The Hausdorff dimension on a separable metric space

Abstract

It is known to hold an inequality which estimates the Hausdorff dimension of a Borel subset of a separable metric space by means of probability measures on that set. We show that this inequality is turned into an equality under some condition. To prove this we construct a net of the space which is a generalization of dyadic decompositions of cubes in the Euclidean space.

Keywords:

• Hausdorff measure
• Hausdorff dimention
• 2000 Mathematics Subject Classification: 28A78

Notes

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Additional information

Notes on contributors

Hisayasu Kurata *

E-mail: [email protected]