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For a BV space function, this is the set where no finite approximate mean limit of a function exists. Its difference from the jump set of a bounded-variation function has zero $(n-1)$-dimensional Hausdorff measure.

A BV space function has two one-sided approximate mean BV traces on a hypersurface at almost every point of an oriented countably rectifiable hypersurface, with respect to Hausdorff measure. They agree outside its approximate discontinuity set. At an approximate jump point, they are the jump values, up to the choice of orientation.

A cover of a set $E$ is a family of sets whose union contains $E$. The members may overlap and may include points outside $E$; intersecting each member with $E$ gives a cover by its subsets. An open cover requires its members to be open sets. In Hausdorff measure, countable covers with a bound on each member's diameter determine geometric size. Constructing one inexpensive cover supplies an upper bound, while a lower bound must control all allowed covers.

Geometric measure theory uses measures and analysis to study geometric objects that need not be smooth. Its measuring rules can distinguish a curve from its surrounding plane or capture the fine structure of a fractal. Hausdorff measure supplies size adapted to an exponent, while Hausdorff dimension identifies the exponent at which that size changes character.

The Cantor set is an elementary starting point: it has zero Lebesgue measure but positive Hausdorff dimension. In Hausdorff dimension of the Cantor set, explicit small covers give an upper bound and a distributed probability measure supplies the matching lower bound. This combination of geometric coverings and analytic control is useful even before more advanced tools for irregular surfaces are introduced.

The Hausdorff dimension is the critical exponent at which Hausdorff measure changes from infinite to zero:with the standard endpoint convention for an empty supremum. This measures fine-scale geometric size rather than the number of coordinates used to describe the set. A line segment has Hausdorff dimension one; a square has dimension two; the middle-third Cantor set has dimension $\log 2/\log 3\approx 0.63093$.

Why can a dimension be fractional? At a scale reduced by a factor of three, the Cantor set needs twice as many construction intervals. The balance $2(1/3{)}^d=1$ gives $d=\log 2/\log 3$, between the growth rates of finitely many points and a whole interval. Making this argument rigorous requires a cover upper bound and the mass distribution principle lower bound, as in Hausdorff dimension of the Cantor set.

At the critical exponent itself, Hausdorff measure can be zero, finite and positive, or infinite; dimension does not specify that endpoint value. For example, finite sets and countably infinite sets both have dimension zero, although their counting measures differ. Dimension is also different from a finite-resolution estimate: box-counting dimension is another asymptotic definition, and fitting a scaling slope over a handful of observed scales does not establish either dimension.

The metric matters. Multiplying every distance by a fixed positive constant multiplies ${\mathcal{H}}^s$ by that constant to the power $s$ and preserves Hausdorff dimension. A Lipschitz map cannot increase it, because mapped cover diameters increase by at most a fixed factor. A bi-Lipschitz map therefore preserves it by applying this fact in both directions. These properties make Hausdorff dimension useful for comparing geometric models while controlling how their measurements change.

The middle-third Cantor set has Hausdorff dimensionIts Lebesgue measure is zero, yet $0<{\mathcal{H}}^d(C)<\infty$. These assertions measure different aspects of size: interval length is too coarse to distinguish this set from a null set, while the critical fractional measuring rule captures its remaining structure.

At level $n$, the construction covers $C$ by $2^n$ intervals of diameter $3^{-n}$. Their total $s$-cost isFor $s>d$ it tends to zero, proving the upper dimension bound. At $s=d$ it equals one, giving ${\mathcal{H}}^d(C)\le 1$. For $s<d$ this particular cover's cost grows, but that alone would not prove a lower bound on Hausdorff measure.

For the matching lower bound, use the Cantor distribution, which assigns mass $2^{-n}$ to each level-$n$ Cantor cylinder. An interval of length $r$ with $3^{-(n+1)}<r\le 3^{-n}$ meets at most two level-$n$ construction intervals, since their gaps are at least $3^{-n}$. Its mass is therefore at mostEvery covering set of diameter $r$ in the line lies in an interval of length $r$, so any small cover must have total $d$-cost at least $1/4$. This is the mass distribution principle in its interval form: $1/4\le {\mathcal{H}}^d(C)\le 1$, sufficient to prove the claimed dimension. The constants are convenient bounds, rather than an assertion of the exact critical Hausdorff measure. In the diagram, the right panel is a finite-stage approximation to the Cantor function, showing how probability is shared while the middle third receives none.

