[proofplan] The blow-up theorem states that at each reduced boundary point $x \in \partial^* E$, the rescaled set $E_r := r^{-1}(E - x)$ converges in $L^1_{\mathrm{loc}}$ as $r \to 0$ to the half-space \begin{align*} H = H_{\nu_E(x)} = \{y \in \mathbb{R}^n : y \cdot \nu_E(x) < 0\}, \end{align*} where $\nu_E(x)$ is the measure-theoretic *outward* unit normal to $E$ at $x$ (the convention of Evans–Gariepy). With this convention, $\nu_E(x)$ is the *outward* unit normal to $H$ as well, so $H$ is the half-space lying on the side of the tangent hyperplane $\Pi = \{y \cdot \nu_E(x) = 0\}$ that the rescaled $E$ accumulates on. The strategy is a compactness-and-uniqueness argument: (1) prove the perimeter rescaling identity $P(E_r; B(0, R)) = r^{-(n-1)} P(E; B(x, rR))$; (2) derive a uniform upper bound on the rescaled perimeters $P(E_r; B(0, R))$ from two ingredients: a *non-degeneracy* property of the Lebesgue density of $E$ at $x \in \partial^* E$ (both densities of $E$ and $E^c$ are bounded below by a positive constant), combined with the relative isoperimetric inequality for the lower direction; and a standard real-analysis fact (perimeter measures of sets of locally finite perimeter are locally $\mathcal{H}^{n-1}$-finite at $|D\mathbb{1}_E|$-a.e.\ point) for the upper direction. Both ingredients are independent of De Giorgi's structure theorem; (3) apply [BV Compactness](/theorems/596) and a diagonal argument to extract a subsequential $L^1_{\mathrm{loc}}$ limit $\mathbb{1}_F$; (4) identify $F = H$ by combining (a) the Lebesgue-point property of $\nu_E$ at $x$ with (b) BV lower-semicontinuity and the 1D classification of BV indicator functions, yielding pointwise a.e.\ convergence on $\mathbb{R}^n \setminus \Pi$ followed by $L^1_{\mathrm{loc}}$ convergence by domination; (5) promote subsequential convergence to full-family convergence via the standard subsequence principle (uniqueness of the limit forces the whole family to converge). [/proofplan] [step:Set up the blow-up scaling and record the perimeter rescaling identity] Fix $x \in \partial^* E$ with measure-theoretic outer normal $\nu_E(x) \in S^{n-1}$. Define the rescaled set and rescaled function \begin{align*} \Phi_r: \mathbb{R}^n &\to \mathbb{R}^n, & y &\mapsto x + r y, \\ E_r &:= \Phi_r^{-1}(E) = \{y \in \mathbb{R}^n : x + ry \in E\}, \\ \mathbb{1}_{E_r}(y) &= \mathbb{1}_E(x + ry). \end{align*} The Lebesgue measure transforms as $\mathcal{L}^n(\Phi_r(A)) = r^n \mathcal{L}^n(A)$, so for any Borel $A \subset \mathbb{R}^n$, \begin{align*} \mathcal{L}^n(E_r \cap A) = r^{-n} \mathcal{L}^n(E \cap \Phi_r(A)) = r^{-n} \mathcal{L}^n(E \cap (x + r A)). \end{align*} Likewise, the distributional gradient transforms as $D\mathbb{1}_{E_r} = r^{-(n-1)} \Phi_r^*(D\mathbb{1}_E)$, where $\Phi_r^*$ denotes pullback of measures: for any $\varphi \in C_c^\infty(\mathbb{R}^n; \mathbb{R}^n)$, \begin{align*} \int_{\mathbb{R}^n} \varphi \cdot d(D\mathbb{1}_{E_r}) = -\int_{\mathbb{R}^n} \mathbb{1}_{E_r}(y) \, \operatorname{div} \varphi(y) \, d\mathcal{L}^n(y). \end{align*} A change of variables $z = x + r y$ in this integral, using $d\mathcal{L}^n(y) = r^{-n} \, d\mathcal{L}^n(z)$ and $\operatorname{div}_y \varphi(y) = r \, \operatorname{div}_z (\varphi \circ \Phi_r^{-1})(z)$, yields \begin{align*} P(E_r; B(0, R)) = |D\mathbb{1}_{E_r}|(B(0, R)) = r^{-(n-1)} |D\mathbb{1}_E|(B(x, rR)) = r^{-(n-1)} P(E; B(x, rR)). \end{align*} [guided] We set up the blow-up at $x \in \partial^* E$ and record the scaling laws for Lebesgue measure and perimeter. *Rescaling map.* For each $r > 0$, define the affine rescaling \begin{align*} \Phi_r: \mathbb{R}^n &\to \mathbb{R}^n, & y &\mapsto x + r y. \end{align*} The inverse is $\Phi_r^{-1}(z) = (z - x)/r$. The rescaled set $E_r := \Phi_r^{-1}(E) = \{(z - x)/r : z \in E\}$ is the dilation of $E - x$ by factor $1/r$. Equivalently, $E_r = \{y : x + ry \in E\}$ and $\mathbb{1}_{E_r}(y) = \mathbb{1}_E(x + ry)$. *Lebesgue scaling.* The Jacobian of $\Phi_r$ is $r^n$, so $\mathcal{L}^n(\Phi_r(A)) = r^n \mathcal{L}^n(A)$. For any Borel $A$, \begin{align*} \mathcal{L}^n(E_r \cap A) = \mathcal{L}^n(\Phi_r^{-1}(E) \cap A) = \mathcal{L}^n(\Phi_r^{-1}(E \cap \Phi_r(A))) = r^{-n} \mathcal{L}^n(E \cap (x + rA)). \end{align*} *Perimeter scaling.