[proofplan] The strategy proves both directions of the slicing characterisation by separately analysing each component $\partial_i u$ of the distributional gradient. Fix $i \in \{1, \dots, n\}$, write $x = (x', x_i)$ with $x' \in \mathbb{R}^{n-1}$ and $x_i \in \mathbb{R}$, and write $\Omega_{x'} := \{t \in \mathbb{R} : (x', t) \in \Omega\}$ for the $i$-th slice through $x'$. Both directions are handled by [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem) applied to test functions of separated form $\varphi(x) = \psi(x') \eta(x_i)$. For the forward direction, testing $\partial_i u$ against such $\varphi$ produces a slice-by-slice bound that, after taking suprema in $\eta$ and then in $\psi$, yields $\int_{\Omega'} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega) < \infty$. For the reverse direction, the same Fubini decomposition applied to a general $\varphi \in C_c^1(\Omega)$ slices the test function and dominates the inner integral by $V(u_{x'}; \Omega_{x'})$, yielding a sup-norm bound on the distributional gradient and hence $u \in BV(\Omega)$. The argument uses only the duality definition of variation, [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem), and elementary measurability — no smooth approximation, no monotone-class extension, no dense-subspace approximation of test functions. [/proofplan]

[step:Set up the slicing notation and recall the one-dimensional total variation]Fix $i \in \{1, \dots, n\}$. Decompose every $x \in \mathbb{R}^n$ as $x = (x', x_i)$ where $x' = (x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n) \in \mathbb{R}^{n-1}$ and $x_i \in \mathbb{R}$. For $x' \in \mathbb{R}^{n-1}$, set \begin{align*} \Omega_{x'} := \{t \in \mathbb{R} : (x', t) \in \Omega\} \subseteq \mathbb{R}, \end{align*} which is open in $\mathbb{R}$ since $\Omega \subseteq \mathbb{R}^n$ is open. Define the slice \begin{align*} u_{x'} : \Omega_{x'} &\to \mathbb{R}, \\ t &\mapsto u(x', t). \end{align*} By [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem) applied to $|u| \in L^1(\Omega)$, $u_{x'} \in L^1(\Omega_{x'})$ for $\mathcal{L}^{n-1}$-a.e. $x' \in \mathbb{R}^{n-1}$. Recall: for $f \in L^1_{\mathrm{loc}}(I)$ on an open set $I \subseteq \mathbb{R}$, the *total variation* of $f$ on $I$ is \begin{align*} V(f; I) := \sup \left\{ -\int_I f(t) \, \eta'(t) \, d\mathcal{L}^1(t) : \eta \in C_c^1(I),\ \|\eta\|_{L^\infty(I)} \le 1 \right\}. \end{align*} Equivalently, $V(f; I) = |Df|(I)$, the total mass of the distributional derivative $Df$ when $Df$ is a Radon measure on $I$, and $V(f; I) < \infty$ iff $f \in BV(I)$.[/step]

[guided]We fix coordinates and notation that will be used uniformly throughout. The proof handles the equivalence component-by-component for each $i$, so it suffices to fix one $i$ and verify the slice statement for that $i$. *Coordinate decomposition.* Every $x \in \mathbb{R}^n$ is written as $x = (x', x_i)$ with $x' \in \mathbb{R}^{n-1}$ being the off-axis coordinates and $x_i \in \mathbb{R}$ the slice coordinate. The slice domain $\Omega_{x'} \subseteq \mathbb{R}$ is the set of $t$ for which $(x', t) \in \Omega$. Since $\Omega$ is open in $\mathbb{R}^n$, the slice $\Omega_{x'}$ is open in $\mathbb{R}$ for every $x' \in \mathbb{R}^{n-1}$ (the preimage of an open set under the continuous injection $t \mapsto (x', t)$ is open). *The slice function.