The reusable pattern is to obtain an upper bound by constructing covers and a lower bound by distributing mass. The same strategy underlies Cartesian powers of the Cantor set and the dimension formula for a finite similarity iterated function system satisfying the open set condition.

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.

Use zero-based indices $0\le i<2^n$; the two index conventions in the printed question enumerate the same partition. Let $O_n(\gamma )$ be the sum of length powers over odd intervals and $H_n(\gamma )$ the sum over all intervals. For odd-address words, the given symbolic potential estimate has $c\le |{\Delta }_i^{(n)}|e^{-H_n^{\mathrm{pot}}(i)}\le C$, with constants independent of $n,i$. Raising these bounds to a fixed real power $\gamma$ and reversing the order when necessary gives finite positive constants $c_{\gamma },C_{\gamma }$ such thatTherefore $n^{-1}\log O_n(\gamma )\to f(\gamma )$. It remains to justify the even intervals, since the supplied estimate covers only half of the partition.

In the standard period-doubling partition, ${\Delta }_0^{(n)}={\beta }^n[-1,1]$ and ${\Delta }_i^{(n)}=g^i({\Delta }_0^{(n)})$. All odd-address intervals lie in the first image restrictive interval, a compact set away from the unique critical point zero. Hence there are uniform constants $0<d\le |g^{\prime }|\le D_1$ there. For an odd $i<2^n-1$, its image is the next even interval, and the mean value theorem gives $d|{\Delta }_i^{(n)}|\le |{\Delta }_{i+1}^{(n)}|\le D_1|{\Delta }_i^{(n)}|$.

For the wraparound pair the image of the last interval is a subinterval of ${\Delta }_0^{(n)}$, not necessarily the whole of it. Iterating the renormalization identity givesThus $|g({\Delta }_{2^n-1}^{(n)})|=|\beta {|}^n(1-\beta )$, while $|{\Delta }_0^{(n)}|=2|\beta {|}^n$. The image is the fixed fraction $(1-\beta )/2$ of the central interval. Combining this with the derivative bounds proves uniform comparability of the last odd interval and the central even interval as well. This establishes the odd-even comparison for Feigenbaum partition lengths for every pair, including wraparound.

Consequently, for each fixed real $\gamma$, the even sum is bounded above and below by constant multiples of $O_n(\gamma )$. The same is true of $H_n(\gamma )$, soFinally take $\gamma >{\gamma }_{\ast }$. Strict decrease gives $f(\gamma )<0$, so this length-power sum tends to zero exponentially. The cover mesh also tends to zero: negative bounded $U$ and the odd-interval estimate give $|{\Delta }_i^{(n)}|\le C{e}^{-bn}$ on odd intervals, and the odd-even comparison transfers this bound to all intervals. Since the intervals cover the Feigenbaum attractor, they are admissible arbitrarily fine covers in the definition of Hausdorff measure, and ${\mathcal{H}}^{\gamma }(F)=0$. Therefore the pressure upper bound for Hausdorff dimension yieldsThe covering argument supplies the requested upper bound; it does not assume a dimension equality or a lower bound not proved here.

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.

Using the diameter normalization, define ${\mathcal{H}}_{\delta }^d(E)=\inf \{{\sum }_j(\mathrm{diam}{U}_j{)}^d:E\subseteq {\bigcup }_j{U}_j,\mathrm{diam}{U}_j\le \delta \}$ and the Hausdorff measure ${\mathcal{H}}^d(E)={\lim }_{\delta \downarrow 0}{\mathcal{H}}_{\delta }^d(E)$. The Hausdorff dimension is $\inf \{d:{\mathcal{H}}^d(E)=0\}$, equivalently $\sup \{d:{\mathcal{H}}^d(E)=\infty \}$.

The Cantor set is the intersection of the sets left by successively removing open middle thirds from $[0,1]$. At level $k$ it has $2^k$ Cantor cylinders of diameter $3^{-k}$. For $s=\log 2/\log 3$, their total $s$-cost is $2^k{3}^{-ks}=1$, so ${\mathcal{H}}^s(C)\le 1$.

The random series $\xi ={\sum }_{n\ge 1}{X}_n{3}^{-n}$ converges absolutely, and its ternary digits are all zero or two. Thus $\xi \in C$ and its law is the Cantor Bernoulli measure. Each level-$k$ cylinder has probability $2^{-k}$. Distinct such intervals have separation at least $3^{-k}$; a set $U$ of smaller diameter meets at most one. If $3^{-(k+1)}\le \mathrm{diam}U<3^{-k}$, thenSets of diameter zero have probability zero, since the measure has no measure atoms; diameters at least one give the bound trivially. For arbitrary nonmeasurable sets use outer probability. Any countable cover of $C$ therefore satisfies $1\le {\sum }_j{\mathbb{P}}^{\ast }(\xi \in U_j)\le 2{\sum }_j(\mathrm{diam}{U}_j{)}^s$. HenceThe last implication follows by multiplying the $s$-costs by powers of the cover diameter: measures above $s$ vanish and those below $s$ are infinite.

