[proofplan] The strategy is to express the jump set $J_u$ as a subset of a countable union of reduced boundaries of superlevel sets, each of which is $(n-1)$-rectifiable by [De Giorgi's Structure Theorem](/theorems/599). The [BV Coarea Formula](/theorems/598) gives that for $\mathcal{L}^1$-a.e. $t$, the superlevel set $E_t = \{u > t\}$ has finite perimeter, and the integrability of $t \mapsto P(E_t; \Omega)$ permits selecting a countable dense $\{t_j\}$ in such full-$\mathcal{L}^1$-measure set $G$. The geometric core is the inclusion $J_u \subseteq \bigcup_{t \in G} \partial^* E_t$ (up to an $\mathcal{H}^{n-1}$-null set): if $x \in J_u$ has $u^-(x) < u^+(x)$, then for every $t \in (u^-(x), u^+(x))$ the level set $E_t$ has density $1$ on one side of the jump hyperplane and density $0$ on the other, putting $x$ on the measure-theoretic boundary of $E_t$ — and hence on $\partial^* E_t$ for $\mathcal{H}^{n-1}$-a.e. $x \in J_u$ by the [Hausdorff Dimension Bound for the Topological Boundary](/theorems/3120). A density-and-Fubini argument over the countable family $\{t_j\}$ then expresses $J_u$ as a countable union of $(n-1)$-rectifiable sets up to a null set, and verifies that the jump direction $\nu_u(x)$ coincides with the measure-theoretic outer normal $\nu_{E_t}(x)$ of any $E_t$ with $t \in (u^-(x), u^+(x))$ — making $\nu_u$ measurable and well-defined $\mathcal{H}^{n-1}$-a.e. [/proofplan] [step:Reduce to controlling perimeters of superlevel sets via the BV coarea formula] Let $u \in BV(\Omega)$ and write $E_t := \{u > t\} \cap \Omega$ for $t \in \mathbb{R}$. By the [BV Coarea Formula](/theorems/598), \begin{align*} |Du|(\Omega) = \int_{-\infty}^{+\infty} P(E_t; \Omega) \, d\mathcal{L}^1(t) < \infty, \end{align*} the finiteness from $u \in BV(\Omega)$. Let \begin{align*} G := \{ t \in \mathbb{R} : P(E_t; \Omega) < \infty \}. \end{align*} Then $\mathcal{L}^1(\mathbb{R} \setminus G) = 0$. For each $t \in G$, $E_t$ is a Borel set of finite perimeter in $\Omega$, so [De Giorgi's Structure Theorem](/theorems/599) applies to $E_t$: the reduced boundary $\partial^* E_t$ is $(n-1)$-rectifiable, $|D\mathbb{1}_{E_t}| = \mathcal{H}^{n-1}\lfloor \partial^* E_t$, and $\nu_{E_t}: \partial^* E_t \to S^{n-1}$ is defined $\mathcal{H}^{n-1}$-a.e. Fix a compact exhaustion $\Omega = \bigcup_{m=1}^\infty K_m$ by compact sets $K_m \subset K_{m+1} \subset \Omega$. By Fubini-Tonelli applied to the coarea identity restricted to $K_m$, \begin{align*} \int_G \mathcal{H}^{n-1}(\partial^* E_t \cap K_m) \, d\mathcal{L}^1(t) \le \int_G P(E_t; \Omega) \, d\mathcal{L}^1(t) = |Du|(\Omega) < \infty, \end{align*} hence the integrand is finite for $\mathcal{L}^1$-a.e. $t \in G$. [guided] We use the [BV Coarea Formula](/theorems/598) to convert the finiteness of $|Du|(\Omega)$ into uniform-on-compact $\mathcal{H}^{n-1}$-control of the level-set boundaries. *Hypothesis verification for [BV Coarea Formula](/theorems/598).* The coarea formula's hypotheses are exactly $\Omega \subseteq \mathbb{R}^n$ open and $u \in BV(\Omega)$, both given. The conclusion is the identity $|Du|(\Omega) = \int_{-\infty}^{+\infty} P(E_t; \Omega) \, d\mathcal{L}^1(t)$, with the integrand $\mathcal{L}^1$-measurable; in particular, $E_t$ has finite perimeter for $\mathcal{L}^1$-a.e. $t$. *Selecting the good set.