Rectifiability

We take the rectifiable dimension \(n\geq1\). Rectifiability describes sets that can be covered, apart from a set of Hausdorff measure zero, by countably many Lipschitz images of Euclidean sets. It retains enough of the structure of a smooth submanifold to support tangent planes and area calculations.

We compare this parametrized description with the existence of approximate tangent planes almost everywhere. The converse direction is the main task: concentration near planes must be turned into Lipschitz graph pieces. We then discuss the part of a set that admits no rectifiable piece of positive measure.

Definition 1. Let \(M \subset \mathbb R^{n+m}\). \(M\) is countably \(n\)-rectifiable if

\[\begin{aligned} M \subset M_0 \cup \bigcup_{j \geq 1} F_j (\mathbb R^n) ~, \end{aligned}\]

where \(\mathcal H^n (M_0) = 0\) and \(F_j : \mathbb R^n \to \mathbb R^{n+m}\) are Lipschitz functions for all \(j\). In the view of the extension theorems, this amounts to saying that for some \(A_j \subset \mathbb R^n\),

\[\begin{aligned} M \subset M_0 \cup \bigcup_{j \geq 1} F_j (A_j) ~, \end{aligned}\]

under the identical conditions as stated above.

Proposition 2. A set \(M\subset\mathbb R^{n+m}\) is countably \(n\)-rectifiable if and only if it is contained in a null set together with countably many embedded \(n\)-dimensional \(C^1\) submanifolds.

Proof. A \(C^1\) parametrization is Lipschitz on each closed ball compactly contained in its coordinate domain. Second countability and a countable exhaustion of each chart give a countable Lipschitz cover. Each restricted map extends componentwise to the whole \(\mathbb R^n\), so this proves one direction.

Conversely, consider a Lipschitz map \(F:\mathbb R^n\to\mathbb R^{n+m}\). The \(C^1\) approximation theorem supplies maps \(G_i:\mathbb R^n\to\mathbb R^{n+m}\) for which

\[\lambda_n\{x:F(x)\ne G_i(x)\text{ or }DF(x)\ne DG_i(x)\}<1/i.\]

Apply the scalar theorem to each output component with error \(1/[i(n+m)]\) and take their union. The exact input is Simon, Chapter 2, Theorem 1.5, which follows from Whitney extension after selecting sets with continuous derivative and uniform differentiability remainder. The common exceptional set \(Z=\mathbb R^n\setminus\bigcup_i\{F=G_i,DF=DG_i\}\) is null, since it has measure at most \(1/i\) for every \(i\). Its Lipschitz image is \(\mathcal H^n\)-null.

For each \(G_i\), the area formula on bounded coordinate balls shows

\[\mathcal H^n\bigl(G_i(\{J_nG_i=0\})\bigr)=0.\]

It is the image of the critical set that is null; the critical set in the domain need not be null. At a regular point, one coordinate \(n\)-minor of \(DG_i\) is nonzero. The inverse function theorem for that coordinate projection writes the nearby image of \(G_i\) as an embedded \(C^1\) graph. Countably many such neighbourhoods cover all regular points. Therefore \(F(\mathbb R^n)\) is covered by countably many embedded submanifolds and a null set. Repeat this for the countably many parametrizations of \(M\). ◻

Remark 3. If \(M\) is measurable, disjointize the cover by \(M_0=M\cap N_0\) and \(M_j=M\cap(N_j\setminus\bigcup_{i<j}N_i)\). The manifold patches can be chosen with finite area in precompact graph charts. This makes the local measure estimates below applicable without assuming that a whole nonproper embedded manifold has finite area in every ambient compact set.

Definition 4. Let \(M\) be a \(\mathcal H^n\)-measurable subset of \(\mathbb R^{n+m}\), with \(\mathcal H^n(M \cap K) < \infty\) for all compact sets \(K\). A \(n\)-dimensional subspace \(P \subset \mathbb R^{n+m}\) is called the approximate tangent space for \(M\) at \(x \in \mathbb R^{n+m}\) if

\[\begin{aligned} \lim\limits_{c \to 0} \int_{\eta_{x,c}(M)} f(y) \dif \mathcal H^n(y) = \int_P f(y) \dif \mathcal H^n(y) \quad \forall f \in \mathscr C_c(\mathbb R^{n+m}) ~. \end{aligned}\]

Here, \(\eta_{x,c}(y) \triangleq \frac1c (y-x)\) for all \(c > 0\) and \(x,y \in \mathbb R^{n+m}\).

