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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 conventionThe 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 costIt 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 $\log 2/\log 3$, 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 ^[ref]. The modern covering and Hausdorff dimension framework is developed in Bishop and Peres, Fractals in Probability and Analysis, §1.2.

The mass distribution principle converts a diffuse measure into a lower bound on Hausdorff dimension. Let $s>0$ and let a Borel probability measure $\mu$ give a Borel set $E$ full mass. If, for some $C,r_0>0$, every ball centered in $E$ satisfiesthen ${\mathcal{H}}^s(E)\ge 1/C>0$ in the diameter-cost convention, and consequently ${\dim }_{H}E\ge s$.

The idea is to keep unit mass from being crowded into cheap covering pieces. For a piece $U$ of positive diameter $r$ meeting $E$, choose $x\in U\cap E$. A ball of radius $r+\epsilon$ contains $U$; letting $\epsilon \downarrow 0$ gives ${\mu }^{\ast }(U)\le C{r}^s$, where ${\mu }^{\ast }$ is the associated outer measure. Zero-diameter pieces have zero mass by the same estimate. For any sufficiently small cover,Thus every cover costs at least $1/C$. This is the lower-bound counterpart to the explicit covers in scale-restricted Hausdorff content. The Cantor distribution supplies the needed mass in Hausdorff dimension of the Cantor set. ^[ref].

For $s\ge 0$ and $\delta >0$, define the scale-restricted Hausdorff content at scale $\delta$ byusing countable covers. The Hausdorff measure is ${\mathcal{H}}^s(F)={\lim }_{\delta \downarrow 0}{\mathcal{H}}_{\delta }^s(F)$; the limit exists by monotonicity. A conventional dimension-dependent normalizing constant does not change the Hausdorff dimensionAt $s=0$, a nonempty singleton has contribution one, so the measure counts points.

Let $\mathcal{T}(K)={\bigcup }_{i=1}^k{S}_i(K)$ and $c={\max }_i{c}_i<1$. For compact nonempty $K,L$, matching a point of $S_i(K)$ with the image of a nearby point of $L$ gives $d_H(\mathcal{T}K,\mathcal{T}L)\le c{d}_H(K,L)$. Choose a sufficiently large closed ball $B$ with every $S_i(B)\subset B$. The nonempty compact sets ${\mathcal{T}}^m(B)$ are nested, so their intersection $I$ is nonempty and compact. The finite union of similarity images commutes with this decreasing compact intersection: a point lying in all unions has some repeated branch index, and compactness supplies a preimage in all the nested sets. Consequently $\mathcal{T}(I)=I$. If $J$ is another invariant compact set, the contraction estimate gives $d_H(I,J)\le c{d}_H(I,J)$, hence $I=J$. This proves existence and uniqueness of the invariant set of a finite similarity iterated function system.

Under the open set condition, there is a nonempty bounded open set $O$ such that the $S_i(O)$ lie inside $O$ and are pairwise disjoint. The dimension formula isOverlaps can invalidate this formula if that assumption is omitted.

For the printed template, subdivide the central straight run of length two into its two collinear unit steps. The eight directed steps, before dividing all coordinates by four, have successive verticesEach step defines a similarity with ratio $1/4$, with rotation matching its direction. The open diamond $O=\{|x-1/2|+|y|<1/2\}$ is an explicit witness for the open set condition. Its eight image diamonds have radius $1/8$ in the ${\ell }^1$ metric, lie in $O$, and their centres have pairwise ${\ell }^1$ distance at least $1/4$; their interiors are therefore disjoint. The Minkowski sausage dimension is consequentlyThis makes the subdivision convention explicit. If the length-two run were instead treated as one unsplit edge replaced at scale $1/2$, there would be six maps of ratio $1/4$ and one of ratio $1/2$. The same diamond separation works, but the equation becomes $64^{-s}+2^{-s}=1$, giving $s={\log }_{2}3$, a different fractal. The requested $3/2$ corresponds to the eight-unit-step interpretation.

For $\delta >0$, scale-restricted Hausdorff content is the infimum of ${\sum }_j(\mathrm{diam}{U}_j{)}^s$ 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, giveIf 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.