Let $C$ be the middle-third Cantor set and choose a positive integer $d$ with $d\log 2/\log 3\ge k$; this is possible for every real $k$, including negative $k$. Set $X=C^d\subset {\mathbb{R}}^d$. It is closed and bounded since each factor is compact. If a subset of $X$ is connected, each coordinate projection is a connected subset of $C$ and hence a singleton; therefore the subset itself is a singleton. Thus $X$ is totally disconnected.

For completeness, the dimension of these Cartesian powers of the Cantor set can be checked directly. At stage $n$ there are $2^{dn}$ cubes of side $3^{-n}$, giving vanishing $s$-dimensional cover cost whenever $s>d\log 2/\log 3$. Conversely put independent equiprobable zero-or-two ternary digits on every coordinate. Each stage-$n$ cube has probability $2^{-dn}$. If $3^{-n}\le r<3^{-n+1}$, a Euclidean ball of radius $r$ meets at most a constant depending on $d$ such cubes. Hence its probability is at most $C_d{r}^{d\log 2/\log 3}$. For any sufficiently fine cover, summing this bound over covering sets shows that its cost at exponent $d\log 2/\log 3$ is bounded below by a positive constant, since total probability is one. These upper and lower Hausdorff measure bounds proveThe construction does not claim arbitrarily large Hausdorff dimension in one fixed ambient Euclidean dimension.

An image segmentation separates an observed image signal into regions that are smooth within themselves and separated by meaningful image edges. Pure image smoothing blurs the very transitions that ought to define these regions. The Mumford–Shah segmentation model instead chooses the reconstruction and its discontinuity set together. For a bounded planar Lipschitz domain and bounded grey-value data $g$, one standard normalization isHere $K$ is the relatively closed edge set and $u\in H^1(\Omega \setminus K)$ can have different traces on its two sides. The first term is quadratic fidelity, the second penalizes variation within regions, and the Hausdorff measure term charges total edge length. It balances fitting, denoising and economical region geometry. The model was developed by David Mumford and Jayant Shah; their 1989 paper formulates this joint variational approach.

All three terms matter. Without fidelity, a constant reconstruction with no edge has zero energy. Without the gradient term, smooth data can be fitted exactly with no edge penalty. Without the length term, fine partitions with nearly constant region means can drive fidelity arbitrarily low while creating excessive boundaries. This is segmentation overfitting without an edge penalty. Increasing $\alpha$ favors flatter regions; increasing $\beta$ makes extra boundaries more expensive and can remove small features. These parameter effects describe a balance, not a guaranteed monotone evolution of every individual boundary.

For fixed $K$, varying $u$ gives the fixed-edge Euler-Lagrange equation for Mumford–Shah:The boundary condition is a separate one-sided Neumann boundary condition on each side of an edge, not continuity of $u$ across it. The fidelity makes this fixed-edge problem strictly convex, so its weak solution is unique by the Lax-Milgram theorem. Optimizing the edge set remains a different geometric problem. The nonconvexity of Mumford–Shah segmentation prevents a general uniqueness assertion or a guarantee that a numerical stationary point is globally optimal.

The piecewise-constant Mumford–Shah problem imposes $u=c_i$ on regions $E_i$. It minimizesThe factor $1/2$ counts each shared internal boundary once. The region means in piecewise-constant segmentation give $c_i=|E_i{|}^{-1}{\int }_{E_i}g$, provided $|E_i|>0$. Thus the remaining optimization concerns the partition. For a smooth interface between $i$ and $j$, moving it in the normal pointing out of $i$ changes fidelity by $[(c_i-g{)}^2-(c_j-g{)}^2]$ per unit displacement, while length changes by its curvature $\kappa$. The resulting segmentation interface curvature balance isWith equal isotropic interface costs, three freely meeting smooth edges satisfy the triple-junction angle in isotropic segmentation: force balance of their unit tangents gives angles of $120$ degrees. These are local stationarity conditions on regular interfaces, not a description of every singular edge configuration.