The next characterization connects countably \(n\)-rectifiability with approximate tangent spaces.

Theorem 5. Suppose \(M\) is \(\mathcal H^n\)-measurable with \(\mathcal H^n(M \cap K) < \infty\) for each compact subset \(K \subset \mathbb R^{n+m}\). Then, \(M\) is countably \(n\)-rectifiable if and only if the approximate tangent space \(T_x M\) exists for \(\mathcal H^n\)-a.e. \(x \in M\).

Proof, rectifiability implies tangents. Use the disjoint pieces \(M_j\subset N_j\) from Remark 3. A \(C^1\) graph, rescaled at \(x\), converges to its tangent plane: in graph coordinates the scaled graph function converges uniformly on bounded sets to its derivative, and the area Jacobian converges uniformly to the linear one. The area formula and dominated convergence then give

\[\int_{\eta_{x,c}(N_j)}f\,d\mathcal H^n\longrightarrow\int_{T_xN_j}f\,d\mathcal H^n.\]

For \(f\) supported in \(B_R(0)\), write the exact identity

\[\int_{\eta_{x,c}(M)}f=\int_{\eta_{x,c}(N_j)}f-\int_{\eta_{x,c}(N_j\setminus M_j)}f+\int_{\eta_{x,c}(M\setminus M_j)}f.\]

The missing part of \(N_j\) is subtracted. By the upper-density theorem applied locally to the last two sets, for almost every \(x\in M_j\) each error in absolute value is at most

\[\|f\|_\infty c^{-n}\mathcal H^n(E\cap B_{cR}(x))\longrightarrow0.\]

The balls are centred at \(x\). Thus the approximate tangent is \(T_xN_j\) at almost every point of \(M_j\), and the countable union proves this direction.

Conversely, tangents imply rectifiability. Write \(\mu=\mathcal H^n\mathbin{\restriction}M\). It is locally finite. Work in a fixed bounded ball, and discard the null set where the tangent does not exist. At each remaining point \(x\), let \(P_x\) be its \(n\)-plane tangent and \(\Pi_x=P_x^\perp\) its \(m\)-dimensional normal plane. If \(q_\Pi\) denotes projection onto \(\Pi^\perp\), define the cone around the normal plane by

\[X_\alpha(\Pi,x)=\{z:|q_\Pi(z-x)|\leq\alpha|z-x|\}.\]

The assumed weak convergence of rescaled measures gives

\[\frac{\mu(B_r(x))}{\omega_nr^n}\to1,\qquad\frac{\mu(X_{1/2}(\Pi_x,x)\cap B_r(x))}{\omega_nr^n}\to0.\]

Indeed the unit sphere has zero \(n\)-measure in \(P_x\), while the closed normal cone meets \(P_x\) only at the origin; continuous upper/lower approximations to these sets justify the two limits.

Fix \(\varepsilon=(4\cdot16^n)^{-1}\). For each integer \(k\), collect the points where, for every \(0<r<1/k\),

\[\mu(B_r(x))\geq(1-\varepsilon)\omega_nr^n,\qquad\mu(X_{1/2}(\Pi_x,x)\cap B_r(x))\leq\varepsilon\omega_nr^n.\tag{1}\]

Their union contains all the remaining points. This countable uniform-scale decomposition supplies the uniform estimates that replace the Egorov step of the classical proof.

The normal-plane Grassmannian is compact in projection-matrix norm. Choose finitely many \(m\)-planes \(\Pi_j\) so every \(\Pi_x\) has \(\|q_{\Pi_x}-q_{\Pi_j}\|<1/16\) for some \(j\). Let \(E_{k,j}\) be the points satisfying (1) and that plane bound. We claim that distinct \(x,y\in E_{k,j}\) with \(r=|y-x|<1/(2k)\) obey

\[|q_{\Pi_j}(y-x)|>r/4.\]

Otherwise, for \(z\in B_{r/8}(y)\),

\[|q_{\Pi_x}(z-x)|\leq(r/4+r/16)+r/8=7r/16\leq\frac12|z-x|,\]

since \(|z-x|\geq7r/8\). Also \(|z-x|<2r\). Thus \(B_{r/8}(y)\subseteq X_{1/2}(\Pi_x,x)\cap B_{2r}(x)\). Applying (1) at \(y\) and \(x\) gives

\[(1-\varepsilon)\omega_n(r/8)^n\leq\varepsilon\omega_n(2r)^n,\]

or \(1-\varepsilon\leq16^n\varepsilon\), contrary to our choice of \(\varepsilon\). The lower-density factor \(1-\varepsilon\) and the change of reference plane are both needed.