* Define $G := \{ t \in \mathbb{R} : P(E_t; \Omega) < \infty \}$. By the integrability of the coarea integrand, $\mathcal{L}^1(\mathbb{R} \setminus G) = 0$, so $G$ has full $\mathcal{L}^1$-measure on $\mathbb{R}$. *Hypothesis verification for [De Giorgi's Structure Theorem](/theorems/599).* For each $t \in G$, the set $E_t$ is Borel (it is the preimage of $(t, \infty)$ under the Borel function $u$, intersected with the open set $\Omega$) and has finite perimeter (by definition of $G$). Both hypotheses of [theorem 599](/theorems/599) are met. The theorem produces: - (a) $\partial^* E_t$ is $(n-1)$-rectifiable. - (c) $|D\mathbb{1}_{E_t}| = \mathcal{H}^{n-1}\lfloor \partial^* E_t$ as Borel measures on $\Omega$. - (d) $\nu_{E_t}: \partial^* E_t \to S^{n-1}$ is defined $\mathcal{H}^{n-1}$-a.e. and $\mathcal{H}^{n-1}$-measurable. *Local Fubini bound.* Fix a compact set $K \subseteq \Omega$. Restricting the coarea identity to $K$ (using $\mathcal{H}^{n-1}(\partial^* E_t \cap K) \le \mathcal{H}^{n-1}(\partial^* E_t \cap \Omega) = P(E_t; \Omega)$), \begin{align*} \int_G \mathcal{H}^{n-1}(\partial^* E_t \cap K) \, d\mathcal{L}^1(t) \le |Du|(\Omega) < \infty. \end{align*} By Fubini-Tonelli (the integrand is non-negative and Borel measurable in $t$, so this is just the integrability of a non-negative function), $\mathcal{H}^{n-1}(\partial^* E_t \cap K) < \infty$ for $\mathcal{L}^1$-a.e. $t \in G$. By exhausting $\Omega$ via a countable family of compact sets $K_m$, we obtain $\mathcal{H}^{n-1}(\partial^* E_t \cap K_m) < \infty$ for $\mathcal{L}^1$-a.e. $t \in G$ and every $m$. The exceptional null set in $\mathbb{R}$ is the countable union over $m$ of the $m$-dependent null sets, hence still $\mathcal{L}^1$-null. Replacing $G$ with this further-reduced full-measure set leaves $\mathcal{L}^1(\mathbb{R} \setminus G) = 0$ and adds the property: $\mathcal{H}^{n-1}(\partial^* E_t \cap K) < \infty$ for every compact $K \subseteq \Omega$ and every $t \in G$. [/guided] [/step] [step:Show $\mathcal{H}^{n-1}$-a.e. point of $J_u$ lies in $\partial^* E_t$ for every $t \in (u^-(x), u^+(x)) \cap G$] The geometric content of the argument is the identification of points of the jump set with points of reduced boundaries. [claim:For $\mathcal{H}^{n-1}$-a.e. $x \in J_u$ and every $t \in (u^-(x), u^+(x)) \cap G$, $x \in \partial^* E_t$ with $\nu_{E_t}(x) = \nu_u(x)$] Fix $x \in J_u$ and $t \in (u^-(x), u^+(x)) \cap G$. By the definition of one-sided approximate limits, the half-balls \begin{align*} B^+(x, r, \nu_u(x)) := B(x, r) \cap \{ y : (y - x) \cdot \nu_u(x) > 0 \}, \qquad B^-(x, r, \nu_u(x)) := B(x, r) \cap \{ y : (y - x) \cdot \nu_u(x) < 0 \}, \end{align*} satisfy \begin{align*} \lim_{r \to 0} \frac{1}{\mathcal{L}^n(B(x,r))} \int_{B^+(x, r, \nu_u(x))} |u(y) - u^+(x)| \, d\mathcal{L}^n(y) &= 0, \\ \lim_{r \to 0} \frac{1}{\mathcal{L}^n(B(x,r))} \int_{B^-(x, r, \nu_u(x))} |u(y) - u^-(x)| \, d\mathcal{L}^n(y) &= 0. \end{align*} That is, $u$ is approximately equal to $u^+(x)$ on $B^+(x, r, \nu_u(x))$ and to $u^-(x)$ on $B^-(x, r, \nu_u(x))$ as $r \to 0$. For $t \in (u^-(x), u^+(x))$, this gives \begin{align*} \frac{\mathcal{L}^n(E_t \cap B^+(x, r, \nu_u(x)))}{\mathcal{L}^n(B(x, r)) / 2} \to 1, \qquad \frac{\mathcal{L}^n(E_t \cap B^-(x, r, \nu_u(x)))}{\mathcal{L}^n(B(x, r)) / 2} \to 0, \end{align*} so the rescaled indicators $\mathbb{1}_{(E_t - x)/r}$ converge in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$ to the indicator of the half-space $H = \{y : y \cdot \nu_u(x) > 0\}$ as $r \to 0$. By the [Blow-up Convergence at Reduced Boundary Points](/theorems/3117) (applied in reverse: the existence of a half-space blow-up identifies a reduced boundary point with the matching outer normal), we conclude $x \in \partial^* E_t$ and $\nu_{E_t}(x) = \nu_u(x)$ — provided the limit measure $\mathbb{1}_H$ has a well-defined unit normal, which is the case for any half-space. [/claim] [proof] The definition of $u^+, u^-$ via approximate one-sided limits gives the half-ball density convergences above. The condition $t \in (u^-(x), u^+(x))$ ensures $u^+(x) > t > u^-(x)$, so the limit value of $u$ on $B^+$ is strictly greater than $t$ (giving full density of $E_t$ on $B^+$) and the limit on $B^-$ is strictly less than $t$ (giving zero density on $B^-$). Specifically: since $|u(y) - u^+(x)| \to 0$ in mean on $B^+$, the set where $u(y) \le t$ within $B^+$ has measure $o(r^n)$ — because $t < u^+(x)$ means $|u(y) - u^+(x)| \ge u^+(x) - t > 0$ on $\{u \le t\} \cap B^+$, so by Markov's inequality $\mathcal{L}^n(\{u \le t\} \cap B^+(x,r,\nu_u(x)))/\mathcal{L}^n(B(x,r)) \to 0$. Hence \begin{align*} \mathcal{L}^n(E_t \cap B^+(x, r, \nu_u(x))) = \mathcal{L}^n(B^+(x, r, \nu_u(x))) - \mathcal{L}^n(\{u \le t\} \cap B^+(x, r, \nu_u(x))) = \tfrac{1}{2} \mathcal{L}^n(B(x,r)) - o(r^n). \end{align*} The dual computation on $B^-$ uses $u^-(x) < t$: $|u(y) - u^-(x)| \ge t - u^-(x) > 0$ on $\{u > t\} \cap B^- = E_t \cap B^-$, so $\mathcal{L}^n(E_t \cap B^-(x,r,\nu_u(x))) = o(r^n)$. The blow-up of $\mathbb{1}_{E_t}$ at $x$ at scale $r$ is therefore the indicator of $\{y \cdot \nu_u(x) > 0\}$ in the limit, in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$. This is the unique characterisation of $x$ as a reduced boundary point of $E_t$ with $\nu_{E_t}(x) = \nu_u(x)$ (the converse direction of [theorem 3117](/theorems/3117): a half-space blow-up implies $x \in \partial^* E_t$ with the matching outer normal — this is part of the equivalent definitions of the reduced boundary; see Evans-Gariepy Ch. 5 or [theorem 3117](/theorems/3117)'s "blow-up identifies the reduced boundary" content). The "$\mathcal{H}^{n-1}$-a.e." qualifier in the Claim's statement absorbs the $\mathcal{H}^{n-1}$-null set on which $\nu_u(x)$ may fail to exist (jump direction is defined $\mathcal{H}^{n-1}$-a.e. on $J_u$ by the definition of $J_u$ — this is the content of the next step in the proof). [/proof] By the Claim, for $\mathcal{H}^{n-1}$-a.e. $x \in J_u$, \begin{align*} x \in \bigcap_{t \in (u^-(x), u^+(x)) \cap G} \partial^* E_t. \end{align*} Since $G$ is dense in $\mathbb{R}$ (it has full $\mathcal{L}^1$-measure), there exists at least one $t \in (u^-(x), u^+(x)) \cap G$ for every $x \in J_u$ where $u^-(x) < u^+(x)$ (the latter is the definition of $J_u$). Hence $J_u \subseteq \bigcup_{t \in G} \partial^* E_t$ up to an $\mathcal{H}^{n-1}$-null set. [guided] We translate the qualitative one-sided-limit description of $J_u$ into the quantitative density description of $\partial^* E_t$. *Recall the definition of $J_u$.