Local Hausdorff Measure

TL;DR

This paper introduces a local Hausdorff dimension and measure on metric spaces, explores their properties, and demonstrates how local measures relate to global dimensions and Ahlfors regular measures.

Contribution

It defines a local Hausdorff measure and shows its connection to global Hausdorff dimension and Ahlfors regular measures on metric spaces.

Findings

Local Hausdorff measure can recover global Hausdorff dimension when finite.

Ahlfors Q-regular measure is strongly equivalent to the local Hausdorff measure.

The local Hausdorff dimension equals the function Q in Ahlfors regular measures.

Abstract

A local Hausdorff dimension is defined on a metric space. We study its properties and use it to define a local Hausdorff measure. We show that in the case that in the local Hausdorff measure is finite we can recover the global Hausdorff dimension from the local one. Lastly, for a variable Ahlfors Q-regular measure on a compact metric space, we show the Ahlfors regular measure is strongly equivalent to the local Hausdorff measure and that the function $Q$ is equal to the local Hausdorff dimension.