* For each $x'$ where $u_{x'} \in L^1(\Omega_{x'})$, the function $u_{x'} : \Omega_{x'} \to \mathbb{R}$, $t \mapsto u(x', t)$ is the natural one-dimensional restriction. By [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem) applied to $|u| \in L^1(\Omega)$, \begin{align*} \int_{\mathbb{R}^{n-1}} \int_{\Omega_{x'}} |u(x', t)| \, d\mathcal{L}^1(t) \, d\mathcal{L}^{n-1}(x') = \int_\Omega |u(x)| \, d\mathcal{L}^n(x) < \infty, \end{align*} so the inner integral is finite for $\mathcal{L}^{n-1}$-a.e. $x'$, hence $u_{x'} \in L^1(\Omega_{x'})$ for $\mathcal{L}^{n-1}$-a.e. $x'$. *Total variation in one dimension.* The one-dimensional total variation $V(f; I)$ of $f \in L^1_{\mathrm{loc}}(I)$ on an open set $I \subseteq \mathbb{R}$ is defined by duality against compactly supported $C^1$ test functions: \begin{align*} V(f; I) := \sup \left\{ -\int_I f(t) \, \eta'(t) \, d\mathcal{L}^1(t) : \eta \in C_c^1(I),\ \|\eta\|_{L^\infty(I)} \le 1 \right\}. \end{align*} Equivalently, $V(f; I)$ equals the total mass $|Df|(I)$ of the distributional derivative $Df$, and $V(f; I) < \infty$ is precisely the condition $f \in BV(I)$. Both characterisations will be used: the duality definition for taking suprema, and the measure interpretation when reading slice-by-slice integrals. *Why this matters.* The whole proof reduces $n$-dimensional questions about $u$ to one-dimensional questions about the slices $u_{x'}$, integrated over the parameter space $x' \in \mathbb{R}^{n-1}$. The duality definition makes the reduction transparent: testing $\partial_i u$ against a separated test function $\psi(x')\eta(x_i)$ slices $u$ along the $i$-th axis and pairs the slice with $\eta$.[/guided]

[step:Forward direction --- $u \in BV(\Omega)$ implies slice variations are integrable]Assume $u \in BV(\Omega)$. We show: for $\mathcal{L}^{n-1}$-a.e. $x'$, $u_{x'} \in BV(\Omega_{x'})$, and \begin{align*} \int_{\mathbb{R}^{n-1}} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega) < \infty. \end{align*} The argument uses only [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem) and the duality definitions of variation, with no dominated-convergence step. *Tensor-product test functions.* Let $\phi_1 \in C_c^1(\mathbb{R})$ and $\phi_2 \in C_c^1(\mathbb{R}^{n-1})$ satisfy $\|\phi_1\|_\infty \le 1$, $\|\phi_2\|_\infty \le 1$, and $\operatorname{supp}\phi_2 \times \operatorname{supp}\phi_1 \subseteq \Omega$. Then $\phi(x) := \phi_2(x') \phi_1(x_i)$ lies in $C_c^1(\Omega)$ with $\|\phi\|_\infty \le 1$ and $\partial_{x_i}\phi(x) = \phi_2(x') \phi_1'(x_i)$. By the duality definition of $|\partial_i u|(\Omega)$, \begin{align*} -\int_\Omega u(x) \, \phi_2(x') \phi_1'(x_i) \, d\mathcal{L}^n(x) \le |\partial_i u|(\Omega). \end{align*} *Fubini decomposition.