A simple piecewise-constant segmentation contrast threshold explains why small objects can disappear. On a domain of area $A$, suppose two constant intensities differ by $d$, occupying areas $a$ and $A-a$ with internal boundary length $L$. Keeping that boundary fits the data exactly and costs $\beta L$. Merging both regions costs the within-region squared error $a(A-a)d^2/A$. Among these two candidates, splitting wins precisely when $\beta L<a(A-a)d^2/A$. Other partitions may beat either candidate, so this is not a universal global segmentation formula.

This original synthetic example compares those two candidate geometries using their least-squares means. It illustrates the edge cost; it does not claim to compute the globally optimal Mumford–Shah segmentation.

Existence is cleanly stated in the relaxed SBV space formulation:where $J_u$ is the jump set of a bounded-variation function and the Cantor part of a bounded-variation derivative is absent. Clip a minimizing sequence to the bounded data range: this cannot increase fidelity, gradient energy or jump length. Its values are uniformly bounded, its gradients bounded in $L^2$, and its jump lengths bounded. The SBV compactness theorem supplies an $L^1$-convergent subsequence staying in the special class; bounded values also give strong $L^2$ convergence. The fidelity then converges, while gradient energy and jump length are lower semicontinuous. The direct method in the calculus of variations produces a minimizer. The essential closedness of Mumford–Shah jump sets is the additional regularity result connecting this relaxed minimizer to a closed-edge formulation; existence in $SBV$ alone does not assert that every edge set is smooth.

A practical continuous approximation is the Ambrosio–Tortorelli approximation, introduced in Ambrosio and Tortorelli's 1990 paper. An auxiliary field $v\in [0,1]$ is near one in regions and near zero at edges. One normalized energy isThe edge field weakens smoothing across a narrow transition, and its own energy approximates interface length. The Gamma-convergence result, together with the required compactness, relates global minimizing sequences to the limiting segmentation energy; it does not make the finite-parameter problem jointly convex. Alternating the two fields solves quadratic elliptic subproblems, but initialization and stopping can affect which local stationary configuration is found. The strengths of the model are joint denoising and segmentation, sharp transitions and a geometric cost. Limitations include competing local minima, sensitivity to scale parameters, loss of fine texture, and boundaries driven by intensity rather than semantic object identity.

A rigorous Mumford–Shah functional permits nonsmooth image signals and free discontinuities. One classical admissible class consists of relatively closed countably rectifiable sets $K\subset \Omega$ with finite Hausdorff measure ${\mathcal{H}}^1(K)$, and $u\in W^{1,2}(\Omega \setminus K)\cap L^{\infty }(\Omega )$ with finite energy. No exterior boundary values are prescribed. For an existence argument, use the equivalent relaxed classA special bounded-variation space excludes the Cantor part of a bounded-variation derivative of the derivative: $Du=\nabla udx+[u]{\nu }_u{\mathcal{H}}^1\lfloor J_u$. The jump set of a bounded-variation function $J_u$ is the relaxed image edge set. Clipping to $[-M,M]$ decreases squared fidelity, does not increase the gradient term, and does not create jumps, so this bound loses no minimizers.

Take a minimizing sequence and compare with a constant image signal. Its $L^2$ gradient norms and jump lengths are bounded. Alsoso the sequence is bounded in $BV$. The SBV compactness theorem for bounded values, superlinear gradient growth and bounded jump measure yields an $L^1$ limit in $SBV$, weak convergence of gradients in $L^2$, and lower semicontinuity of both the Dirichlet term and the jump measure. The uniform value bound upgrades convergence to $L^2$, so fidelity converges. This proves existence of a relaxed minimizer. Essential closedness of Mumford–Shah jump sets then supplies a relatively closed representative $K=\overline{J_u}\cap \Omega$, without added length, and $u\in W^{1,2}(\Omega \setminus K)$. This completes the outline for the classical pair problem. Arbitrary Hausdorff convergence of image edge sets alone is not an adequate substitute for these compactness and regularity results. No uniqueness is claimed for segmentation.

As $\alpha \to \infty$ with $\beta$ fixed, bounded energy forces $\nabla u\to 0$ in $L^2$. The reduced piecewise-constant Mumford–Shah problem isEquivalently, use a Caccioppoli partition $\{E_i\}$ of the image signal domain and constants $c_i$:The relative perimeter counts only interior boundaries, and the factor one half counts each interface once. Adjacent equal-valued regions can be merged, removing unnecessary boundaries.

For fixed $K$, let $E_i$ be its positive-area regions. Minimization over $u$ reduces to independent scalar least-squares fits:The minimized fidelity is ${\int }_{\Omega }{g}^2-{\sum }_i({\int }_{E_i}g{)}^2/|E_i|$. Thus region means in piecewise-constant segmentation give the optimal grey values for a fixed segmentation.

