Hausdorff Measure

Definition

Let $(X,\rho )$ be a metric space. We define the diameter of a subset $U\subset X$ by

$$\text{diam}(U)=\sup \{\rho (x,y)\mid x,y\in U\},$$

i.e., the distance between the farthest pair of points. By definition we say that $\text{diam}\varnothing =0$.

For any real number $\delta >0$ and subset $S\subset X$, we define

$$H_{\delta }^d(S)=\inf \left\{\sum _{i=1}^{\infty }(\text{diam}{U}_i{)}^d:\bigcup _{i=1}^{\infty }{U}_i\supset S,\text{diam}{U}_i<\delta \right\}.$$

We then define the outer measure

$$H_{\ast }^d(S)=\underset{\delta \to 0}{\lim }{H}_{\delta }^d(S).$$

By the Caratheodory’s extension theorem, $H_{\ast }^d$ restricted the $\sigma$-algebra of sets $E$ that satisfy

$$H_{\ast }^d(A)=H_{\ast }^d(E\cap A)+H_{\ast }^d(E^c\cap A)$$

for all $A\subset X$, is a measure which we call the $d$-dimensional Hausdorff measure.

Properties

The following properties hold for the Hausdorff outer measure, and consequently, the Hausdorff measure:

1. Monotonicity: If $E_1\subset E_2$, then $H^d(E_1)\le H^d(E_2)$.
2. Sub-additivity: For a countable family of sets $E_1,E_2,\dots$, $H^d({\bigcup }_{j=1}^{\infty }{E}_j)\le {\sum }_{j=1}^{\infty }{H}^d(E_j)$.
3. If $\rho (E_1,E_2)>0$, then $H^d(E_1\cup E_2)=H^d(E_1)+H^d(E_2)$.
4. $H^d(E+h)=H^d(E)$ for all $h\in X$ (whenever this operation makes sense).
5. $H^d(\lambda E)={\lambda }^d{H}^d(E)$ for all $\lambda >0$ (whenever this operation makes sense).

Thm

If $X={\mathbb{R}}^d$ with the usual metric, then $H^d=m$. That is, the Hausdorff measure and the Lebesgue measure are equivalent.