= Scale-restricted Hausdorff content
{title2=$\mathcal H_\delta^s(F)$}
For $\delta>0$, <scale-restricted Hausdorff content> is the <infimum> of $\sum_j(\operatorname{diam}U_j)^s$ over <countable> <covers> of $F$ by <sets> of <diameter> at most $\delta$, using the convention in <Hausdorff measure>. It increases as $\delta$ decreases; its <limit> is <Hausdorff measure>. Unrestricted Hausdorff content imposes no <diameter> bound and can behave differently.
Any explicit <cover> supplies an upper bound on <scale-restricted Hausdorff content>, rather than its exact value. The level-$n$ intervals of the <Cantor set>, for example, give
$$
\mathcal H_{3^{-n}}^s(C)\leq2^n3^{-ns}.
$$
If this right-hand side tends to zero, then $\mathcal H^s(C)=0$. If it grows, that alone proves no positive lower bound: a different <cover> might be cheaper. The <mass distribution principle> addresses precisely this gap by bounding the cost of every sufficiently small <cover>.
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