= Hausdorff measure
{c}
{title2=$\mathcal H^s$}
{wiki}
Hausdorff measure measures the geometric size of a <set> using <covers> by increasingly small pieces. The exponent $s$ chooses the measuring scale: $s=1$ is length-like, $s=2$ is area-like, and noninteger exponents can measure <fractals>. Unlike a count of occupied grid cells, the construction allows pieces of different sizes and considers every admissible <cover>.
For $s>0$ and a <subset> $E$ of a <metric space>, use the diameter-cost convention
$$
\mathcal H_\delta^s(E)=\inf\left\{
\sum_{j=1}^\infty(\operatorname{diam}U_j)^s:
E\subseteq\bigcup_jU_j,\quad\operatorname{diam}U_j\leq\delta
\right\},\qquad
\mathcal H^s(E)=\lim_{\delta\downarrow0}\mathcal H_\delta^s(E).
$$
The first quantity is <scale-restricted Hausdorff content>. Shrinking $\delta$ removes options from the <infimum>, so these contents increase to a <limit>, possibly infinite. The resulting <outer measure> restricts to a <measure> on the <Borel sets>. For $s=0$, counting each nonempty covering piece with cost one gives <counting measure>.
A different common convention multiplies these quantities by $c_s=\pi^{s/2}/(2^s\Gamma(1+s/2))$, using the <Gamma function>. At integer $s$, that normalization agrees with ordinary Euclidean $s$-dimensional volume. Here $c_1=1$, so both conventions make $\mathcal H^1$ agree with <arc length> on <smooth curves> in <Euclidean space>. Positive normalization constants never change <Hausdorff dimension>. A definition must specify this constant when numerical values of <Hausdorff measure> matter.
For a line segment of <length> $L$, splitting it into $n$ equal pieces gives the <cover> cost
$$
n(L/n)^s=L^s n^{1-s}.
$$
It tends to zero for $s>1$, stays at $L$ for $s=1$, and grows for $s<1$. The matching lower bound comes from the fact that any interval <cover> must have total <length> at least $L$. The exponent one is therefore the transition between a measuring rule too coarse to see the segment and one that assigns it infinite size. For a <Cantor set>, the transition is instead at $\log2/\log3$, as shown in <Hausdorff dimension of the Cantor set>.
This makes <Hausdorff measure> useful when ordinary <Lebesgue measure> sees too little: a <curve> in the plane has zero planar area but can have positive <arc length>, and the <Cantor set> has zero line <length> but positive geometric size at its own dimension. It also separates local covering estimates from global cardinality: a <countable> dense <set> has zero <Hausdorff dimension>, despite meeting every open region of its closure.
An original source for the construction is Felix Hausdorff's https://link.springer.com/article/10.1007/BF01457179[Dimension und äußeres Maß, Mathematische Annalen 79, pp. 157–179]{ref}. The modern covering and <Hausdorff dimension> framework is developed in https://www.math.stonybrook.edu/~bishop/fractalbook.pdf[Bishop and Peres, Fractals in Probability and Analysis, §1.2].
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