For a fixed full spatial function $u$, the image edge set must contain its jumps; any extra curve only adds length. The optimal choice is its essential jump set, with a relatively closed representative when appropriate. There is no independent relocation of boundaries while that full function is held fixed. A different common alternating step fixes the values $c_i$ but allows the labels $E_i$ to move. It minimizes the fidelity-plus-perimeter partition functional above. Without the perimeter term each point takes its nearest grey value; with it, interface length is penalized. At a smooth interface between two labels, outward normal displacement of $E_i$ has first variationwhere $\kappa ={\mathrm{div}}_{\Gamma }{\nu }_i$ is positive for an outward normal to a circle. The stationary segmentation interface curvature balance isThis is the geometric interpretation of optimizing boundaries with fixed grey levels, and distinguishes it from fixing the whole spatial image signal.

As $\beta \to \infty$ with $\alpha$ fixed, a constant competitor bounds the minimum independently of $\beta$, forcing ${\mathcal{H}}^1(J_u)\to 0$. compactness in the relaxed formulation leaves no jump or Cantor part of a bounded-variation derivative, so the limit is in $W^{1,2}(\Omega )$ on the connected rectangle. The reduced edge-free Mumford–Shah limit isA set of zero length can be omitted; this does not impose a zero image signal or a Dirichlet boundary value. Comparison with any fixed $H^1$ competitor and lower semicontinuity justify the limit minimization.

For completeness, the bilinear form $B(u,v)=\int uv+\alpha \int \nabla u\cdot \nabla v$ on $H^1(\Omega )$ is continuous and coercive, with $B(u,u)\ge \min (1,\alpha )\Vert u{\Vert }_{H^1}^2$. The right-hand side $\int gv$ is bounded because $g\in L^2$ on the bounded rectangle. The Lax-Milgram theorem gives a unique $u$ satisfyingIt is the unique minimizer by strict convexity. Formally,with the Neumann condition understood through this weak formulation. Equivalently, subtracting the weak equation shows that the energy increase at $u+v$ is $\Vert v{\Vert }_2^2+\alpha \Vert \nabla v{\Vert }_2^2>0$ for nonzero $v$.

Take the indicator function $u(x_1,x_2)={\chi }_{\{x_1>1/2\}}$ inside $(0,1{)}^2$. For a compactly supported test vector field, integration in $x_1$ and then $x_2$ givesThe vertical derivative term integrates to zero. The absolute pairing is at most one, and test vector fields with ${\xi }_1=-1$ along all but arbitrarily short endpoint pieces approach one. Thus $|Du|(\Omega )=1$. More explicitly, the distributional derivative is $Du=e_1{\mathcal{H}}^1\upharpoonright \{(1/2,x_2):0<x_2<1\}$, where ${\mathcal{H}}^1$ is Hausdorff measure along the segment. Also $\Vert u{\Vert }_1=1/2$.

A Sobolev space function in $W^{1,1}(\Omega )$ has each weak derivative represented by an $L^1$ density with respect to planar Lebesgue measure. The nonzero segment measure here is singular with respect to that measure, so this requirement fails. Equivalently, the Sobolev fundamental theorem of calculus on lines would require an absolutely continuous representative on almost every horizontal slice, whereas each slice has a jump. ThereforeThis is the bounded-variation step outside W11 example.

The perimeter of a measurable set relative to an open domain is the total variation seminorm of its indicator function there. For a smooth region it equals the Hausdorff measure of the boundary lying inside the domain. Global perimeter uses the entire space as the domain; an internal interface shared by two regions appears in both relative perimeters and must be counted only once in a segmentation energy.

Here the sets of finite perimeter $E,F$ differ only compactly inside $U$. A minimizing set with bounded volume forcing satisfies this inequality. Interior regularity theorems give graph patches outside a singular set negligible for surface Hausdorff measure; bounded forcing allows the graph estimates used in the noncontact of ROF level boundaries.

Suppose shrinking interval covers of $F$ have length-power sums with logarithmic growth rate $P(\gamma )$, and $P$ is strictly decreasing with zero ${\gamma }_{\ast }$. For every $\gamma >{\gamma }_{\ast }$ those sums tend to zero exponentially. The definition of Hausdorff measure then gives ${\mathcal{H}}^{\gamma }(F)=0$, hence the displayed upper bound. A matching lower bound requires additional information and does not follow from this covering argument alone.

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.