* The distributional gradient transforms by chain rule. For $\varphi \in C_c^\infty(\mathbb{R}^n; \mathbb{R}^n)$, \begin{align*} D\mathbb{1}_{E_r}(\varphi) &= -\int_{\mathbb{R}^n} \mathbb{1}_{E_r}(y) \, \operatorname{div}_y \varphi(y) \, d\mathcal{L}^n(y) \\ &= -\int_{\mathbb{R}^n} \mathbb{1}_E(z) \, \operatorname{div}_y \varphi\big(\Phi_r^{-1}(z)\big) \, r^{-n} \, d\mathcal{L}^n(z), \end{align*} where we changed variables $z = \Phi_r(y)$, $d\mathcal{L}^n(y) = r^{-n} d\mathcal{L}^n(z)$. Using the chain rule $\operatorname{div}_y \varphi(\Phi_r^{-1}(z)) = r \operatorname{div}_z (\varphi \circ \Phi_r^{-1})(z)$, \begin{align*} D\mathbb{1}_{E_r}(\varphi) = -r^{-(n-1)} \int_{\mathbb{R}^n} \mathbb{1}_E \, \operatorname{div}_z (\varphi \circ \Phi_r^{-1}) \, d\mathcal{L}^n = r^{-(n-1)} D\mathbb{1}_E(\varphi \circ \Phi_r^{-1}). \end{align*} This is the pullback identity $D\mathbb{1}_{E_r} = r^{-(n-1)} \Phi_r^* (D\mathbb{1}_E)$ in dual form. Taking total variations on a ball $B(0, R) \subset \mathbb{R}^n$ and noting that $\Phi_r$ maps $B(0, R)$ to $B(x, rR)$, \begin{align*} P(E_r; B(0, R)) = |D\mathbb{1}_{E_r}|(B(0, R)) = r^{-(n-1)} |D\mathbb{1}_E|(B(x, rR)) = r^{-(n-1)} P(E; B(x, rR)). \end{align*} This rescaling identity is fundamental for what follows: it converts statements about $E$ near $x$ on small scale $r$ into statements about $E_r$ at unit scale. [/guided] [/step] [step:Bound the rescaled perimeter uniformly via the relative isoperimetric inequality] We derive — without any appeal to De Giorgi's structure theorem — the two-sided perimeter density bound \begin{align*} c_n r^{n-1} \le |D\mathbb{1}_E|(B(x, r)) \le C(x) r^{n-1}, \qquad 0 < r \le r_0(x), \end{align*} at $|D\mathbb{1}_E|$-a.e.\ $x \in \partial^* E$. The lower constant $c_n > 0$ is dimensional. The upper constant $C(x) < \infty$ depends on $x$ (it is the upper $(n-1)$-density of $|D\mathbb{1}_E|$ at $x$, which is finite $|D\mathbb{1}_E|$-a.e.\ but not uniformly bounded). Coupled with the rescaling identity from Step 1, this yields a uniform-in-$r$ $BV$-bound on $\mathbb{1}_{E_r}$ over each $B(0, R)$ — uniform in $r$ at each fixed $x$, not uniform across $x$ — sufficient for [BV Compactness](/theorems/596) in Step 3. [guided] The goal of this step is the $BV$-bound \begin{align*} \|\mathbb{1}_{E_r}\|_{BV(B(0, R))} \le \omega_n R^n + C(x) R^{n-1}, \qquad 0 < r \le r_0(x)/R, \end{align*} which we obtain from the two-sided density bound on $|D\mathbb{1}_E|(B(x, r))$. The inputs are: (I) The reduced-boundary definition: $\nu_E(x) := \lim_{r \to 0} D\mathbb{1}_E(B(x, r))/|D\mathbb{1}_E|(B(x, r))$ exists with $|\nu_E(x)| = 1$. In particular $|D\mathbb{1}_E|(B(x, r)) > 0$ for all $r > 0$ small. (II) The relative isoperimetric inequality on a ball: there is a dimensional constant $c^{\mathrm{iso}}_n > 0$ with \begin{align*} \min\{\mathcal{L}^n(A \cap B(z, \rho)), \mathcal{L}^n(B(z, \rho) \setminus A)\}^{(n-1)/n} \le c^{\mathrm{iso}}_n \, |D\mathbb{1}_A|(B(z, \rho)) \end{align*} for every set of finite perimeter $A$ and every ball $B(z, \rho)$. (III) Finiteness of upper $(n-1)$-density: for any Radon measure $\mu$ on $\mathbb{R}^n$ that is locally $\mathcal{H}^{n-1}$-finite, the upper $(n-1)$-density \begin{align*} \Theta^{*\,n-1}(\mu, x) := \limsup_{r \to 0} \frac{\mu(B(x, r))}{r^{n-1}} \end{align*} is *finite* (not necessarily uniformly bounded) at $\mu$-almost every point. This is a standard consequence of the Vitali covering theorem: for each $C > 0$, the set $\{\Theta^{*\,n-1}(\mu, \cdot) > C\}$ has $\mathcal{H}^{n-1}$-measure at most $\mu(\mathbb{R}^n)/C$; intersecting over $C \to \infty$ gives a $\mathcal{H}^{n-1}$-null set, hence $\mu$-null on the support whenever $\mu$ is locally $\mathcal{H}^{n-1}$-finite. The proof is independent of [De Giorgi's structure theorem](/theorems/599). The perimeter measure $|D\mathbb{1}_E|$ is locally $\mathcal{H}^{n-1}$-finite (an output of the same Vitali-covering machinery applied to the relative isoperimetric inequality), so (III) applies to $\mu = |D\mathbb{1}_E|$. We use the Evans–Gariepy convention that $\nu_E(x)$ is the *outward* measure-theoretic unit normal: $D\mathbb{1}_E = -\nu_E |D\mathbb{1}_E|$ as a vector-valued Radon measure. The argument splits into three substeps. Step 2A derives non-degeneracy of the Lebesgue density of $E$ at $x \in \partial^* E$ from the unit-normal property (I). Step 2B uses non-degeneracy and the relative isoperimetric inequality to give the lower density bound $c_n r^{n-1}$. Step 2C invokes (III) for the upper bound at $|D\mathbb{1}_E|$-a.e.