[proofplan] The identity $P(E; \Omega) = \mathcal{H}^{n-1}(\partial^* E \cap \Omega)$ is a direct corollary of [De Giorgi's Structure Theorem](/theorems/599)(c), which asserts the equality of measures \begin{align*} |D\mathbb{1}_E| = \mathcal{H}^{n-1}\lfloor \partial^* E. \end{align*} Evaluating both sides on the open set $\Omega$ gives the claim. The proof has two short steps: first, recall the definition $P(E; \Omega) := |D\mathbb{1}_E|(\Omega)$; second, apply the measure identity from De Giorgi to evaluate the right-hand side as $\mathcal{H}^{n-1}(\Omega \cap \partial^* E)$. The hypothesis that $E$ has finite perimeter is what makes $|D\mathbb{1}_E|$ a finite Radon measure to which De Giorgi's theorem applies, and this in turn ensures $\mathcal{H}^{n-1}(\partial^* E)$ is locally finite. [/proofplan] [step:Recall the definition of perimeter as the total variation of the indicator] For a Borel set $E \subseteq \mathbb{R}^n$ of finite perimeter and an open set $\Omega \subseteq \mathbb{R}^n$, the perimeter of $E$ in $\Omega$ is by definition \begin{align*} P(E; \Omega) := |D\mathbb{1}_E|(\Omega), \end{align*} the total variation of the distributional gradient $D\mathbb{1}_E$ — viewed as a finite vector-valued Radon measure on $\mathbb{R}^n$ — evaluated on $\Omega$. The hypothesis that $E$ has finite perimeter is the assertion that $|D\mathbb{1}_E|(\mathbb{R}^n) < \infty$, which makes $|D\mathbb{1}_E|$ a finite (hence Radon) measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$. [guided] The starting point is purely definitional. A Borel set $E \subseteq \mathbb{R}^n$ has finite perimeter if and only if $\mathbb{1}_E \in BV(\mathbb{R}^n)$, which means the distributional gradient $D\mathbb{1}_E$ is a finite vector-valued Radon measure on $\mathbb{R}^n$. The perimeter of $E$ in any open subset $\Omega \subseteq \mathbb{R}^n$ is then defined to be \begin{align*} P(E; \Omega) := |D\mathbb{1}_E|(\Omega), \end{align*} where $|D\mathbb{1}_E|$ is the total variation measure of the vector-valued $D\mathbb{1}_E$. Equivalently, by the variational characterisation, \begin{align*} P(E; \Omega) = \sup \left\{ \int_E \operatorname{div} \varphi \, d\mathcal{L}^n : \varphi \in C_c^1(\Omega; \mathbb{R}^n), \ |\varphi| \le 1 \right\}. \end{align*} The hypothesis "$E$ has finite perimeter" means $P(E; \mathbb{R}^n) = |D\mathbb{1}_E|(\mathbb{R}^n) < \infty$. Hence $|D\mathbb{1}_E|$ is a finite Radon measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$, in particular on $\Omega$ for any open $\Omega \subseteq \mathbb{R}^n$. This regularity is what allows the application of the structure theorem in the next step. The conclusion of this step is the formal identification of $P(E; \Omega)$ with $|D\mathbb{1}_E|(\Omega)$ — a renaming, but one we use directly. [/guided] [/step] [step:Apply De Giorgi's theorem to identify $|D\mathbb{1}_E|$ as $\mathcal{H}^{n-1}\lfloor \partial^* E$] By [De Giorgi's Structure Theorem](/theorems/599), part (c), the perimeter measure of a Borel set $E$ of finite perimeter satisfies \begin{align*} |D\mathbb{1}_E| = \mathcal{H}^{n-1}\lfloor \partial^* E, \end{align*} that is, for every Borel set $A \subseteq \mathbb{R}^n$, \begin{align*} |D\mathbb{1}_E|(A) = \mathcal{H}^{n-1}(A \cap \partial^* E). \end{align*} Applying this measure identity to the open set $A = \Omega$ (open sets are Borel), \begin{align*} |D\mathbb{1}_E|(\Omega) = \mathcal{H}^{n-1}(\Omega \cap \partial^* E) = \mathcal{H}^{n-1}(\partial^* E \cap \Omega). \end{align*} Combining with Step 1, \begin{align*} P(E; \Omega) = |D\mathbb{1}_E|(\Omega) = \mathcal{H}^{n-1}(\partial^* E \cap \Omega), \end{align*} the desired identity. The hypothesis of the theorem — that $E$ has finite perimeter — is precisely the hypothesis of De Giorgi's theorem, and the open set $\Omega$ may be any open subset of $\mathbb{R}^n$. [guided] The result is a direct consequence of De Giorgi's structure theorem. Let us spell out the application. *Hypotheses of [De Giorgi's Structure Theorem](/theorems/599).* The theorem requires $E \subseteq \mathbb{R}^n$ to be a Borel-measurable set of finite perimeter. Both hypotheses are explicit in the corollary's statement: "If $E \subset \mathbb{R}^n$ has finite perimeter, then \ldots", and finite-perimeter sets are taken to be Borel-measurable in our setting (one can always replace a Lebesgue-measurable set with a Borel set differing by a $\mathcal{L}^n$-null set, and this does not change the distributional gradient $D\mathbb{1}_E$ since $D\mathbb{1}_E$ is well-defined modulo $\mathcal{L}^n$-a.e. equality). *Conclusion of the structure theorem.* Part (c) of the theorem gives the perimeter measure identity \begin{align*} |D\mathbb{1}_E|(A) = \mathcal{H}^{n-1}(A \cap \partial^* E) \quad \text{for every Borel } A \subseteq \mathbb{R}^n. \end{align*} This is an equality of finite Borel measures on $\mathbb{R}^n$: the LHS is $|D\mathbb{1}_E|$ evaluated on $A$, the RHS is the restriction $\mathcal{H}^{n-1}\lfloor \partial^* E$ evaluated on $A$. (Note: $\mathcal{H}^{n-1}$ in $\mathbb{R}^n$ is not finite as a measure, but its restriction to the $(n-1)$-rectifiable set $\partial^* E$ is, because the structure theorem's part (a) guarantees rectifiability and hence $\mathcal{H}^{n-1}(\partial^* E \cap K) < \infty$ for every compact $K$, with total mass $|D\mathbb{1}_E|(\mathbb{R}^n) < \infty$ globally.) *Evaluating at $A = \Omega$.* Open sets are Borel-measurable. Setting $A = \Omega$ in the structure theorem's identity, \begin{align*} |D\mathbb{1}_E|(\Omega) = \mathcal{H}^{n-1}(\Omega \cap \partial^* E). \end{align*} The intersection is symmetric: $\Omega \cap \partial^* E = \partial^* E \cap \Omega$. So \begin{align*} |D\mathbb{1}_E|(\Omega) = \mathcal{H}^{n-1}(\partial^* E \cap \Omega). \end{align*} *Combining with Step 1.* By Step 1, $P(E; \Omega) = |D\mathbb{1}_E|(\Omega)$. Substituting, \begin{align*} P(E; \Omega) = \mathcal{H}^{n-1}(\partial^* E \cap \Omega). \end{align*} This is the corollary's claim. The proof is complete. *Remark on the structure of the argument.* The substantive content lies entirely in the structure theorem itself, which carries the deep analytic-geometric work: it asserts that the perimeter measure $|D\mathbb{1}_E|$ — defined a priori as the total variation of a distributional gradient — coincides with the $(n-1)$-dimensional Hausdorff measure on the reduced boundary, a purely geometric set. The corollary then specialises this measure identity by evaluating both sides on any open set $\Omega$. Conceptually, the corollary says: the analytic notion of perimeter (total variation of $D\mathbb{1}_E$) and the geometric notion (Hausdorff measure of $\partial^* E$) coincide on every open set, which is the bridge between BV theory and classical geometric measure theory. [/guided] [/step]