Results 1 to 5 of 5

Math Help - Zero content subset

  1. #1
    Junior Member RaisinBread's Avatar
    Joined
    Mar 2011
    Posts
    37

    Zero content subset

    I'm trying to find the proof of the following proposition.

    Let S^n be the unit sphere in R^n. If K \subset  S^n is n-1 dimensional, then K has Lebesgue measure zero.

    I'd b very thankful if any of you could give me the general idea behind the proof or the link to a textbook or internet source where the proof of this is explained.
    Last edited by RaisinBread; May 28th 2011 at 10:00 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by RaisinBread View Post
    I'm trying to find the proof of the following proposition.

    Let S^n be the unit sphere in R^n. If K \subset  S^n is n-1 dimensional, then K has Lebesgue measure zero.

    I'd b very thankful if any of you could give me the general idea behind the proof or the link to a textbook or internet source where the proof of this is explained.
    Are you aware of the theorem that states that if M is an open submanifold of \mathbb{R}^m with m<n and F:M\to\mathbb{R}^n is smooth then F(M) has measure zero?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    What's the definition of "n-1 dimensional"?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member RaisinBread's Avatar
    Joined
    Mar 2011
    Posts
    37
    Hm, now that I think about it, the formulation I have written up there may be a little sloppy. The whole situation is the following;

    I'm trying to prove that a random matrix is always full rank, or, in other words, that if we are in R^n, a set of vectors with random coefficients will always be linearly independent, so long as we don't choose more than n vectors.

    Now I can easily show this in 2 dimensions, and the idea is that, if we choose one vector at random, V_1=(r,\theta_1), we can calculate the probability that a second vector chosen at random will linearly dependent with V_1 by calculating the probability that the vector will be on the subspace of 1 dimension spanned by V_1. You can then show that this is going to happen if the second vector has an angle of \theta_1 or \theta_1 + \pi , and this set on the unit circle has measure zero, and thus an integral of any probability distribution over this set is zero.

    I'm now trying to generalize this idea on R^n.
    Last edited by RaisinBread; May 28th 2011 at 01:23 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    I'd first show that, generically, you won't ever have the zero vector as a column in your random matrix.

    Then go by induction: change the basis so that the first column becomes the first standard basis vector (1,0,...,0). Now unless the second column is all zero except for possibly the first entry, you can change the basis again so that it becomes the second standard basis vector (0,1,0,...,0). And so forth.

    You're just using over and over again the fact that a point chosen "at random" from R^n won't be the origin.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof on Openness of a Subset and a Function of This Subset
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 24th 2010, 10:04 PM
  2. content zero
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: July 30th 2010, 09:14 PM
  3. Zero content
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 19th 2009, 01:19 PM
  4. subset U subset is NOT subspace of Superset?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 20th 2009, 05:17 PM
  5. content
    Posted in the LaTeX Help Forum
    Replies: 1
    Last Post: October 23rd 2008, 04:09 AM

Search Tags


/mathhelpforum @mathhelpforum