Results 1 to 5 of 5

Math Help - a property of point of density

  1. #1
    Newbie
    Joined
    Nov 2009
    From
    Italy
    Posts
    5

    a property of point of density

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

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

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

    Can someone help me? Thanks

    ( |\cdot| denote the measure of Lebesgue.)
    Last edited by miccoli; November 6th 2009 at 11:24 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by miccoli View Post
    I have a problem, that I can't do.

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

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

    Can someone help me? Thanks

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

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

    The main observation is that, for all x, since d(x)\leq\|x\| (this is because 0\in E), we have 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: |B(0,2\|x\|)\cap E|\leq |B(0,2\|x\|)|-|B(x,d(x))|, you divide by 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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    From
    Italy
    Posts
    5
    Hello!

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

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

    Thanks you for your availability!
    Last edited by miccoli; November 7th 2009 at 05:45 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by miccoli View Post
    I have a question: 0 is a point of density for E, but this doesn't implicate that 0 belongs to E. If 0\notin E, can I say d(x)\leq||x|| for all x\in\mathbb{R}^n?
    That's a good point. However, 0 must be a limit point of E. Otherwise, for B small enough around 0, B\cap E=\emptyset hence \frac{|B\cap E|}{|B|}=0, which is false. Thus it doesn't really matter. But the safest way is to consider B(0,2\|x\|)\setminus B(x,\frac{d(x)}{2}) !

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

  5. #5
    Newbie
    Joined
    Nov 2009
    From
    Italy
    Posts
    5
    Ok! Now, it's clear! Thanks very much!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fixed point property.
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: May 3rd 2011, 11:52 AM
  2. Finding Marrginal Density Fcn From Join Density Fcn
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 26th 2010, 08:25 PM
  3. Proving the second property of Density Functions
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: May 24th 2009, 02:30 AM
  4. standard normal density special property?
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: December 3rd 2008, 08:06 PM
  5. marginal density, conditional density, and probability
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 24th 2008, 06:50 PM

Search Tags


/mathhelpforum @mathhelpforum