14 January 2010 (UTC)

Infinite version statement

This was previously stated with 3 in place of 5, which was incorrect, as the following example shows. In one dimension, take

$Z:=\{B(x,r):|x|<1/2,\,|x|<r<(|x|+1)/3\}.$

Then every ball in $Z$ contains $0$, and hence every disjoint subcollection of $Z$ consists of just one ball. But for any one ball $B\in Z$, the expanded ball $3B$ does not cover $\bigcup Z=(-1,1)$.

One can replace 5 with any number larger than 3 (but not 3), but this does not seem to be worth the effort. Oded (talk) 00:43, 5 May 2008 (UTC)

Vitali covering theorem for the Hausdorff measure

In the statement of the theorem, I added that the sets in the Vitali class ${\mathcal {V}}$ are closed, because it is so in the Falconer reference, and because no other reference or proof is given; anyway a minimal regularity should be assumed I suppose. Bdmy (talk) 11:44, 14 October 2008 (UTC)

Also, I found disturbing that the celebrated Vitali covering theorem was given in the present article only in terms of the Hausdorff measure, a notion that even professional mathematicians are not necessarily familiar with. So I added a corollary using the Lebesgue measure. Bdmy (talk) 10:52, 16 October 2008 (UTC)

In the same vein, I found wrong that an article about Vitali's theorem did not indicate what Vitali actually proved. Bdmy (talk) 13:06, 20 October 2008 (UTC)

Split?

The present structure is very awkward. It seems that Vitali covering theorem should be an article on its own. Arcfrk (talk) 04:39, 26 January 2009 (UTC)

What do you mean exactly by "covering theorem"? I worked on this article quite a bit, but it was already well advanced before I came. The "covering lemma" (as it is named in this article) comes essentially from Evans-Gariepy, where it is more or less the only mention to Vitali's result, and is perhaps even called "Vitali theorem" there. Then I insisted on having the "classical" statement of Vitali, and later the fact that the "Lemma" implies the "Theorem" was added. What would you suggest about the structure? --Bdmy (talk) 08:14, 26 January 2009 (UTC)

I tried to make a better division in sections. However, I don't think that a splitting is a good thing to do here, as the so-called "Vitali lemma" is a (nice, in my opinion) reformulation of an important part of the original proof by Vitali of the "Vitali theorem". As such, I believe that it belongs to an article on the Vitali theorem. On the other hand, I have no objection to a renaming of the article as "Vitali theorem", but this is just beyond my wiki-abilities. --Bdmy (talk) 12:27, 31 January 2009 (UTC)

I like the name as it is, or at least I like having the word "covering" a part of the title. But I don't think it is beyond your wiki-abilities. There is a move tab next to the history tab. Just click it and type in what the new title should be. If I remember correctly it automatically redirects the old page title to the new one. Thenub314 (talk) 21:22, 31 January 2009 (UTC)

Well, of course I meant to say "Vitali covering theorem", I have no intention to remove the word covering. You see that the point is that what Vitali really proved is the possibility to cover, up to a negligible set, a given set by a certain disjoint collection. Let say that this is the theorem. On the other hand, this article had put at the beginning a very strong emphasis on the combinatorial lemma, which is, for example, the only thing you need when proving the Hardy-Littlewood maximal inequality. This lemma was not really stated by Vitali, although one can say that Vitali essentially proved it, on the way to the theorem. So the suggestion of Arcfrk to split makes sense, though I don't really agree. Thanks for the wiki-indications anyway. --Bdmy (talk) 22:13, 31 January 2009 (UTC)