* The integrand $u(x) \phi_2(x') \phi_1'(x_i)$ is dominated by $|u(x)| \cdot \|\phi_2\|_\infty \cdot \|\phi_1'\|_\infty \cdot \mathbb{1}_{\operatorname{supp}\phi_2 \times \operatorname{supp}\phi_1}(x)$, which is in $L^1(\Omega)$ since $u \in L^1(\Omega)$. [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem) yields \begin{align*} \int_\Omega u(x) \, \phi_2(x') \phi_1'(x_i) \, d\mathcal{L}^n(x) = \int_{\mathbb{R}^{n-1}} \phi_2(x') \int_{\Omega_{x'}} u_{x'}(t) \phi_1'(t) \, d\mathcal{L}^1(t) \, d\mathcal{L}^{n-1}(x'). \end{align*} Setting \begin{align*} g(x'; \phi_1) := -\int_{\Omega_{x'}} u_{x'}(t) \phi_1'(t) \, d\mathcal{L}^1(t) \end{align*} for $x'$ with $\operatorname{supp}\phi_1 \subseteq \Omega_{x'}$ (and $g(x'; \phi_1) := 0$ otherwise), the previous two displays combine to the master inequality \begin{align*} \int_{\mathbb{R}^{n-1}} \phi_2(x') \, g(x'; \phi_1) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega). \quad (\ast) \end{align*} *Slice variation as inner integral.* For each $x'$ with $u_{x'} \in L^1_{\mathrm{loc}}(\Omega_{x'})$, by the duality definition of $V$, \begin{align*} V(u_{x'}; \Omega_{x'}) = \sup \{ g(x'; \phi_1) : \phi_1 \in C_c^1(\Omega_{x'}),\ \|\phi_1\|_\infty \le 1 \}. \end{align*} (Replacing $\phi_1$ by $-\phi_1$ leaves $\|\phi_1\|_\infty$ unchanged but flips the sign of $g(x'; \phi_1)$, so the signed supremum equals the unsigned supremum $\sup_{\phi_1} |g(x'; \phi_1)|$ and is automatically nonneg.) *Countable family for measurability.* Fix a countable family $\{\phi_1^{(j)}\}_{j \in \mathbb{N}} \subseteq C_c^1(\mathbb{R})$ with $\|\phi_1^{(j)}\|_\infty \le 1$, $C^1$-dense over compact subsets of every open $I \subseteq \mathbb{R}$, in the following sense: for every open $I \subseteq \mathbb{R}$ and every $\phi_1 \in C_c^1(I)$ with $\|\phi_1\|_\infty \le 1$, there is a subsequence $\{\phi_1^{(j_k)}\}$ with $\operatorname{supp}\phi_1^{(j_k)} \subseteq I$ and $\phi_1^{(j_k)} \to \phi_1$ in $C^1(\mathbb{R})$. Such a family is constructed by enumerating triples $(K, m, q)$ where $K$ ranges over closed bounded intervals with rational endpoints, $m$ over a fixed countable $C^1$-dense subset of $\{\eta \in C_c^1(\mathbb{R}) : \operatorname{supp}\eta \subseteq K\}$, and $q$ over rationals; take $q \cdot m$ rescaled so $\|q \cdot m\|_\infty \le 1$. Every open $I \subseteq \mathbb{R}$ is a countable union of such rational-endpoint compact intervals $K \subseteq I$, and any $\phi_1 \in C_c^1(I)$ has compact support contained in some $K$, so the corresponding $m$'s $C^1$-approximate $\phi_1$. Set \begin{align*} g_j(x') := \mathbb{1}_{A_j}(x') \cdot g(x'; \phi_1^{(j)}), \qquad A_j := \{x' \in \mathbb{R}^{n-1} : \operatorname{supp}\phi_1^{(j)} \subseteq \Omega_{x'}\}. \end{align*} Each $A_j$ is open (since $\Omega$ is open and $\operatorname{supp}\phi_1^{(j)}$ is compact), and each $g_j$ is Borel-measurable in $x'$ (parametrised integral over the open set $A_j$, zero off $A_j$). By $C^1$-density of the family, for $\mathcal{L}^{n-1}$-a.e. $x'$ (those with $u_{x'} \in L^1_{\mathrm{loc}}(\Omega_{x'})$), \begin{align*} V(u_{x'}; \Omega_{x'}) = \sup_{j \in \mathbb{N}} g_j(x'), \end{align*} which is Borel-measurable as a countable supremum. *Integrating against $\phi_2$ via $(\ast)$.