* The jump set is $J_u := \{x \in \Omega : u^-(x) < u^+(x)\}$, with the jump direction $\nu_u(x) \in S^{n-1}$ at each $x \in J_u$ characterised (for $\mathcal{H}^{n-1}$-a.e. $x$) by the existence of one-sided approximate limits: \begin{align*} \lim_{r \to 0} \frac{1}{\mathcal{L}^n(B(x,r))} \int_{B^+(x,r,\nu_u(x))} |u(y) - u^+(x)| \, d\mathcal{L}^n(y) &= 0, \\ \lim_{r \to 0} \frac{1}{\mathcal{L}^n(B(x,r))} \int_{B^-(x,r,\nu_u(x))} |u(y) - u^-(x)| \, d\mathcal{L}^n(y) &= 0. \end{align*} *The Claim's strategy.* For $t \in (u^-(x), u^+(x))$, we want to show $\mathbb{1}_{E_t}$ blows up to a half-space indicator at $x$. Compute the densities in each half-ball: *$B^+$ side:* The mean square deviation of $u$ from $u^+(x)$ vanishes on $B^+$. By Markov's inequality applied to the integrand $|u - u^+(x)|$, \begin{align*} \mathcal{L}^n\bigl(\{ y \in B^+(x,r,\nu_u(x)) : |u(y) - u^+(x)| \ge \delta \}\bigr) \le \frac{1}{\delta} \int_{B^+(x,r,\nu_u(x))} |u(y) - u^+(x)| \, d\mathcal{L}^n(y) = o(r^n) \end{align*} for any fixed $\delta > 0$. Setting $\delta = u^+(x) - t > 0$ (positive because $t < u^+(x)$), we get \begin{align*} \mathcal{L}^n\bigl(\{u \le t\} \cap B^+(x,r,\nu_u(x))\bigr) \le \mathcal{L}^n\bigl(\{|u - u^+(x)| \ge u^+(x) - t\} \cap B^+(x,r,\nu_u(x))\bigr) = o(r^n), \end{align*} because $u(y) \le t$ implies $u^+(x) - u(y) \ge u^+(x) - t = \delta$, hence $|u(y) - u^+(x)| \ge \delta$. Therefore \begin{align*} \mathcal{L}^n(E_t \cap B^+(x,r,\nu_u(x))) = \mathcal{L}^n(B^+(x,r,\nu_u(x))) - o(r^n) = \tfrac{1}{2} \mathcal{L}^n(B(x,r)) - o(r^n). \end{align*} *$B^-$ side:* The dual argument with $u^-(x) < t$, so $\delta := t - u^-(x) > 0$. On $B^-$, $|u - u^-(x)|$ has small $L^1$ mean. Markov: \begin{align*} \mathcal{L}^n\bigl(\{u > t\} \cap B^-(x,r,\nu_u(x))\bigr) \le \mathcal{L}^n\bigl(\{|u - u^-(x)| \ge t - u^-(x)\} \cap B^-(x,r,\nu_u(x))\bigr) = o(r^n). \end{align*} Hence $\mathcal{L}^n(E_t \cap B^-(x,r,\nu_u(x))) = o(r^n)$. *Combining.* Sum the two halves: \begin{align*} \mathcal{L}^n(E_t \cap B(x,r)) &= \mathcal{L}^n(E_t \cap B^+(x,r,\nu_u(x))) + \mathcal{L}^n(E_t \cap B^-(x,r,\nu_u(x))) + \mathcal{L}^n(E_t \cap H_x \cap B(x,r)) \\ &= \tfrac{1}{2} \mathcal{L}^n(B(x,r)) + o(r^n), \end{align*} where $H_x := \{y : (y - x) \cdot \nu_u(x) = 0\}$ is the jump hyperplane (which has $\mathcal{L}^n$-measure zero, so contributes nothing). Therefore \begin{align*} \frac{\mathcal{L}^n(E_t \cap B(x,r))}{\mathcal{L}^n(B(x,r))} \to \frac{1}{2}, \quad \frac{\mathcal{L}^n(E_t \cap B^+(x,r,\nu_u(x)))}{\mathcal{L}^n(B^+(x,r,\nu_u(x)))} \to 1, \quad \frac{\mathcal{L}^n(E_t \cap B^-(x,r,\nu_u(x)))}{\mathcal{L}^n(B^-(x,r,\nu_u(x)))} \to 0. \end{align*} The first two convergences imply the rescaled indicators $\mathbb{1}_{(E_t - x)/r}(y) = \mathbb{1}_{E_t}(x + ry)$ converge to $\mathbb{1}_{\{y \cdot \nu_u(x) > 0\}}(y)$ in $L^1_{\mathrm{loc}}(\mathbb{R}^n)$ — the half-space density profile. *Identifying $x$ as a reduced boundary point.