### Archive

Posts Tagged ‘hausdorff’

## Thoughts on Hausdorff Measure and Hausdorff Dimension

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.

## Domains convergence

Define $\Omega=(0,3)\times (0,1)$ and $\Omega_n=\{ (x,y) : 0. Study if $\Omega_n$ converges to $\Omega$ in the three types of convergence mentioned here.

Categories: Real Analysis, Topology

## Hausdorff convergence and frontier convergence

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 $\partial K \subset L \subset K$.

Categories: Problem Solving, Topology

## 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.)

Categories: Uncategorized