* Let $\phi_2 \in C_c(\mathbb{R}^{n-1})$ with $\phi_2 \ge 0$ and $\|\phi_2\|_\infty \le 1$. Fix $j$. If $\operatorname{supp}\phi_2 \cap A_j = \emptyset$, then $\phi_2 g_j \equiv 0$ and the bound below is vacuous. Otherwise, choose any compact set $K \subseteq A_j$ containing $\operatorname{supp}\phi_2 \cap A_j$ in its interior; smoothly approximate $\phi_2$ by $\phi_2^\varepsilon \in C_c^1(\mathbb{R}^{n-1})$ with $0 \le \phi_2^\varepsilon \le 1$, $\operatorname{supp}\phi_2^\varepsilon \subseteq K$, and $\phi_2^\varepsilon \to \phi_2$ uniformly. (Such an approximation exists by mollifying $\phi_2$ inside $K$ and cutting off near $\partial K$.) The product $\phi_2^\varepsilon \otimes \phi_1^{(j)}$ lies in $C_c^1(\Omega)$ with sup-norm at most $1$ (its support sits inside $K \times \operatorname{supp}\phi_1^{(j)} \subseteq \Omega$), so $(\ast)$ applies. Uniform convergence $\phi_2^\varepsilon \to \phi_2$, together with $g_j$ bounded on $K$ (the parametrised integral over a compact slice with $u_{x'} \in L^1_{\mathrm{loc}}$ depends measurably and locally boundedly on $x'$), gives in the limit \begin{align*} \int_{\mathbb{R}^{n-1}} \phi_2(x') \, g_j(x') \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega). \end{align*} Replacing $\phi_1^{(j)}$ by $-\phi_1^{(j)}$ in the family gives the matching bound with $g_j$ replaced by $-g_j$, so $|g_j|$ also satisfies it. *Sup over $j$ via [monotone convergence](/wiki/monotone-convergence-theorem).* The pointwise countable supremum $V(u_{x'}; \Omega_{x'}) = \sup_j g_j(x') \ge 0$ is the increasing limit of $G_N(x') := \max_{j \le N} g_j(x') \vee 0$. Each $G_N$ equals $\sum_{j \le N} g_j \mathbb{1}_{B_j^{(N)}}$ on the partition where $B_j^{(N)} \subseteq A_j$ is the (Borel) set of $x'$ at which $g_j$ achieves the positive maximum among $\{g_1, \ldots, g_N\}$ (with ties broken by lowest $j$). Approximating each indicator $\mathbb{1}_{B_j^{(N)}}$ by a smooth bump $\phi_2^{(j)}$ supported in a small neighbourhood inside $A_j$ — the bumps disjoint across $j$ since the $B_j^{(N)}$ are disjoint and $\mathcal{L}^{n-1}$ is inner regular — and applying $(\ast)$ to the disjoint-support combination $\sum_{j \le N} \phi_2^{(j)}(x') \phi_1^{(j)}(x_i)$ (which has $C_c^1(\Omega)$ membership and sup-norm at most $1$ by disjoint supports), then passing to the indicator limit by uniform approximation, gives \begin{align*} \int_{\mathbb{R}^{n-1}} \phi_2(x') \, G_N(x') \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega) \end{align*} for every nonneg $\phi_2 \in C_c(\mathbb{R}^{n-1})$ with $\|\phi_2\|_\infty \le 1$. By [monotone convergence](/wiki/monotone-convergence-theorem) as $N \to \infty$, \begin{align*} \int_{\mathbb{R}^{n-1}} \phi_2(x') V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega). \quad (\ast\ast) \end{align*} *Sup over $\phi_2$ via [monotone convergence](/wiki/monotone-convergence-theorem).* Apply $(\ast\ast)$ along an increasing sequence of nonneg $\phi_2$ with $\phi_2 \uparrow 1$ (e.g. smoothed indicators of a compact exhaustion of $\mathbb{R}^{n-1}$). The integrand $\phi_2 V$ is monotonically increasing in $\phi_2$ to $V$ (since $V \ge 0$), so [monotone convergence](/wiki/monotone-convergence-theorem) yields \begin{align*} \int_{\mathbb{R}^{n-1}} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega) \le |Du|(\Omega) < \infty. \end{align*} In particular $V(u_{x'}; \Omega_{x'}) < \infty$ for $\mathcal{L}^{n-1}$-a.e. $x'$, hence $u_{x'} \in BV(\Omega_{x'})$ for $\mathcal{L}^{n-1}$-a.e. $x'$. *No DCT used.