Intersect \(E_{k,j}\) with a countable cover by balls of diameter below \(1/(2k)\). On each piece projection onto \(\Pi_j^\perp\) is injective, and its inverse has Lipschitz constant at most four. After an orthogonal coordinate change the piece is a graph over a subset of \(\mathbb R^n\); its normal-coordinate function has Lipschitz constant at most \(\sqrt{15}\). Extend each component to all of \(\mathbb R^n\) using the formula

\[\widetilde u(a)=\inf_{b\in D}\{u(b)+L|a-b|\},\]

which is \(L\)-Lipschitz and equals \(u\) on its original domain \(D\). The resulting vector map is Lipschitz, so its graph is a Lipschitz image of \(\mathbb R^n\). There are only countably many pieces over all \(k,j\) and covering balls. Exhaust ambient space by bounded balls and add the discarded null set. This proves countable rectifiability. ◻

The following construction generalizes \(n\)-rectifiability, by permitting an index function to each point of a set \(M\).

Definition 6. Let \(M\) be \(\mathcal H^n\)-measurable and let \(0<\theta<\infty\) almost everywhere on \(M\), with \(\int_{M\cap K}\theta\,d\mathcal H^n<\infty\) for every compact \(K\). A plane \(P_x\) is an approximate tangent with multiplicity \(\theta(x)\) at a point \(x\in M\) where \(0<\theta(x)<\infty\) if

\[\lim_{c\downarrow0}\int_{\eta_{x,c}(M)}f(y)\theta(x+cy)\,d\mathcal H^n(y)=\theta(x)\int_{P_x}f\,d\mathcal H^n\quad(f\in C_c(\mathbb R^{n+m})).\]

Remark 7. To apply Lusin’s theorem, first restrict to \(M\cap\overline B_j(0)\cap\{1/j\leq\theta\leq j\}\). Its \(\mathcal H^n\) measure is at most \(j\int_{M\cap\overline B_j}\theta\,d\mathcal H^n<\infty\). Compact approximation and Lusin then give countably many compact subsets covering \(M\) up to a null set, on each of which \(\theta\) is continuous. Positivity on a compact piece gives a positive minimum, so finite weighted measure implies finite unweighted measure there. Continuity without positivity or this localization would not suffice.

Definition 8. \(P \subset \mathbb R^{n+m}\) is purely \(n\)-unrectifiable if it contains no countably \(n\)-rectifiable subset of positive \(\mathcal H^n\) measure.

Lemma 9. Every \(\mathcal H^n\)-measurable, \(\mathcal H^n\)-sigma-finite set \(A\) admits a disjoint decomposition \(A=R\sqcup P\), where \(R\) is countably \(n\)-rectifiable and \(P\) is purely \(n\)-unrectifiable. The set \(R\) may be chosen Borel.

Proof. Write \(A=\bigcup_jA_j\) with each \(A_j\) measurable of finite measure, and put

\[\alpha_j=\sup\{\mathcal H^n(S):S\subseteq A_j,\ S\text{ measurable and countably }n\text{-rectifiable}\}.\]

Choose such sets approaching \(\alpha_j\), and then compact subsets \(K_{ij}\) with \(\mathcal H^n(K_{ij})>\alpha_j-1/i\) whenever \(\alpha_j>0\); take the empty set when \(\alpha_j=0\). Compact approximation is available for each finite restricted Hausdorff measure in Euclidean space, and a subset of a rectifiable set is rectifiable. Set \(R=\bigcup_{i,j}K_{ij}\). This is Borel and rectifiable, and \(\mathcal H^n(R\cap A_j)=\alpha_j\): it is at least the supremum limit and at most \(\alpha_j\) by definition.

If \(P=A\setminus R\) had a measurable rectifiable subset \(S\) of positive measure, then \(\mathcal H^n(S\cap A_j)>0\) for some \(j\). The disjoint union \((R\cap A_j)\cup(S\cap A_j)\) would be a rectifiable subset of \(A_j\) with measure greater than \(\alpha_j\), a contradiction. For a possibly nonmeasurable rectifiable subset of \(P\), cover it by a Borel rectifiable carrier using Proposition 2; its measurable intersection with \(P\) contains the subset and has at least its outer measure. This reduces to the same contradiction. Thus \(P\) is purely unrectifiable. ◻

Lemma 10. If every coordinate projection of \(S\subset\mathbb R^{n+m}\) onto an \(n\)-dimensional coordinate plane has zero \(\mathcal H^n\) outer measure, then \(S\) is purely \(n\)-unrectifiable.