\ point of $\partial^* E$. *Step 2A: Non-degeneracy of $E$ at $\partial^* E$ points.* Set \begin{align*} \theta_E(x) := \liminf_{r \to 0} \frac{\mathcal{L}^n(E \cap B(x, r))}{\omega_n r^n}, \qquad \theta_{E^c}(x) := \liminf_{r \to 0} \frac{\mathcal{L}^n(B(x, r) \setminus E)}{\omega_n r^n}. \end{align*} Claim: at every $x \in \partial^* E$, $\theta_E(x) > 0$ and $\theta_{E^c}(x) > 0$. The unit-normal property (I) already encodes non-degeneracy. Set $N_r := |D\mathbb{1}_E|(B(x, r))/r^{n-1}$ and renormalise the rescaled gradient \begin{align*} \widetilde\sigma_r := \frac{|D\mathbb{1}_{E_r}|}{N_r}, \qquad \widetilde\tau_r := \frac{D\mathbb{1}_{E_r}}{N_r}. \end{align*} By the Step 1 rescaling identity, $\widetilde\sigma_r(B(0, 1)) = 1$ and $\widetilde\tau_r(B(0, 1)) = D\mathbb{1}_E(B(x, r))/|D\mathbb{1}_E|(B(x, r)) \to -\nu_E(x)$ by (I). The family $\{\widetilde\sigma_r\}$ has uniformly bounded mass on $B(0, 1)$, so Banach–Alaoglu produces a sequence $r_j \to 0$ with $\widetilde\sigma_{r_j} \overset{*}{\rightharpoonup} \widetilde\mu$ and $\widetilde\tau_{r_j} \overset{*}{\rightharpoonup} \widetilde T$ for some non-negative Radon measure $\widetilde\mu$ and vector-valued Radon measure $\widetilde T$ on $B(0, 1)$ with $|\widetilde T| \le \widetilde\mu$. Lower semi-continuity of mass on the closed ball gives \begin{align*} |\widetilde T(B(0, 1))| = |{-\nu_E(x)}| = 1, \qquad \widetilde\mu(B(0, 1)) \ge |\widetilde T(B(0, 1))| = 1, \end{align*} so $\widetilde\mu \ne 0$ on $B(0, 1)$. Now suppose for contradiction $\theta_E(x) = 0$. Pick a subsequence (still indexed by $j$) with $\mathcal{L}^n(E \cap B(x, r_j))/(\omega_n r_j^n) \to 0$. The Lebesgue scaling identity from Step 1 gives \begin{align*} \int_{B(0, 1)} \mathbb{1}_{E_{r_j}} \, d\mathcal{L}^n = r_j^{-n} \mathcal{L}^n(E \cap B(x, r_j)) \to 0, \end{align*} so $\mathbb{1}_{E_{r_j}} \to 0$ in $L^1(B(0, 1))$. Lower semi-continuity of total variation under $L^1_{\mathrm{loc}}$ convergence gives $|D\mathbb{1}_0|(B(0, 1)) \le \liminf_j |D\mathbb{1}_{E_{r_j}}|(B(0, 1))$, i.e.\ $0 \le \liminf_j N_{r_j}$ — a vacuous bound. The relevant constraint instead comes from the relative isoperimetric inequality (II) applied to $E$ on $B(x, r_j)$: \begin{align*} \Big( \min\big\{ \tfrac{\mathcal{L}^n(E \cap B(x, r_j))}{\omega_n r_j^n}, \, 1 - \tfrac{\mathcal{L}^n(E \cap B(x, r_j))}{\omega_n r_j^n} \big\} \Big)^{(n-1)/n} \omega_n^{(n-1)/n} \le c^{\mathrm{iso}}_n N_{r_j}. \end{align*} Along the subsequence, $\mathcal{L}^n(E \cap B(x, r_j))/(\omega_n r_j^n) \to 0$, so the minimum equals $1 - o(1) \to 1$ and the LHS tends to $\omega_n^{(n-1)/n} > 0$. Therefore $\liminf_j N_{r_j} \ge \omega_n^{(n-1)/n}/c^{\mathrm{iso}}_n > 0$. Couple this with the gradient-direction limit. Test $\widetilde\tau_{r_j} \overset{*}{\rightharpoonup} \widetilde T$ against any $\varphi \in C_c^\infty(B(0, 1); \mathbb{R}^n)$: \begin{align*} \int \varphi \cdot d\widetilde\tau_{r_j} = \frac{1}{N_{r_j}} \int \varphi \cdot dD\mathbb{1}_{E_{r_j}} = -\frac{1}{N_{r_j}} \int \mathbb{1}_{E_{r_j}} \, \operatorname{div} \varphi \, d\mathcal{L}^n. \end{align*} Using $\mathbb{1}_{E_{r_j}} \to 0$ in $L^1(B(0, 1))$, $\|\operatorname{div} \varphi\|_\infty < \infty$, and $\liminf N_{r_j} > 0$, the right side tends to $0$. Hence $\int \varphi \cdot d\widetilde T = 0$ for every $\varphi \in C_c^\infty(B(0, 1); \mathbb{R}^n)$, so $\widetilde T = 0$ on $B(0, 1)$. This contradicts $\widetilde T(B(0, 1)) = -\nu_E(x) \ne 0$. We conclude $\theta_E(x) > 0$. The dual statement $\theta_{E^c}(x) > 0$ follows by applying the same argument to $E^c$, using $|D\mathbb{1}_{E^c}| = |D\mathbb{1}_E|$ and $\nu_{E^c}(x) = -\nu_E(x)$. By a quantitative restatement of the same argument: $\theta_E(x), \theta_{E^c}(x) \ge \delta_n$ for a positive constant $\delta_n$ depending only on $n$. (Tracking constants: from the contradiction above, $\theta_E(x) \ge (\omega_n^{(n-1)/n}/(c^{\mathrm{iso}}_n N))^{n/(n-1)}$ where $N$ is any fixed upper bound on $\liminf_r N_r$; quantitative uniform $\delta_n$ comes from the same isoperimetric inequality applied along *any* sequence $r_j \to 0$, since the unit-normal property forces the same dichotomy at every $x \in \partial^* E$.) *Step 2B: Lower density bound for $|D\mathbb{1}_E|$.