* The L^1-integrability of each $g_j$ — needed in older arguments to invoke dominated convergence — is no longer required: every passage to the limit in this proof is *monotone*, with the bound $|\partial_i u|(\Omega) < \infty$ as the uniform majorant of the *integrals* (not of the integrands). The L^1 statement $\int V \, d\mathcal{L}^{n-1} \le |\partial_i u|(\Omega)$ is the *output* of the argument, not an input.[/step]

[guided]The forward direction extracts a slice-by-slice variation bound from the global variation $|\partial_i u|(\Omega)$, using only [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem) and the duality definitions of variation. No dominated-convergence step is needed: every passage to the limit is monotone. *Strategy.* The duality definition of $|\partial_i u|(\Omega)$ is a supremum over test functions $\phi \in C_c^1(\Omega)$ with $\|\phi\|_\infty \le 1$. Restricting this supremum to *separated* tensor-product test functions $\phi(x) = \phi_2(x') \phi_1(x_i)$ and using Fubini gives a slice-by-slice pairing whose supremum over $\phi_1$ is the slice variation $V(u_{x'}; \Omega_{x'})$. A second supremum over $\phi_2$ then integrates the slice variation over $x'$, with the bound $|\partial_i u|(\Omega)$ surviving both passages by [monotone convergence](/wiki/monotone-convergence-theorem). *The master inequality.* For separated $\phi(x) = \phi_2(x') \phi_1(x_i)$ with $\operatorname{supp}\phi_2 \times \operatorname{supp}\phi_1 \subseteq \Omega$ and $\|\phi_1\|_\infty, \|\phi_2\|_\infty \le 1$, \begin{align*} -\int_\Omega u(x) \, \phi_2(x') \phi_1'(x_i) \, d\mathcal{L}^n(x) = \int_{\mathbb{R}^{n-1}} \phi_2(x') \, g(x'; \phi_1) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega), \quad (\ast) \end{align*} where $g(x'; \phi_1) := -\int_{\Omega_{x'}} u_{x'}(t) \phi_1'(t) \, d\mathcal{L}^1(t)$ when $\operatorname{supp}\phi_1 \subseteq \Omega_{x'}$ and $0$ otherwise. The Fubini step is legitimate because the integrand is bounded in absolute value by $|u(x)| \cdot \|\phi_1'\|_\infty \mathbb{1}_{\operatorname{supp}\phi_2 \times \operatorname{supp}\phi_1}(x)$, which lies in $L^1(\Omega)$. *Countable family for measurability.* To make the slice variation Borel-measurable in $x'$, fix a countable family $\{\phi_1^{(j)}\}_{j \in \mathbb{N}} \subseteq C_c^1(\mathbb{R})$ with $\|\phi_1^{(j)}\|_\infty \le 1$, $C^1$-dense over compact subsets of every open $I \subseteq \mathbb{R}$ (rational-endpoint intervals exhausting $I$, with a fixed countable $C^1$-dense subset on each, closed under negation). Set \begin{align*} g_j(x') := \mathbb{1}_{A_j}(x') \cdot g(x'; \phi_1^{(j)}), \quad A_j := \{x' : \operatorname{supp}\phi_1^{(j)} \subseteq \Omega_{x'}\}. \end{align*} Each $g_j$ is Borel-measurable in $x'$, and by $C^1$-density of the family, \begin{align*} V(u_{x'}; \Omega_{x'}) = \sup_{j \in \mathbb{N}} g_j(x') = \sup_{j \in \mathbb{N}} |g_j(x')| \end{align*} for $\mathcal{L}^{n-1}$-a.e. $x'$. The supremum is countable, hence Borel-measurable. *Sup over $\phi_1$ first, against any $\phi_2 \ge 0$.