Techniques to Calculate Hausdorff Measure

Mass Distribution

A measure on a bounded subset of ${\mathbb{R}}^n$ for which $0<\mu ({\mathbb{R}}^n)<\infty$ will be called a mass distribution.

Mass Distribution Principle

Let $\mu$ be a mass distribution on $F\subset {\mathbb{R}}^n$, and suppose for some $s>0$, there are numbers $c>0$ and $\epsilon >0$ such that $\mu (U)\le c|U{|}^s$for all sets $U$ with $|U|\le \epsilon$. Then \newcommand{\H}{\mathcal{H}}\H^{s}(F) \geq\mu(F)/ c and $s\le {\dim }_{\mathrm{H}}F\le {\underline{\dim }}_{\mathrm{B}}F\le {\overline{\dim }}_{\mathrm{B}}F.$

Proof Suppose $\{U_i{\}}_{i=1}^{\infty }$ is a cover for $F$, $F\subset {\cup }_i^{\infty }{U}_i$, then $0<\mu (F)\le \mu ({\cup }_i^{\infty }{U}_i)\le {\sum }_{i=1}^{\infty }\mu (U_i)\le c{\sum }_i|U_i{|}^s.$Taking infimum, we have \H^{s}_{\delta}(F)\geq \mu(F)/c for sufficiently small $\delta$. Thus, \H^{s}(F)\geq \mu(F)/c. In particular, since $\mu (F)>0$, we get ${\dim }_{\mathrm{H}}F\ge s$. $\square$

Potential Theoretic Methods

$s$-Potential

For some $s\ge 0$, the $s$-potential at a point $x\in {\mathbb{R}}^n$, resulting from the mass distribution $\mu$ on ${\mathbb{R}}^n$ is defined as ${\varphi }_s(x):=\int \frac{1}{|x-y{|}^s}\text{dd}\mu (y).$

This is a Generalization of The Gravitational Potential

If we are working in ${\mathbb{R}}^3$ and $s=1$, then this is essentially the familiar Newtonian gravitational potential.

$s$-Energy

The $s$-energy of a mass distribution $\mu$ on ${\mathbb{R}}^n$ is defined as $I_s(\mu ):=\int {\varphi }_s(x)\text{dd}\mu (x)=\iint \frac{1}{|x-y{|}^s}\text{dd}\mu (x)\text{dd}\mu (y).$

Theorem

Let $F$ be a subset of ${\mathbb{R}}^n$.

• If there is a mass distribution $\mu$ on $F$ with $I_s(\mu )<\infty$, then \H^{s}(F) = \infty and ${\dim }_{H}F\ge s$.
• If $F$ is a Borel set with 0 < \H^{s}(F) < \infty, then for all $0<t<s$, there exists a mass distribution $\mu$ on $F$ with $I_t(\mu )<\infty$.

Proof