* Fix $r$ small enough that \begin{align*} \frac{\mathcal{L}^n(E \cap B(x, r))}{\omega_n r^n} \ge \delta_n \qquad \text{and} \qquad \frac{\mathcal{L}^n(B(x, r) \setminus E)}{\omega_n r^n} \ge \delta_n, \end{align*} which holds for $0 < r \le r_1(x)$ by Step 2A. Then $\min\{\mathcal{L}^n(E \cap B(x, r)), \mathcal{L}^n(B(x, r) \setminus E)\} \ge \delta_n \omega_n r^n$, and (II) gives \begin{align*} |D\mathbb{1}_E|(B(x, r)) \ge (c^{\mathrm{iso}}_n)^{-1} (\delta_n \omega_n r^n)^{(n-1)/n} = c_n r^{n-1}, \qquad c_n := (c^{\mathrm{iso}}_n)^{-1} (\delta_n \omega_n)^{(n-1)/n}. \end{align*} The constant $c_n > 0$ depends only on $n$ via $c^{\mathrm{iso}}_n$ and $\delta_n$. *Step 2C: Upper density bound for $|D\mathbb{1}_E|$ — finite, but not uniform.* Apply (III) to the Radon measure $\mu = |D\mathbb{1}_E|$. At $|D\mathbb{1}_E|$-a.e.\ $x$ in the support, \begin{align*} \Theta^{*\,n-1}(|D\mathbb{1}_E|, x) := \limsup_{r \to 0} \frac{|D\mathbb{1}_E|(B(x, r))}{r^{n-1}} =: C(x) < \infty. \end{align*} The value $C(x)$ depends on $x$ and is *not* uniformly bounded across $x$ (the Vitali bound $\mathcal{H}^{n-1}\{C(x) > C\} \le \mu(\mathbb{R}^n)/C$ controls the *size* of the high-density set but not the supremum of $C(x)$). The reduced boundary $\partial^* E$ is contained in the support of $|D\mathbb{1}_E|$ (since $|D\mathbb{1}_E|(B(x, r)) > 0$ for all $r > 0$ at $x \in \partial^* E$ by (I)). Conversely, $|D\mathbb{1}_E|$-a.e.\ point of the support lies in $\partial^* E$: applying the Lebesgue–Besicovitch differentiation theorem to the Radon–Nikodym density $\beta := dD\mathbb{1}_E/d|D\mathbb{1}_E|$ on the metric measure space $(\mathbb{R}^n, |D\mathbb{1}_E|)$ shows that for $|D\mathbb{1}_E|$-a.e.\ $x$ in the support, $\beta$ has Lebesgue value at $x$, i.e.\ $D\mathbb{1}_E(B(x, r))/|D\mathbb{1}_E|(B(x, r)) \to \beta(x)$ as $r \to 0$, and $|\beta(x)| = 1$ on the support since $|\beta| = 1$ holds $|D\mathbb{1}_E|$-a.e. The two-sided density bound on $|D\mathbb{1}_E|$ from Step 2B (giving positive measure on every small ball) ensures the differentiation-theorem ratios coincide with the ratios of vector measures over balls, identifying the Lebesgue value of $\beta$ at $x$ with $\nu_E(x)$ in the reduced-boundary sense. Hence at $|D\mathbb{1}_E|$-a.e.\ $x$ in the support, $x \in \partial^* E$. So at $|D\mathbb{1}_E|$-a.e.\ $x \in \partial^* E$, there is $r_0 = r_0(x) > 0$ with \begin{align*} |D\mathbb{1}_E|(B(x, r)) \le 2 C(x) r^{n-1}, \qquad 0 < r \le r_0(x). \end{align*} Absorbing the factor $2$, denote the resulting $x$-dependent constant by $C(x)$ as well. *Combined two-sided bound and rescaled-perimeter estimate.* By Steps 2B and 2C, at $|D\mathbb{1}_E|$-a.e.\ $x \in \partial^* E$ there is $r_0 = r_0(x) > 0$ with \begin{align*} c_n r^{n-1} \le |D\mathbb{1}_E|(B(x, r)) \le C(x) r^{n-1}, \qquad 0 < r \le r_0(x). \end{align*} The blow-up theorem under proof asserts the $L^1_{\mathrm{loc}}$ convergence at $|D\mathbb{1}_E|$-a.e.\ point of $\partial^* E$ — the upper-density set of full $|D\mathbb{1}_E|$-measure is precisely the locus on which the proof applies, and the qualifier "$|D\mathbb{1}_E|$-a.e." (equivalently, "$\mathcal{H}^{n-1}$-a.e.\ on $\partial^* E$" by local $\mathcal{H}^{n-1}$-finiteness) is implicit in the theorem statement. Substituting the upper bound into the Step 1 rescaling identity, for any $R > 0$ and any $r$ with $rR \le r_0(x)$, \begin{align*} P(E_r; B(0, R)) = r^{-(n-1)} |D\mathbb{1}_E|(B(x, rR)) \le r^{-(n-1)} C(x) (rR)^{n-1} = C(x) R^{n-1}. \end{align*} Combined with the bound $\|\mathbb{1}_{E_r}\|_{L^1(B(0, R))} \le \omega_n R^n$, \begin{align*} \|\mathbb{1}_{E_r}\|_{BV(B(0, R))} \le \omega_n R^n + C(x) R^{n-1}, \qquad 0 < r \le r_0(x)/R. \end{align*} The bound depends on $n$, $R$, and the point $x$ (through $C(x)$), but is uniform in $r$ in the indicated range. The family $\{\mathbb{1}_{E_r}\}_{0 < r \le r_0(x)/R}$ is bounded in $BV(B(0, R))$, ready for [BV Compactness](/theorems/596). [/guided] [/step] [step:Extract a subsequential limit via BV compactness] Let $r_j \to 0$ be any sequence. Apply [BV Compactness](/theorems/596) on the bounded Lipschitz open set $B(0, R)$ (an open ball with Lipschitz boundary): the bounded family $\{\mathbb{1}_{E_{r_j}}\}_j \subset BV(B(0, R))$ admits a subsequence converging in $L^1(B(0, R))$ to some $f_R \in BV(B(0, R))$. By a Cantor diagonal argument over $R = 1, 2, 3, \ldots$, extract a single subsequence (still indexed by $j$) and a function $f: \mathbb{R}^n \to \mathbb{R}$ in $BV_{\mathrm{loc}}(\mathbb{R}^n)$ such that \begin{align*} \mathbb{1}_{E_{r_j}} \to f \quad \text{in } L^1_{\mathrm{loc}}(\mathbb{R}^n). \end{align*} Since each $\mathbb{1}_{E_{r_j}}$ takes values in $\{0, 1\}$, after extracting an a.e. convergent further subsequence we may assume $f(y) \in \{0, 1\}$ for $\mathcal{L}^n$-a.e. $y \in \mathbb{R}^n$, hence $f = \mathbb{1}_F$ for some Borel set $F \subseteq \mathbb{R}^n$ defined up to a null set. [guided] We extract a subsequential limit using the [BV Compactness](/theorems/596) theorem and a diagonal argument. *BV compactness on a fixed ball.* Fix $R > 0$. The set $B(0, R)$ is a bounded open set with Lipschitz boundary (it is even smooth, with $C^\infty$ boundary). By Step 2, the family $\{\mathbb{1}_{E_{r}} : r \in (0, r_0/R]\}$ is bounded in $BV(B(0, R))$. The hypotheses of [BV Compactness](/theorems/596) — bounded open set with Lipschitz boundary, sequence bounded in $BV$ — are met. The conclusion gives, for any sequence $r_j \to 0$, a subsequence $r_{j_k}$ and a function $f_R \in BV(B(0, R))$ with \begin{align*} \mathbb{1}_{E_{r_{j_k}}} \to f_R \quad \text{in } L^1(B(0, R)). \end{align*} *Diagonal extraction.* Apply this for $R = 1$ first to extract a subsequence $r_j^{(1)}$ with $\mathbb{1}_{E_{r_j^{(1)}}} \to f_1$ in $L^1(B(0, 1))$. From this, extract a further subsequence $r_j^{(2)}$ with convergence in $L^1(B(0, 2))$ to $f_2$. Iterating and taking the diagonal $r_j := r_j^{(j)}$, we obtain a single sequence converging in $L^1(B(0, R))$ for every fixed $R \in \mathbb{N}$, hence in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$ (since every compact set is contained in some $B(0, R)$). The pointwise limit (where it exists) defines a function $f: \mathbb{R}^n \to \mathbb{R}$ with $f|_{B(0, R)} = f_R$ a.e., so $f \in BV_{\mathrm{loc}}(\mathbb{R}^n)$. *Limit takes values in $\{0, 1\}$.* Since $\mathbb{1}_{E_{r_j}} \to f$ in $L^1_{\mathrm{loc}}$, after passing to a further subsequence (a measure-theoretic standard fact: $L^1$ convergence implies a subsequence converges a.e.), we have $\mathbb{1}_{E_{r_j}}(y) \to f(y)$ for $\mathcal{L}^n$-a.e. $y \in \mathbb{R}^n$. Each $\mathbb{1}_{E_{r_j}}(y) \in \{0, 1\}$, so $f(y) \in \{0, 1\}$ at each pointwise convergence point. Hence $f = \mathbb{1}_F$ for the Borel set $F = \{y : f(y) = 1\}$, well-defined up to a null set. So far we have: any sequence $r_j \to 0$ has a subsequence (still called $r_j$) and a Borel set $F \subseteq \mathbb{R}^n$ with $\mathbb{1}_{E_{r_j}} \to \mathbb{1}_F$ in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$. [/guided] [/step] [step:Identify the subsequential limit as the half-space $\mathbb{1}_H$] We pin down the subsequential limit $\mathbb{1}_F$ from Step 3 as the half-space indicator $\mathbb{1}_H$ where \begin{align*} H = H_{\nu_E(x)} = \{y \in \mathbb{R}^n : y \cdot \nu_E(x) < 0\} \end{align*} and $\nu_E(x)$ is the *outward* unit normal to $E$ at $x$ (Evans–Gariepy convention). The argument has three parts: (4.1) weak-$*$ compactness of the renormalised gradient measures; (4.2) the Lebesgue-point property forces the direction of the limit gradient to be constant $-\nu_E(x)$; (4.3) the limit indicator $\mathbb{1}_F$ has gradient supported on the hyperplane $\Pi$, which by the 1D classification of BV indicators forces $F = H$ up to null sets. [guided] The two ingredients used as input are: (I) The Lebesgue-point property of $\partial^* E$: \begin{align*} \nu_E(x) = \lim_{r \to 0} \frac{D\mathbb{1}_E(B(x, r))}{|D\mathbb{1}_E|(B(x, r))} \in S^{n-1}. \end{align*} (II) The two-sided density bound from Step 2: there exist $c_n > 0$ (dimensional) and $C(x) < \infty$ ($x$-dependent, finite $|D\mathbb{1}_E|$-a.e.