Proof. Suppose \(S\) contained a rectifiable set of positive measure. Proposition 2 gives an embedded \(n\)-manifold patch \(N\) for which \(N\cap S\) has positive outer measure. At every \(x\in N\), a basis matrix for \(T_xN\) has rank \(n\), hence some coordinate \(n\)-minor is nonzero. The corresponding projection restricted to \(N\) is a local \(C^1\) diffeomorphism. Shrink to precompact coordinate patches so that its inverse is Lipschitz. A countable collection of these patches covers \(N\), and one has intersection with \(S\) of positive outer measure. The Lipschitz inverse inequality then forces its projection to have positive measure, contradicting the hypothesis. We need a positive-measure subset of a manifold, not an entire manifold contained in \(S\). ◻

Example 11. Let \(C_0 = [0,1] \times [0,1]\). Let \(C_1\) be the union of 4 sub-squares of \(C_0\) such that one of each squares’ corners is a corner of \(C_0\). Consider the line \(y=\frac12 x\). We can explicitly calculate the projection of each sub-square onto the line. For instance, the top sub-square has coordinates \(\{(0,\frac34),(0,1),(\frac14,\frac34),(\frac14,1)\}\); call these points \(\{i,ii,iii,iv\}\) respectively. The orthogonal projections are

\[\begin{aligned} P_{y=\frac12x}(i) &= \frac{(1,\frac12) \cdot (0,\frac34)}{(1,\frac12) \cdot (1,\frac12)} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{\frac38}{\frac54} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{3}{10} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} \\ P_{y=\frac12x}(ii) &= \frac{(1,\frac12) \cdot (0,1)}{(1,\frac12) \cdot (1,\frac12)} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{\frac12}{\frac54} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{2}{5} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} \\ P_{y=\frac12x}(iii) &= \frac{(1,\frac12) \cdot (\frac14,\frac34)}{(1,\frac12) \cdot (1,\frac12)} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{\frac14 + \frac38}{\frac54} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac12 \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} \\ P_{y=\frac12x}(iv) &= \frac{(1,\frac12) \cdot (\frac14,1)}{(1,\frac12) \cdot (1,\frac12)} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{\frac14 + \frac12}{\frac54} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{3}{5} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} \end{aligned}\]

and doing calculations for all sub-squares gives the furthest orthogonal projection along the line is the point

\[\begin{aligned} =\frac{(1,\frac12) \cdot (1,1)}{(1,\frac12) \cdot (1,\frac12)} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac{\frac32}{\frac54} \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} = \frac65 \begin{pmatrix} 1 \\ \frac12 \end{pmatrix} ~. \end{aligned}\]

Thus the projection of \(C_1\) onto the line has length \(3/\sqrt5\). Define \(C_k\) by retaining the four corner squares of relative side \(1/4\) at every step. Then \(C_{k+1}\subseteq C_k\), each \(C_k\) consists of \(4^k\) squares of side \(4^{-k}\), and \(C=\bigcap_kC_k\) is compact.

To verify that the oblique projection has no gaps even in the limit, write \(C=K\times K\), where \(K=\{\sum_{j\geq1}a_j4^{-j}:a_j\in\{0,3\}\}\). Projection distance along the unit vector \((2,1)/\sqrt5\) is \((2x+y)/\sqrt5\). Its possible digits are \(2a_j+b_j\in\{0,3,6,9\}\). Every number in \([0,3]\) has an expansion \(\sum_{j\geq1}3d_j4^{-j}\) with \(d_j\in\{0,1,2,3\}\), and each digit \(3d_j\) is realized by one pair \((a_j,b_j)\). Therefore the projection of \(C\) is exactly an interval of length \(3/\sqrt5\).

The Lipschitz projection bound and the level-\(k\) square cover give

\[\frac3{\sqrt5}\leq\mathcal H^1(C)\leq\lim_{k\to\infty}4^k(\sqrt2\,4^{-k})=\sqrt2.\]

Meanwhile, either coordinate projection is covered by \(2^k\) intervals of length \(4^{-k}\), so its one-dimensional Lebesgue measure is at most \(2^{-k}\) for every \(k\), hence zero. These are unions of intervals, not one full interval. On each coordinate line \(\mathcal H^1=\lambda_1\), and Lemma 10 proves that \(C\) is purely 1-unrectifiable. Positive projection in one exceptional direction is consistent with the almost-everywhere theorem below.