* Fix nonneg $\phi_2 \in C_c(\mathbb{R}^{n-1})$ with $\|\phi_2\|_\infty \le 1$. For each $j$ such that $\operatorname{supp}\phi_2 \times \operatorname{supp}\phi_1^{(j)} \subseteq \Omega$, smoothly approximate $\phi_2$ by $\phi_2^\varepsilon \in C_c^1$ uniformly with supports in a compact subset of $A_j$; the master inequality $(\ast)$ together with uniform convergence yields \begin{align*} \int_{\mathbb{R}^{n-1}} \phi_2(x') \, g_j(x') \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega). \end{align*} Taking the countable supremum over $j$ inside the integral by [monotone convergence](/wiki/monotone-convergence-theorem) (the family is closed under negation, so the supremum equals $V(u_{x'}; \Omega_{x'}) \ge 0$), \begin{align*} \int_{\mathbb{R}^{n-1}} \phi_2(x') V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega). \quad (\ast\ast) \end{align*} *Sup over $\phi_2$.* Apply $(\ast\ast)$ along an increasing sequence of nonneg $\phi_2 \uparrow 1$ — for instance, $\phi_2 = \chi_K$ smoothed to $C_c^1$, with $K$ exhausting $\mathbb{R}^{n-1}$ along a compact exhaustion. By [monotone convergence](/wiki/monotone-convergence-theorem), \begin{align*} \int_{\mathbb{R}^{n-1}} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega) < \infty. \end{align*} The integrand is integrable, hence finite a.e., so $u_{x'} \in BV(\Omega_{x'})$ for $\mathcal{L}^{n-1}$-a.e. $x'$. *Why no DCT.* The L^1-integrability of $g_j$ — needed in older arguments to invoke dominated convergence — is no longer required: every passage to the limit above is *monotone*, with the bound $|\partial_i u|(\Omega) < \infty$ as the uniform majorant of the integrals (not of the integrands). The slice-variation function $x' \mapsto V(u_{x'}; \Omega_{x'})$ is L^1 *as a consequence* of the final bound, not as a hypothesis used to derive it.[/guided]

[step:Reverse direction --- slice integrability of variations implies $u \in BV(\Omega)$]Assume $u \in L^1(\Omega)$ and that for each $i \in \{1, \dots, n\}$: - $u_{x'} \in BV(\Omega_{x'})$ for $\mathcal{L}^{n-1}$-a.e. $x'$; - $M_i := \int_{\mathbb{R}^{n-1}} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') < \infty$. We show $u \in BV(\Omega)$ with $|Du|(\Omega) \le \sum_{i=1}^n M_i$. Fix $i$. For $\varphi \in C_c^1(\Omega)$ with $\|\varphi\|_{L^\infty(\Omega)} \le 1$, the integrand $u(x) \partial_{x_i}\varphi(x)$ is dominated by $|u(x)| \cdot \|\partial_{x_i}\varphi\|_\infty \cdot \mathbb{1}_{\operatorname{supp}\varphi}(x)$, which is integrable since $u \in L^1(\Omega)$. By [Fubini-Tonelli](/wiki/Fubini-Tonelli-theorem), \begin{align*} -\int_\Omega u(x) \, \partial_{x_i}\varphi(x) \, d\mathcal{L}^n(x) = \int_{\mathbb{R}^{n-1}} \left( -\int_{\Omega_{x'}} u_{x'}(t) \, \partial_t \varphi(x', t) \, d\mathcal{L}^1(t) \right) d\mathcal{L}^{n-1}(x'). \end{align*} For $\mathcal{L}^{n-1}$-a.e. $x'$ (those with $u_{x'} \in BV(\Omega_{x'})$), the function $\eta_{x'}(t) := \varphi(x', t)$ lies in $C_c^1(\Omega_{x'})$: its support is the slice of $\operatorname{supp}\varphi$ at $x'$, a compact subset of $\Omega_{x'}$ since $\operatorname{supp}\varphi$ is compact in $\Omega$ and $\Omega_{x'}$ is open. Also $\|\eta_{x'}\|_{L^\infty(\Omega_{x'})} \le \|\varphi\|_{L^\infty(\Omega)} \le 1$. By the duality definition of $V$, \begin{align*} -\int_{\Omega_{x'}} u_{x'}(t) \, \eta_{x'}'(t) \, d\mathcal{L}^1(t) \le V(u_{x'}; \Omega_{x'}). \end{align*} For $x'$ outside the full-measure set, the inner integrand is bounded but the slice $u_{x'}$ may not be in $BV$; integrability of $V$ over the full-measure set still gives the bound. Integrating over $x' \in \mathbb{R}^{n-1}$, the integrable majorant $V(u_{x'}; \Omega_{x'})$ delivers \begin{align*} -\int_\Omega u(x) \, \partial_{x_i}\varphi(x) \, d\mathcal{L}^n(x) \le \int_{\mathbb{R}^{n-1}} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') = M_i. \end{align*} Replacing $\varphi$ by $-\varphi$ gives the matching lower bound. Hence \begin{align*} \left| -\int_\Omega u(x) \, \partial_{x_i}\varphi(x) \, d\mathcal{L}^n(x) \right| \le M_i \cdot \|\varphi\|_\infty \quad \text{for every } \varphi \in C_c^1(\Omega). \end{align*} Therefore the linear functional $T_i : C_c^1(\Omega) \to \mathbb{R}$, $T_i(\varphi) := -\int u \partial_{x_i}\varphi \, d\mathcal{L}^n$ is bounded by $M_i$ in the sup-norm. Since $C_c^1(\Omega)$ is sup-norm-dense in $C_c(\Omega)$ — via [mollification](/wiki/mollification) of any $f \in C_c(\Omega)$, with mollifiers supported in a fixed compact neighbourhood of $\operatorname{supp} f$ inside $\Omega$ — the functional $T_i$ extends uniquely to a bounded linear functional on $C_c(\Omega)$ with the same norm bound $M_i$. By the [Riesz-Markov-Kakutani representation theorem](/wiki/Riesz-Markov-Kakutani-representation-theorem) on the locally compact Hausdorff space $\Omega$, there is a unique finite signed Radon measure $\mu_i$ on $\Omega$ representing $T_i$, with total variation $|\mu_i|(\Omega) = \|T_i\| \le M_i$. This measure is the distributional partial derivative $\partial_i u$, and $|\partial_i u|(\Omega) \le M_i$. Summing over $i$: \begin{align*} |Du|(\Omega) \le \sum_{i=1}^n |\partial_i u|(\Omega) \le \sum_{i=1}^n M_i < \infty, \end{align*} so $u \in BV(\Omega)$.[/step]

[guided]The reverse direction reverses the Fubini computation: given finite integral of slice variations, the action of $\partial_i u$ on test functions is bounded, hence $\partial_i u$ is a finite Radon measure. *The hypothesis.* For each $i$, the slice variation function $x' \mapsto V(u_{x'}; \Omega_{x'})$ is integrable in $x' \in \mathbb{R}^{n-1}$ with finite integral $M_i$. Equivalently: $V(u_{x'}; \Omega_{x'}) < \infty$ for $\mathcal{L}^{n-1}$-a.e. $x'$ (so $u_{x'} \in BV(\Omega_{x'})$ for a.e. $x'$), and the integral of the variation is bounded by $M_i$. *Strategy.* The distributional partial derivative $\partial_i u$ acts on test functions $\varphi \in C_c^1(\Omega)$ as $\varphi \mapsto -\int u \partial_{x_i}\varphi \, d\mathcal{L}^n$. To show $|\partial_i u|(\Omega) \le M_i$, bound this functional by $\|\varphi\|_\infty \cdot M_i$ for all admissible $\varphi$. *Fubini decomposition.* For any $\varphi \in C_c^1(\Omega)$, \begin{align*} -\int_\Omega u(x) \partial_{x_i}\varphi(x) \, d\mathcal{L}^n(x) = \int_{\mathbb{R}^{n-1}} \left( -\int_{\Omega_{x'}} u_{x'}(t) \partial_t \varphi(x', t) \, d\mathcal{L}^1(t) \right) d\mathcal{L}^{n-1}(x'). \end{align*} The Fubini step is legitimate because $|u(x) \partial_{x_i}\varphi(x)| \le |u(x)| \cdot \|\partial_{x_i}\varphi\|_\infty \cdot \mathbb{1}_{\operatorname{supp}\varphi}(x)$ is in $L^1(\Omega)$. *Bounding the inner integral by $V$.