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Thoughts on Hausdorff Measure and Hausdorff Dimension

April 13, 2012 Leave a comment

The Hausdorff measure is a good way to study the fine properties of sets since it is more ‘sensible’ than the Lebesgue measure. For example in \Bbb{R}^3 the surfaces and paths have zero Lebesgue measure, but the Hausdorff dimension captures what Lebesgue measure misses for zero-measure sets, and it confirms our intuition that a path is in fact  one dimensional and a surface is two dimensional. The Hausdorff dimension of a set is the exact value of \alpha such that the \beta-dimensional Hausdorff measure is zero for \beta > \alpha and infinity for \beta <\alpha. The problem of finding the Hausdorff dimension of a set is not very simple, since the dimension doesn’t need to be an integer. There exist sets with fractional and other bizare dimensions. I will present a few problems below which can be approached using Hausdorff measure and Hausdorff dimension. A good reference for starters can be found in: Real Analysis.

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Domains convergence

November 25, 2010 Leave a comment

Define \Omega=(0,3)\times (0,1) and \Omega_n=\{ (x,y) : 0<x<3, \ 0<y<2+\sin(nx)\}. Study if \Omega_n converges to \Omega in the three types of convergence mentioned here.
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Hausdorff convergence and frontier convergence

November 25, 2010 2 comments

1) Prove giving a counterexample that the Hausdorff convergence of a sequence of open sets \Omega_n does not imply the Hausdorff convergence of the boundary of \Omega_n to the boundary of \Omega.
2) Let K_n be a sequence of compact sets converging in the Hausdorff distance to K. Prove that the set \partial K_n has at least a point of accumulation and every such point L satisfies \partial K \subset L \subset K.
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Open sets convergence

November 21, 2010 1 comment

Consider the sequence of open sets of [0,1] defined by
a) \displaystyle \Omega_n:= \bigcup_{k=0}^{2^{n-1}-1} \left(\frac{2k}{2^n},\frac{2k+1}{2^n}\right).

b) \displaystyle \Omega_n:= \bigcup_{k=0}^{2^{n}-1} \left(\frac{k}{2^n},\frac{k+1}{2^n}\right).

Study the convergence of the sets \Omega_n in the following types of convergence:

  • weak convergence of characteristic functions;
  • convergence in the topology defined by the Hausdorff distance;
  • convergence in the way of compact sets.

(Each of the previous types of convergence will be presented below.)

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