## 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 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 such that the -dimensional Hausdorff measure is zero for and infinity for . 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 and . Study if converges to in the three types of convergence mentioned here.

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## Hausdorff convergence and frontier convergence

1) Prove giving a counterexample that the Hausdorff convergence of a sequence of open sets does not imply the Hausdorff convergence of the boundary of to the boundary of .

2) Let be a sequence of compact sets converging in the Hausdorff distance to . Prove that the set has at least a point of accumulation and every such point satisfies .

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## Open sets convergence

Consider the sequence of open sets of defined by

a) .

b) .

Study the convergence of the sets 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.)