Besicovitch Covering Lemma

[ArcanaNoir]

Homework Statement

Let {Q} be a collection of cubes covering a set E in R^n. Prove that there is a countable sub collection {Q}' of these cubes which covers E and $\cup (\frac{1}{2}){Q} \subseteq \cup {Q}'$, and the number of cubes in the subcollection containing any given point of E is less than something depending only on the dimension.
Oh and the sup of the side lengths of the cubes is bounded, but E is not bounded.

Homework Equations

$\frac{1}{2}Q$ means the cube inside Q with side length 1/2 of the side length of Q.
In other words, the countable subcovering needs to cover the middle sections of the original cover.

The Attempt at a Solution

I really like this proof http://books.google.com/books?id=5d...=onepage&q=besicovitch covering cubes&f=false

But in this proof the set E is the centers of the cubes, so we only end up covering centers, not middle halves, and also E is bounded in that proof. My professor said I could modify this proof but I don't see how. *sad face* He also said the first half of my most recent attempt at this proof was badly written and the second half was worse. :(:(

The idea for dealing with the unbounded set E is to take "layers" of cubes which are about the same size. So something like sup{S(Q)}=R where S(Q) is the side length of Q and then $(1-\epsilon)^kR< S(q)\le R$ for the first layer, then take a maximal set of points in E which are some specific distance apart (this is a piece I messed up on, no matter what distance I separate my points by there is always some flaw in my choice). Then take the cubes containing those points and discard some cubes (which ones?). Rinse and repeat.

I tried doing some optimal arrangement of separated points around a specific point for the bound, but I was told I did that wrong too :( (that was the "worse" part apparently.)

 

[mighty2000]

it is important to approach problems with a clear and logical mindset. Let's break down the problem and see if we can come up with a solution.

First, we are given a collection of cubes {Q} that cover a set E in R^n. This means that every point in E is contained within one of the cubes in {Q}. However, we need to find a countable subcollection {Q}' that covers E and also satisfies the condition that the middle sections of the original cubes are covered.

To start, we can consider a specific cube Q in {Q}. We know that the cube Q has a side length S(Q). We can also define a smaller cube with side length (1/2)S(Q), denoted as (1/2)Q. This cube is contained within Q and is the "middle section" of Q.

Now, we want to find a countable subcollection {Q}' that covers E and also satisfies the condition that (1/2)Q is contained within {Q}'. We can do this by selecting a maximal set of points in E that are separated by a specific distance. Let's call this distance d.

Next, we can take the cubes in {Q} that contain these selected points and add them to our subcollection {Q}'. We can also discard any cubes in {Q} that do not contain these points, as they will not contribute to covering the middle sections of the original cubes.

We can repeat this process for each cube Q in {Q}. This will give us a countable subcollection {Q}' that covers E and satisfies the condition that the middle sections of the original cubes are covered.

Now, we need to address the fact that E is unbounded. To do this, we can consider "layers" of cubes with side lengths that are approximately the same size. For example, we can define a layer with side length R, where R is the supremum of all the side lengths of the cubes in {Q}. We can then take a maximal set of points in E that are separated by a specific distance d, as before.

We can continue this process for each layer, taking a maximal set of points separated by d and adding the cubes that contain these points to our subcollection {Q}'. This will ensure that we cover all the points in E, even though it is unbounded.

Finally, we need to address the condition that