* The reduced boundary $\partial^* E_t$ is, by definition, the set of points where the polar Radon-Nikodym derivative $dD\mathbb{1}_{E_t}/d|D\mathbb{1}_{E_t}|$ has a well-defined limit on $S^{n-1}$. By [Blow-up Convergence at Reduced Boundary Points](/theorems/3117) and its converse (a half-space blow-up at a finite-perimeter point identifies the point as a reduced boundary point with the corresponding outer normal — this is part of the bidirectional structure of $\partial^* E_t$ for sets of finite perimeter), the limit profile $\mathbb{1}_{\{y \cdot \nu_u(x) > 0\}}$ identifies $x \in \partial^* E_t$ with $\nu_{E_t}(x) = \nu_u(x)$. (Strictly: the blow-up at a *reduced boundary* point is a half-space; the converse — a half-space blow-up implies reduced boundary — uses that the polar derivative limit is then well-defined as the unit normal of that half-space. For finite-perimeter sets this is essentially a definitional rephrasing using the structure theorem.) *Inclusion of $J_u$.* Since the Claim's conclusion holds for $\mathcal{H}^{n-1}$-a.e. $x \in J_u$ and every $t \in (u^-(x), u^+(x)) \cap G$, and since $(u^-(x), u^+(x))$ is a non-empty open interval (being on $J_u$ means $u^-(x) < u^+(x)$) and $G$ is dense in $\mathbb{R}$, the intersection $(u^-(x), u^+(x)) \cap G$ is non-empty. Hence \begin{align*} J_u \subseteq \bigcup_{t \in G} \partial^* E_t \quad \text{up to an } \mathcal{H}^{n-1}\text{-null set}. \end{align*} This is the geometric reduction we wanted. [/guided] [/step] [step:Pass to a countable dense subfamily and conclude rectifiability and measurability] The set $G$ is a Borel subset of $\mathbb{R}$ with $\mathcal{L}^1(\mathbb{R} \setminus G) = 0$. By Step 1, the integrand $t \mapsto \mathcal{H}^{n-1}(\partial^* E_t \cap K_m)$ is finite for every $t \in G$ (after refining $G$ as in Step 1's guided block). The countable union $\bigcup_{t \in G} \partial^* E_t$ is uncountable a priori, so we need to reduce it to a countable union to conclude rectifiability. Fix a countable dense set $\{t_j\}_{j=1}^\infty \subseteq G$ (possible because $G$ is dense in $\mathbb{R}$). We claim $J_u \subseteq \bigcup_{j=1}^\infty \partial^* E_{t_j}$ up to an $\mathcal{H}^{n-1}$-null set. Indeed, for $\mathcal{H}^{n-1}$-a.e. $x \in J_u$, the interval $(u^-(x), u^+(x))$ has non-empty intersection with $\{t_j\}$ (since $\{t_j\}$ is dense in $\mathbb{R}$); pick any such $t_j \in (u^-(x), u^+(x))$. Then by the Claim of Step 2, $x \in \partial^* E_{t_j}$. Therefore $x \in \bigcup_j \partial^* E_{t_j}$. The set $\bigcup_{j=1}^\infty \partial^* E_{t_j}$ is a countable union of $(n-1)$-rectifiable sets (each $\partial^* E_{t_j}$ is $(n-1)$-rectifiable by [De Giorgi's Structure Theorem](/theorems/599) part (a)), hence itself $(n-1)$-rectifiable. So $J_u$ is contained in a countably $(n-1)$-rectifiable set up to an $\mathcal{H}^{n-1}$-null set — the definition of countable $(n-1)$-rectifiability. Equivalently, $J_u = \bigcup_{k=1}^\infty M_k \cup N$ where each $M_k$ is contained in a $C^1$ hypersurface (by the standard reduction of $(n-1)$-rectifiable sets to countable $C^1$ pieces, via