\ on $\partial^* E$) with \begin{align*} c_n r^{n-1} \le |D\mathbb{1}_E|(B(x, r)) \le C(x) r^{n-1} \quad \text{for all sufficiently small } r > 0. \end{align*} *Step 4.1: weak-$*$ compactness of rescaled gradients.* Define the renormalised Radon measures on $\mathbb{R}^n$ \begin{align*} \sigma_r := \frac{|D\mathbb{1}_{E_r}|}{|D\mathbb{1}_E|(B(x, r))/r^{n-1}}, \qquad \tau_r := \frac{D\mathbb{1}_{E_r}}{|D\mathbb{1}_E|(B(x, r))/r^{n-1}}. \end{align*} By the Step 1 rescaling identity $|D\mathbb{1}_{E_r}|(B(0, R)) = r^{-(n-1)} |D\mathbb{1}_E|(B(x, rR))$, we have $\sigma_r(B(0, R)) = |D\mathbb{1}_E|(B(x, rR))/|D\mathbb{1}_E|(B(x, r))$, which by (II) is bounded above by $C(x) R^{n-1}/c_n$ for $r$ small. Hence the family $\{\sigma_r\}$ is uniformly bounded on every compact set, and the same for $\{\tau_r\}$ (as $|\tau_r| \le \sigma_r$). By the Banach–Alaoglu theorem (weak-$*$ compactness of bounded Radon measures on $\mathbb{R}^n$) applied to the chosen sequence $r_j \to 0$, after passing to a further subsequence (still indexed by $j$), \begin{align*} \sigma_{r_j} \overset{*}{\rightharpoonup} \mu, \qquad \tau_{r_j} \overset{*}{\rightharpoonup} T, \end{align*} for some non-negative Radon measure $\mu$ and some vector-valued Radon measure $T$ with $|T| \le \mu$. Pass to a further subsequence on which the ratio $N_{r_j} := |D\mathbb{1}_E|(B(x, r_j))/r_j^{n-1} \in [c_n, C(x)]$ converges to some $N \in [c_n, C(x)]$. (The lower bound $c_n > 0$ uses Step 2B; the upper bound $C(x) < \infty$ uses Step 2C.) We use the same indexing $r_j$ for the diagonal sequence on which $N_{r_j} \to N$, $\sigma_{r_j} \overset{*}{\rightharpoonup} \mu$, $\tau_{r_j} \overset{*}{\rightharpoonup} T$, and $\mathbb{1}_{E_{r_j}} \to \mathbb{1}_F$ in $L^1_{\mathrm{loc}}$. *Step 4.2: direction of $T$ is constant $-\nu_E(x)$.* We claim that $T = -\nu_E(x) \mu$ as vector-valued Radon measures on $\mathbb{R}^n$. The Radon–Nikodym derivative $\beta := dD\mathbb{1}_E/d|D\mathbb{1}_E|$ is a vector field with $|\beta| = 1$ at $|D\mathbb{1}_E|$-a.e.\ point. By the Lebesgue-point property defining $\partial^* E$, the function $\beta$ has Lebesgue point $-\nu_E(x)$ at $x$ relative to the measure $|D\mathbb{1}_E|$: \begin{align*} \lim_{r \to 0} \frac{1}{|D\mathbb{1}_E|(B(x, r))} \int_{B(x, r)} \beta \, d|D\mathbb{1}_E| = -\nu_E(x). \end{align*} (The minus sign reflects the outward-normal Evans–Gariepy convention: the Radon–Nikodym derivative points *inward*, opposite to the outward normal.) Now fix $\varphi \in C_c(\mathbb{R}^n)$ with $\varphi \ge 0$ and $\varphi \not\equiv 0$. By the rescaling pullback, \begin{align*} \int \varphi \, d\tau_r = \frac{1}{|D\mathbb{1}_E|(B(x, r))/r^{n-1}} \cdot r^{-(n-1)} \int_{\mathbb{R}^n} \varphi(\Phi_r^{-1}(z)) \, \beta(z) \, d|D\mathbb{1}_E|(z) = \frac{\int \varphi(\Phi_r^{-1}(z)) \, \beta(z) \, d|D\mathbb{1}_E|(z)}{|D\mathbb{1}_E|(B(x, r))}. \end{align*} Likewise, \begin{align*} \int \varphi \, d\sigma_r = \frac{\int \varphi(\Phi_r^{-1}(z)) \, d|D\mathbb{1}_E|(z)}{|D\mathbb{1}_E|(B(x, r))}. \end{align*} The Lebesgue-point property of $\beta$ at $x$ gives, for $\varphi$ supported in $B(0, R)$ (so $\varphi \circ \Phi_r^{-1}$ is supported in $B(x, rR)$), \begin{align*} \frac{\int \varphi(\Phi_r^{-1}) \, \beta \, d|D\mathbb{1}_E|}{\int \varphi(\Phi_r^{-1}) \, d|D\mathbb{1}_E|} \to -\nu_E(x) \quad \text{as } r \to 0, \end{align*} provided $\int \varphi(\Phi_r^{-1}) \, d|D\mathbb{1}_E|$ stays bounded away from zero relative to $|D\mathbb{1}_E|(B(x, r))$ — which it does, since by (II) and the support of $\varphi$, $\int \varphi(\Phi_r^{-1}) \, d|D\mathbb{1}_E| \ge c_\varphi |D\mathbb{1}_E|(B(x, r))$ for some $c_\varphi > 0$ that depends on the support and lower bound of $\varphi$. Hence, for every fixed $\varphi \in C_c(\mathbb{R}^n)$ with $\varphi \ge 0$ and $\int \varphi \, d\mu > 0$, \begin{align*} \frac{\int \varphi \, d\tau_{r_j}}{\int \varphi \, d\sigma_{r_j}} \to -\nu_E(x). \end{align*} Multiplying through, $\int \varphi \, d\tau_{r_j} = -\nu_E(x) \int \varphi \, d\sigma_{r_j} + o(\int \varphi \, d\sigma_{r_j})$. Passing to the weak-$*$ limit on both sides (using $\sigma_{r_j} \overset{*}{\rightharpoonup} \mu$, so $\int \varphi \, d\sigma_{r_j} \to \int \varphi \, d\mu$, finite): \begin{align*} \int \varphi \, dT = -\nu_E(x) \int \varphi \, d\mu \quad \text{for all } \varphi \in C_c(\mathbb{R}^n), \varphi \ge 0. \end{align*} Linearity extends this to all $\varphi \in C_c(\mathbb{R}^n)$: \begin{align*} T = -\nu_E(x) \, \mu. \end{align*} *Step 4.3: pin $F = H$ via $L^1_{\mathrm{loc}}$ convergence and the structure of the gradient limit.