Hausdorff content

Remarks

The map $\mathcal{H}^s_\infty$ is an outer measure on $X$. It has the same null sets as the Hausdorff measure $\mathcal{H}^s$, so $\dim_H E = \inf \set{s > 0 : \mathcal{H}^s_\infty(E) = 0}$, with the convention $\inf \emptyset = \infty$. Example: in $\R$, $\mathcal{H}^{1}_\infty([0,1]) = 1$. The cover by $[0,1]$ itself gives at most $1$, and any countable cover has total diameter at least the Lebesgue measure $1$. Non-example of additivity: in $\R$ with $s = 1/2$, each of $[0,1]$ and $[1,2]$ has content $1$. Indeed, a cover that uses a set of diameter at least $1$ has total at least $1$. In a cover with all diameters below $1$, one has $(\operatorname{diam} U_i)^{1/2} \ge \operatorname{diam} U_i$, and the diameters add up to at least $1$. But the single set $[0,2]$ covers their union, so the union has content at most $\sqrt 2 < 2$.