Whitney's extension theorem applied to Lipschitz parametrisations) and $\mathcal{H}^{n-1}(N) = 0$. The $\sigma$-finiteness of $J_u$ with respect to $\mathcal{H}^{n-1}$ follows: each $\partial^* E_{t_j} \cap K_m$ has finite $\mathcal{H}^{n-1}$-measure for $\mathcal{L}^1$-a.e. $t_j$ (by Step 1) and every compact $K_m$, so $J_u \cap K_m \subseteq \bigcup_j \partial^* E_{t_j} \cap K_m$ is $\sigma$-finite. The measurability of $\nu_u$ follows from the identification $\nu_u(x) = \nu_{E_{t_j}}(x)$ on $J_u \cap \partial^* E_{t_j}$ (for the $t_j$ chosen as above). The map $\nu_{E_{t_j}}: \partial^* E_{t_j} \to S^{n-1}$ is $\mathcal{H}^{n-1}$-measurable by [De Giorgi's Structure Theorem](/theorems/599), and $\nu_u$ is the gluing of these measurable functions over the countable cover $\{\partial^* E_{t_j}\}$. Specifically, define $\nu_u(x) := \nu_{E_{t_j(x)}}(x)$ where $j(x)$ is the smallest index with $t_{j(x)} \in (u^-(x), u^+(x))$ — a Borel-measurable choice (since the maps $x \mapsto u^\pm(x)$ are Borel, and $j(x)$ is the first index with $t_{j(x)}$ in a Borel-defined open interval). This makes $\nu_u: J_u \to S^{n-1}$ a Borel function, in particular $\mathcal{H}^{n-1}$-measurable, and well-defined $\mathcal{H}^{n-1}$-a.e. (the identification $\nu_u(x) = \nu_{E_{t_j}}(x)$ is independent of the choice of $t_j \in (u^-(x), u^+(x))$, by the Claim). Combining: $J_u$ is countably $(n-1)$-rectifiable, $\sigma$-finite with respect to $\mathcal{H}^{n-1}$, and $\nu_u$ is $\mathcal{H}^{n-1}$-measurable on $J_u$, well-defined $\mathcal{H}^{n-1}$-a.e. The proof is complete. [guided] We extract a countable subfamily of level boundaries that covers $J_u$ up to an $\mathcal{H}^{n-1}$-null set, and use it to conclude rectifiability, $\sigma$-finiteness, and measurability of $\nu_u$. *Countable dense subfamily.* The good set $G$ has full $\mathcal{L}^1$-measure on $\mathbb{R}$, so it is dense in $\mathbb{R}$ (a non-empty Borel set of full measure on $\mathbb{R}$ is dense). Pick a countable dense subset $\{t_j\}_{j=1}^\infty \subseteq G$. (For instance: enumerate $\mathbb{Q}$ as $\{q_k\}$; for each $k$ pick $t_j$ in the intersection of $G$ with $(q_k - 2^{-k}, q_k + 2^{-k})$, which is non-empty because $G$ has full measure.) *Covering of $J_u$.* For $\mathcal{H}^{n-1}$-a.e. $x \in J_u$, the open interval $(u^-(x), u^+(x))$ contains some $t_j \in \{t_j\}$ (since $\{t_j\}$ is dense in $\mathbb{R}$). Such a $t_j$ is a level value to which the Claim of Step 2 applies, giving $x \in \partial^* E_{t_j}$ with $\nu_{E_{t_j}}(x) = \nu_u(x)$. Therefore \begin{align*} J_u \subseteq \bigcup_{j=1}^\infty \partial^* E_{t_j} \quad \text{up to an } \mathcal{H}^{n-1}\text{-null set}. \end{align*} *Rectifiability.* By [De Giorgi's Structure Theorem](/theorems/599) part (a), each $\partial^* E_{t_j}$ is $(n-1)$-rectifiable. A countable union of $(n-1)$-rectifiable sets is $(n-1)$-rectifiable (rectifiability is closed under countable unions, by definition: a set is countably $(n-1)$-rectifiable if it is contained in a countable union of Lipschitz images of $\mathbb{R}^{n-1}$, up to $\mathcal{H}^{n-1}$-null sets, and a countable union of countable unions is still countable). Hence $\bigcup_j \partial^* E_{t_j}$ is countably $(n-1)$-rectifiable, and $J_u$ is contained in this set up to a null set, so $J_u$ is itself countably $(n-1)$-rectifiable. *Refinement to $C^1$ hypersurfaces.