# Exact Hausdorff measures of Cantor sets

## TL;DR

This paper extends Hausdorff measures by allowing gauge functions to depend on midpoints, providing a method to determine exact measures of certain Cantor sets, including those with previously unsupported measures.

## Contribution

It introduces a midpoint-dependent extension of Hausdorff measures and establishes a theorem for calculating exact measures of regular Cantor sets.

## Key findings

- Derived a theorem for Hausdorff measure of regular Cantor sets
- Extended Hausdorff measures to depend on midpoints of intervals
- Provided a solution for Cantor sets with unsupported measures

## Abstract

Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try to resolve this problem by studying an extension of the Hausdorff measures \( \mu_h\) on \(\mathbb{R}\), allowing gauge functions to depend on the midpoint of the covering intervals instead of only on the diameter. As a main result, a theorem about the Hausdorff measure of any regular enough Cantor set, with respect to a chosen gauge function, is obtained.