Mathematische Annalen

, Volume 319, Issue 4, pp 817–836

# Scaling properties of Hausdorff and packing measures

• Marianna Csörnyei
• R. Daniel Mauldin
Original article

## Abstract.

Let $$m\in{\mathbb N}$$. Let $$\theta$$ be a continuous increasing function defined on $${\mathbb R}^+$$, for which $$\theta(0)=0$$ and $$\theta(t)/t^m$$ is a decreasing function of t. Let $$\|\cdot\|$$ be a norm on $${\mathbb R}^m$$, and let $$\rho$$, $${\mathcal H}^\theta ={\mathcal H}^\theta_\rho$$, $${\mathcal P}^\theta ={\mathcal P}^\theta_\rho$$ denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions $$\theta$$ such that the corresponding Hausdorff or packing measure scales with exponent $$\alpha$$ by showing it must be of the form $$\theta(t) = t^\alpha L(t)$$, where L is slowly varying. We also show that for continuous increasing functions $$\theta$$ and $$\eta$$ defined on $${\mathbb R}^+$$, for which $$\theta(0)=\eta(0)=0$$, $${\mathcal H}^\theta={\mathcal P}^\eta$$ is either trivially true or false: we show that if $${\mathcal H}^\theta={\mathcal P}^\eta$$, then $${\mathcal H}^\theta={\mathcal P}^\eta=c\cdot\lambda$$ for a constant c, where $$\lambda$$ is the Lebesgue measure on $${\mathbb R}^m$$.

Mathematics Subject Classification (2000): 28A12, 28A80