Hausdorff measure in a metric space

For $s>0$ and $E$ in a metric space, define

\mathcal H^s_\delta(E)=\inf\left\{\sum_j(\operatorname{diam}U_j)^s: E\subseteq\bigcup_jU_j,\ \operatorname{diam}U_j\le\delta\right\}.

The Hausdorff outer measure is $\mathcal H^s(E)=\lim_{\delta\downarrow0}\mathcal H^s_\delta(E)$ . The limit exists because the covering infimum increases as the allowed diameters shrink. Its restriction to Borel sets is a measure. This convention omits a normalization constant; null sets are unaffected by multiplying by a fixed positive constant.

Dimension and metric

The associated Hausdorff dimension records the threshold where these measures vanish. The metric is part of the definition: a different scaling of time and space can change both the measure and the dimension. For $s=0$ , the conventional Hausdorff measure is counting measure.