# Thread: a property of point of density

1. ## a property of point of density

I have a problem, that I can't do.

Let be $\displaystyle E$ a Lebesgue-measurable set of $\displaystyle R^n$, and let be $\displaystyle 0$ a point of density (of Lebesgue) for $\displaystyle E$, i.e., for all set Lebesgue-measurable $\displaystyle B$ which contains $\displaystyle 0$, we have
$\displaystyle \lim_{B\to0}\frac{|E\cap B|}{|B|}=1.$

Let be a sequence $\displaystyle {x_k}\rightarrow0$. Proof that
$\displaystyle \lim_{k\to\infty}\frac{D(x_k)}{||x_k||}=0,$
where $\displaystyle D(y)$ is the distance of $\displaystyle y$ from the set $\displaystyle E$.

Can someone help me? Thanks

($\displaystyle |\cdot|$ denote the measure of Lebesgue.)

2. Originally Posted by miccoli
I have a problem, that I can't do.

Let be $\displaystyle E$ a Lebesgue-measurable set of $\displaystyle R^n$, and let be $\displaystyle 0$ a point of density (of Lebesgue) for $\displaystyle E$, i.e., for all set Lebesgue-measurable $\displaystyle B$ which contains $\displaystyle 0$, we have
$\displaystyle \lim_{B\to0}\frac{|E\cap B|}{|B|}=1.$

Let be a sequence $\displaystyle {x_k}\rightarrow0$. Proof that
$\displaystyle \lim_{k\to\infty}\frac{D(x_k)}{||x_k||}=0,$
where $\displaystyle D(y)$ is the distance of $\displaystyle y$ from the set $\displaystyle E$.

Can someone help me? Thanks

($\displaystyle |\cdot|$ denote the measure of Lebesgue.)
Hi,

Let me denote by $\displaystyle B(x,r)$ the ball of center $\displaystyle x$ and radius $\displaystyle r$.

The main observation is that, for all $\displaystyle x$, since $\displaystyle d(x)\leq\|x\|$ (this is because $\displaystyle 0\in E$), we have $\displaystyle B(0,2\|x\|)\cap E\subset B(0,2\|x\|)\setminus B(x,d(x))$. After some thought, this inclusion should become obvious.

Then you take the measure of the subsets: $\displaystyle |B(0,2\|x\|)\cap E|\leq |B(0,2\|x\|)|-|B(x,d(x))|$, you divide by $\displaystyle B(0,2\|x\|)$, and you should be able to conclude very shortly.

Don't hesitate telling us what you tried and what failed if you want more help.

3. Hello!

Thanks very much!! But I have a question: $\displaystyle 0$ is a point of density for $\displaystyle E$, but this doesn't implicate that 0 belongs to $\displaystyle E$. If $\displaystyle 0\notin E$, can I say $\displaystyle d(x)\leq||x||$ for all $\displaystyle x\in\mathbb{R}^n$?

Another question: What $\displaystyle |B(0,2||x_k||)|$ and $\displaystyle |B(x_k,d(x_k))|$ are equal to?

4. Originally Posted by miccoli
I have a question: $\displaystyle 0$ is a point of density for $\displaystyle E$, but this doesn't implicate that 0 belongs to $\displaystyle E$. If $\displaystyle 0\notin E$, can I say $\displaystyle d(x)\leq||x||$ for all $\displaystyle x\in\mathbb{R}^n$?
That's a good point. However, 0 must be a limit point of $\displaystyle E$. Otherwise, for $\displaystyle B$ small enough around 0, $\displaystyle B\cap E=\emptyset$ hence $\displaystyle \frac{|B\cap E|}{|B|}=0$, which is false. Thus it doesn't really matter. But the safest way is to consider $\displaystyle B(0,2\|x\|)\setminus B(x,\frac{d(x)}{2})$ !

Another question: What $\displaystyle |B(0,2||x_k||)|$ and $\displaystyle |B(x_k,d(x_k))|$ are equal to?
You have of course $\displaystyle |B(x,r)|=C_n r^n$ where $\displaystyle C_n$ is the volume of the unit ball, and the good news is you don't need the value of $\displaystyle C_n$, it should simplify.

5. Ok! Now, it's clear! Thanks very much!!