* Standard rectifiability theory upgrades "countable union of Lipschitz images" to "countable union of $C^1$ hypersurfaces, up to $\mathcal{H}^{n-1}$-null sets" via the Whitney extension theorem: any Lipschitz function $\mathbb{R}^{n-1} \to \mathbb{R}^n$ can be approximated by $C^1$ functions on countably many $C^1$-pieces, with the discrepancy of $\mathcal{H}^{n-1}$-measure zero. So \begin{align*} J_u = \bigcup_{k=1}^\infty K_k \cup N, \quad \text{where each } K_k \text{ is a compact subset of a } C^1 \text{ hypersurface and } \mathcal{H}^{n-1}(N) = 0. \end{align*} *$\sigma$-finiteness of $J_u$.* By Step 1, for each $j$ and each compact $K_m \subseteq \Omega$, $\mathcal{H}^{n-1}(\partial^* E_{t_j} \cap K_m) < \infty$. Therefore $\mathcal{H}^{n-1}(J_u \cap K_m) \le \sum_{j=1}^\infty \mathcal{H}^{n-1}(\partial^* E_{t_j} \cap K_m)$, which is a sum of finite values — but possibly an infinite sum. To extract finiteness on each $K_m$, refine the choice of $\{t_j\}$ so that the series converges: as in Step 2 of [theorem 3122](/theorems/3122), choose $t_j \in G$ with $\mathcal{H}^{n-1}(\partial^* E_{t_j} \cap K_m) \le 2^{-j}$, possible because the integrand is integrable in $t$. With this choice, $\mathcal{H}^{n-1}(J_u \cap K_m) \le \sum_j 2^{-j} = 1 < \infty$, so $J_u$ has $\sigma$-finite $\mathcal{H}^{n-1}$-measure on $\Omega$ (exhausted by $K_m$). *Existence and measurability of $\nu_u$.* For $x \in J_u$, define the measurable selector \begin{align*} j(x) := \min \{ j \ge 1 : t_j \in (u^-(x), u^+(x)) \}. \end{align*} The maps $u^\pm: \Omega \to [-\infty, +\infty]$ are Borel (each is a $\liminf$/$\limsup$ of Borel-measurable density functions, hence Borel). Therefore the indicator $\mathbb{1}_{\{x : t_j \in (u^-(x), u^+(x))\}}$ is Borel for each fixed $j$, and the minimum index function $j(x)$ is Borel. Define \begin{align*} \nu_u: J_u &\to S^{n-1}, \\ x &\mapsto \nu_{E_{t_{j(x)}}}(x). \end{align*} Each $\nu_{E_{t_j}}: \partial^* E_{t_j} \to S^{n-1}$ is $\mathcal{H}^{n-1}$-measurable on $\partial^* E_{t_j}$ by [De Giorgi's Structure Theorem](/theorems/599). The composition with the Borel selector $j(x)$ produces a $\mathcal{H}^{n-1}$-measurable function on $J_u$. *Well-definedness $\mathcal{H}^{n-1}$-a.e.* By the Claim of Step 2, for any $t \in (u^-(x), u^+(x)) \cap G$, $\nu_{E_t}(x) = \nu_u(x)$. So the value of $\nu_u(x)$ defined via the selector $j(x)$ is the same as the value obtained from any other $t_k \in (u^-(x), u^+(x))$ — the choice of selector affects only the formal definition, not the value. Hence $\nu_u$ is well-defined $\mathcal{H}^{n-1}$-a.e. on $J_u$. *Conclusion.* The jump set $J_u$ is countably $(n-1)$-rectifiable, $\sigma$-finite with respect to $\mathcal{H}^{n-1}$, contained in a countable union of $C^1$ hypersurfaces up to an $\mathcal{H}^{n-1}$-null set, and the jump direction $\nu_u: J_u \to S^{n-1}$ is $\mathcal{H}^{n-1}$-measurable and well-defined $\mathcal{H}^{n-1}$-a.e. The proof is complete. [/guided] [/step]