Results 1 to 1 of 1

Math Help - Lower bound of 1-hausdorff measure

  1. #1
    Dec 2009

    Lower bound of 1-hausdorff measure

    Let K to be a Cantor middle-half set. Consider KXK.
    figure 2 shows the 1-stage of this cantor set construction.

    diam(B)=sup{|x-y|:x,ybelongs to B}

    Want to show that
    Ans from the article:
    The lower bound can be derived by considering the probability measure μ on K^2 such that μ(K^2 ∩ Q) = 4^(−n) for each of the 4^n squares Q arising at stage n of the construction. This measure satisfies μ(B) ≤ 9diam(B) for any closed set B, since B can be covered by at most nine dyadic squares with side-length smaller than diam(B). Therefore, for any cover {Bi } of K^2 by disks, \sum_{i}diam(Bi )>= \sum_{i}μ(Bi )/9>=μ(K^2)/9 =1/9

    I do not understand why μ(B) ≤ 9diam(B). I do not understand why probability measure is related to the diameter.
    Please help me!
    Last edited by ldpsong; April 6th 2010 at 09:25 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Hausdorff Measure
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 22nd 2010, 07:44 AM
  2. Replies: 0
    Last Post: February 19th 2010, 01:06 AM
  3. Greatest lower bound and lower bounds
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 13th 2009, 02:26 PM
  4. Upper bound/Lower bound?
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: September 13th 2009, 10:48 AM
  5. least upper bound and greatest lower bound
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 22nd 2007, 09:59 AM

Search Tags

/mathhelpforum @mathhelpforum