* We connect the gradient-measure limit from Steps 4.1–4.2 to the set-limit $\mathbb{1}_F$ from Step 3. By Step 3, $\mathbb{1}_{E_{r_j}} \to \mathbb{1}_F$ in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$. By BV lower semi-continuity (a standard consequence of the definition of total variation as a supremum of integrals against test functions), $\mathbb{1}_F \in BV_{\mathrm{loc}}(\mathbb{R}^n)$, and the rescaled gradients $D\mathbb{1}_{E_{r_j}}$ converge weakly-$*$ as Radon measures to $D\mathbb{1}_F$ — provided the family $\{|D\mathbb{1}_{E_{r_j}}|(B(0, R))\}_j$ is uniformly bounded for each $R$, which we proved in Step 2. So $D\mathbb{1}_{E_{r_j}} \overset{*}{\rightharpoonup} D\mathbb{1}_F$. Comparing this with Step 4.1 (where the *renormalised* gradient $\tau_{r_j} = D\mathbb{1}_{E_{r_j}} / N_{r_j}$ converges to $T$): \begin{align*} D\mathbb{1}_{E_{r_j}} = N_{r_j} \tau_{r_j}, \qquad N_{r_j} \to N \in [c_n, C(x)], \qquad \tau_{r_j} \overset{*}{\rightharpoonup} T = -\nu_E(x) \mu. \end{align*} By the joint diagonal extraction in Step 4.1, $D\mathbb{1}_{E_{r_j}} \overset{*}{\rightharpoonup} N \cdot T = -N \nu_E(x) \mu$. By uniqueness of weak-$*$ limits, \begin{align*} D\mathbb{1}_F = -N \nu_E(x) \mu. \end{align*} In particular, $D\mathbb{1}_F$ is parallel to the constant vector $-\nu_E(x)$ as a vector-valued Radon measure. Equivalently, the orthogonal complement vector measures vanish: for every vector $v \perp \nu_E(x)$, \begin{align*} v \cdot D\mathbb{1}_F = 0 \quad \text{as a scalar Radon measure on } \mathbb{R}^n. \end{align*} *1D classification.* The condition "$v \cdot D\mathbb{1}_F = 0$ for every $v \perp \nu_E(x)$" means $\mathbb{1}_F$ is invariant under translation in directions orthogonal to $\nu_E(x)$ (since its distributional partial derivative in any orthogonal direction vanishes). Up to $\mathcal{L}^n$-null sets, $\mathbb{1}_F(y) = g(y \cdot \nu_E(x))$ for some Borel function $g: \mathbb{R} \to \{0, 1\}$. Since $\mathbb{1}_F \in BV_{\mathrm{loc}}(\mathbb{R}^n)$ and $g$ takes values in $\{0, 1\}$, $g$ is a $\{0,1\}$-valued $BV_{\mathrm{loc}}$ function on $\mathbb{R}$. The set $S = g^{-1}(1) \subset \mathbb{R}$ has locally finite perimeter in $\mathbb{R}$. *Pointwise convergence and orientation.* We now use the rescaling structure to determine $S$. By Step 4.2, the limit gradient $D\mathbb{1}_F = -N \nu_E(x) \mu$ has direction $-\nu_E(x)$ everywhere on its support. Translating to the slicing $g$: the distributional derivative $g'$ as a Radon measure on $\mathbb{R}$ has constant sign, equal to that of $-\nu_E(x) \cdot \nu_E(x) = -1$, meaning $g$ is *non-increasing* on $\mathbb{R}$. Since $g \in \{0, 1\}$ and is non-increasing, $g(t) = \mathbb{1}_{(-\infty, t_0)}(t)$ or $g(t) = \mathbb{1}_{(-\infty, t_0]}(t)$ for some $t_0 \in \mathbb{R} \cup \{\pm\infty\}$ (the two choices differ on a single point and yield the same $\mathbb{1}_F$ up to $\mathcal{L}^n$-null sets). *Pinning $t_0 = 0$.* The Lebesgue-point property (I) at the original point $x$ forces the limit measure $\mu$ to place positive mass on every neighbourhood of the origin in the rescaled space. To see this: by Step 4.1, $\sigma_{r_j} \overset{*}{\rightharpoonup} \mu$, and $\sigma_{r_j}(B(0, R)) = |D\mathbb{1}_E|(B(x, r_j R))/|D\mathbb{1}_E|(B(x, r_j))$. By (II), this ratio is bounded below by $c_n R^{n-1}/C(x) > 0$ for every fixed $R > 0$. By weak-$*$ lower semi-continuity of mass on open sets, \begin{align*} \mu(B(0, R)) \ge \liminf_j \sigma_{r_j}(B(0, R)) \ge \frac{c_n}{C(x)} R^{n-1} > 0 \end{align*} for every $R > 0$. In particular, $0 \in \operatorname{supp}(\mu)$. By Step 4.3, $D\mathbb{1}_F = -N \nu_E(x) \mu$, so $|D\mathbb{1}_F| = N \mu$ on $\mathbb{R}^n$, and $\operatorname{supp}(|D\mathbb{1}_F|) = \operatorname{supp}(\mu)$ (since $N > 0$). The 1D classification gives $F = \{y : y \cdot \nu_E(x) < t_0\}$ up to null sets, so $|D\mathbb{1}_F|$ is supported on the hyperplane $\Pi_{t_0} := \{y : y \cdot \nu_E(x) = t_0\}$. Combining, \begin{align*} 0 \in \operatorname{supp}(\mu) = \operatorname{supp}(|D\mathbb{1}_F|) \subseteq \Pi_{t_0} = \{y : y \cdot \nu_E(x) = t_0\}. \end{align*} Setting $y = 0$: $0 = 0 \cdot \nu_E(x) = t_0$. Hence $t_0 = 0$, and (up to $\mathcal{L}^n$-null sets, which do not distinguish $(-\infty, 0)$ from $(-\infty, 0]$), \begin{align*} F = \{y : y \cdot \nu_E(x) < 0\} = H. \end{align*} *Step 4.4: conclusion.* The subsequential limit $\mathbb{1}_F$ equals $\mathbb{1}_H$ a.e., where $H = \{y \in \mathbb{R}^n : y \cdot \nu_E(x) < 0\}$ is the open half-space whose *outward* unit normal is $\nu_E(x)$. The orientation is consistent: $D\mathbb{1}_H = -\nu_E(x) \mathcal{H}^{n-1} \lfloor \Pi$, the gradient of $\mathbb{1}_H$ points from $H^c$ into $H$, i.e.\ in the $-\nu_E(x)$ direction. [/guided] [/step] [step:Promote subsequential convergence to full-family convergence] Every sequence $r_j \to 0$ has a subsequence along which $\mathbb{1}_{E_{r_j}} \to \mathbb{1}_H$ in $L^1_{\mathrm{loc}}$. Since the limit $\mathbb{1}_H$ is the same for every subsequence, the full family converges: \begin{align*} \lim_{r \to 0} \int_{B(0, R)} |\mathbb{1}_{E_r}(z) - \mathbb{1}_H(z)| \, d\mathcal{L}^n(z) = 0 \quad \text{for every } R > 0. \end{align*} Equivalently, $\mathcal{L}^n((E_r \triangle H) \cap B(0, R)) \to 0$ as $r \to 0$, completing the proof. [guided] We promote the subsequential convergence (any sequence has a subsequence convergent to $\mathbb{1}_H$) to convergence of the full family. *Standard subsequential principle.* In any topological space, if every sequence has a subsequence convergent to a fixed limit $L$, then the full family converges to $L$. The contrapositive: suppose $\mathbb{1}_{E_r} \not\to \mathbb{1}_H$ in $L^1_{\mathrm{loc}}$. Then there exist $\delta > 0$, $R > 0$, and $r_j \to 0$ with \begin{align*} \|\mathbb{1}_{E_{r_j}} - \mathbb{1}_H\|_{L^1(B(0, R))} \ge \delta \quad \text{for all } j. \end{align*} But by Step 3 applied to $(r_j)$, there is a subsequence $r_{j_k}$ along which $\mathbb{1}_{E_{r_{j_k}}} \to \mathbb{1}_H$ in $L^1_{\mathrm{loc}}$, in particular in $L^1(B(0, R))$. This contradicts the lower bound. Hence the full family converges: \begin{align*} \lim_{r \to 0} \mathbb{1}_{E_r} = \mathbb{1}_H \quad \text{in } L^1_{\mathrm{loc}}(\mathbb{R}^n). \end{align*} *Restatement in measure-theoretic form.* The $L^1_{\mathrm{loc}}$ statement is, for every $R > 0$, \begin{align*} \lim_{r \to 0} \int_{B(0, R)} |\mathbb{1}_{E_r}(z) - \mathbb{1}_H(z)| \, d\mathcal{L}^n(z) = 0. \end{align*} Since $\mathbb{1}_{E_r}, \mathbb{1}_H \in \{0, 1\}$, the integrand $|\mathbb{1}_{E_r} - \mathbb{1}_H| = \mathbb{1}_{E_r \triangle H}$, so \begin{align*} \int_{B(0, R)} \mathbb{1}_{E_r \triangle H}(z) \, d\mathcal{L}^n(z) = \mathcal{L}^n((E_r \triangle H) \cap B(0, R)). \end{align*} Hence $\mathcal{L}^n((E_r \triangle H) \cap B(0, R)) \to 0$ as $r \to 0$, the symmetric-difference form of the blow-up convergence. *Summary of the argument.* Step 1 set up the rescaling; Step 2 derived two-sided density bounds on $|D\mathbb{1}_E|(B(x, r))$ self-contained from $x \in \partial^* E$ and the relative isoperimetric inequality, with no input from [De Giorgi's structure theorem](/theorems/599) (the lower direction from non-degeneracy of the Lebesgue density of $E$ at $\partial^* E$ points combined with the relative isoperimetric inequality; the upper direction from the standard real-analysis fact that perimeter measures are locally $\mathcal{H}^{n-1}$-finite at $|D\mathbb{1}_E|$-a.e.\ point); Step 3 applied [BV Compactness](/theorems/596) and a diagonal argument to extract a subsequential $L^1_{\mathrm{loc}}$ limit $\mathbb{1}_F$; Step 4 used the Lebesgue-point property at $x$ to identify the limit gradient as parallel to $-\nu_E(x)$, then invoked the 1D classification of $\{0,1\}$-valued $BV$ functions to conclude $F = H$; Step 5 promoted subsequential convergence to full convergence by the uniqueness of the limit. The proof is complete. [/guided] [/step]