Theorem 12. Let \(Q\subset\mathbb R^{n+m}\) be measurable, purely \(n\)-unrectifiable and \(\mathcal H^n\)-sigma-finite. For almost every \(n\)-plane \(V\) under the rotation-invariant probability measure on the Grassmannian \(G(n+m,n)\),

\[\mathcal H^n(P_VQ)=0.\]

This is the Besicovitch–Federer projection theorem. The measure is the pushforward of Haar probability on the orthogonal group; the collection of projections is not itself a group.

This theorem is an external input: Simon, Chapter 3, Theorem 3.6, p.80, states the exact sigma-finite projection result. Sigma-finiteness is a hypothesis, not a consequence of pure unrectifiability. Its finite-measure form extends to the stated form by a countable union of exceptional sets of directions.

Theorem 13. Let \(A\) be measurable and \(\mathcal H^n\)-sigma-finite. Suppose every measurable \(B\subseteq A\) of positive \(\mathcal H^n\) measure has \(\mathcal H^n(P_VB)>0\) on a positive-measure set of directions \(V\in G(n+m,n)\). Then \(A\) is countably \(n\)-rectifiable.

Proof. Decompose \(A=R\sqcup P\) by Lemma 9. If \(\mathcal H^n(P)>0\), take \(B=P\) in the hypothesis. Theorem 12 says its projection is null for almost every direction, a contradiction. Thus \(P\) is null, and a rectifiable set together with a null set is rectifiable. ◻

Remark 14. The hypothesis tests every positive-measure subset, because a set can contain both rectifiable and purely unrectifiable parts. Testing only \(A\) would miss a positive unrectifiable part hidden beside a rectifiable one.

1. A Class of Rectifiable Sets: Locally Finite Perimeter

Perimeter is a codimension-one notion. Work now in an open \(U\subset\mathbb R^{n+1}\), with a Lebesgue measurable set \(S\). By the bounded-variation definition, locally finite perimeter means that \(D\mathbb1_S\) is a vector Radon measure. Write its polar decomposition as \(D\mathbb1_S=\nu\mu_S\), where \(\mu_S=|D\mathbb1_S|\) and \(|\nu|=1\) almost everywhere for \(\mu_S\). Thus

\[\int_S\operatorname{div}\varphi\,dx=-\int_U\varphi\cdot\nu\,d\mu_S\qquad(\varphi\in C_c^1(U;\mathbb R^{n+1})).\]

For a smooth boundary the divergence theorem identifies \(\mu_S=\mathcal H^n\mathbin{\restriction}\partial S\) and \(\nu\) as the inward normal, because the usual outward-normal formula has the opposite sign.

Define the reduced boundary by

\[\partial^*S=\left\{x\in U\cap\operatorname{spt}\mu_S:\lim_{r\downarrow0}\frac{1}{\mu_S(B_r(x))}\int_{B_r(x)}\nu\,d\mu_S\text{ exists and has norm }1\right\}.\]

The support condition ensures positive denominators for sufficiently small balls contained in \(U\). Denote the limit by \(\nu_S(x)\).

Theorem 15. For a set of locally finite perimeter in \(U\subset\mathbb R^{n+1}\), the reduced boundary is countably \(n\)-rectifiable and

\[\mu_S=\mathcal H^n\mathbin{\restriction}\partial^*S.\]

At every \(x\in\partial^*S\), the rescaled perimeter measures converge with multiplicity one to \(\mathcal H^n\) on the plane \(T_x=\{z:z\cdot\nu_S(x)=0\}\), and

\[\mathbb1_{(S-x)/r}\longrightarrow\mathbb1_{\{z:z\cdot\nu_S(x)>0\}}\quad\text{in }L^1_{\mathrm{loc}}(\mathbb R^{n+1})\quad(r\downarrow0).\]

This is De Giorgi’s structure theorem, used here as a named theorem rather than inferred from the definition alone. The exact form, including the every-reduced-boundary-point blow-up, is Simon, Chapter 3, Theorem 4.3, pp.81–84. In ambient dimension \(d\), the same statement has boundary dimension \(d-1\); it is not an arbitrary-codimension \(n\)-dimensional conclusion in \(\mathbb R^{n+m}\). The convergence is of characteristic functions, and the sign of the limiting half-space matches the inward normal in the integration-by-parts identity.

Reading: Carmo (1992), Riemannian geometry.