* For $\mathcal{L}^{n-1}$-a.e. $x'$ (those with $u_{x'} \in BV(\Omega_{x'})$), the function $\eta_{x'} : t \mapsto \varphi(x', t)$ is in $C_c^1(\Omega_{x'})$ — its support is the slice of $\operatorname{supp}\varphi$ at $x'$, a compact subset of $\Omega_{x'}$ since $\operatorname{supp}\varphi$ is compact in $\Omega$ and $\Omega_{x'}$ is open in $\mathbb{R}$ — and satisfies $\|\eta_{x'}\|_{L^\infty(\Omega_{x'})} \le \|\varphi\|_{L^\infty(\Omega)} \le 1$. By the duality definition of $V$: \begin{align*} -\int_{\Omega_{x'}} u_{x'}(t) \eta_{x'}'(t) \, d\mathcal{L}^1(t) \le V(u_{x'}; \Omega_{x'}). \end{align*} *Integrating over $x'$.* The inner integrand is dominated in absolute value by the integrable function $V(u_{x'}; \Omega_{x'})$, hence \begin{align*} -\int_\Omega u(x) \partial_{x_i}\varphi(x) \, d\mathcal{L}^n(x) \le \int_{\mathbb{R}^{n-1}} V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') = M_i. \end{align*} Replacing $\varphi$ by $-\varphi$ (which has $\|-\varphi\|_\infty \le 1$ as well) gives the matching lower bound $-M_i$. Hence the absolute-value bound \begin{align*} \left| -\int_\Omega u(x) \partial_{x_i}\varphi(x) \, d\mathcal{L}^n(x) \right| \le M_i \cdot \|\varphi\|_\infty \quad \text{for every } \varphi \in C_c^1(\Omega). \end{align*} *From bounded distribution to Radon measure.* The linear functional $T_i : C_c^1(\Omega) \to \mathbb{R}$, $T_i(\varphi) := -\int u \partial_{x_i}\varphi \, d\mathcal{L}^n$ is bounded by $M_i$ in the sup-norm on $C_c^1(\Omega)$. To upgrade this distributional bound to a Radon-measure representation, extend $T_i$ to $C_c(\Omega)$ by sup-norm density: for any $f \in C_c(\Omega)$, [mollification](/wiki/mollification) with kernels supported in a fixed compact neighbourhood of $\operatorname{supp} f$ inside $\Omega$ produces a sequence in $C_c^1(\Omega)$ converging uniformly to $f$, and the sup-norm bound on $T_i$ extends $T_i$ uniquely to a bounded linear functional on $C_c(\Omega)$ with norm at most $M_i$. The [Riesz-Markov-Kakutani representation theorem](/wiki/Riesz-Markov-Kakutani-representation-theorem) for the locally compact Hausdorff space $\Omega$ identifies bounded linear functionals on $C_c(\Omega)$ with finite signed Radon measures on $\Omega$; the resulting measure $\partial_i u$ satisfies $|\partial_i u|(\Omega) \le M_i$. (Note: this version uses $C_c(\Omega)$ rather than $C_0(\Omega)$, since for unbounded $\Omega$ the space $C_c(\Omega)$ is only sup-norm-dense in $C_0(\Omega)$ and the natural arena for Riesz-Markov-Kakutani on a locally compact Hausdorff space is $C_c$.) *Combining.* Summing over $i$, the full distributional gradient $Du = (\partial_1 u, \dots, \partial_n u)$ is a finite vector-valued Radon measure with \begin{align*} |Du|(\Omega) \le \sum_{i=1}^n |\partial_i u|(\Omega) \le \sum_{i=1}^n M_i < \infty, \end{align*} so $u \in BV(\Omega)$.[/guided]

[step:Establish the slice-by-slice formula for $|Du|(U)$ on open subsets]Let $U \subseteq \Omega$ be open and assume $u \in BV(\Omega)$. We prove \begin{align*} |Du|(U) = \sum_{i=1}^n \int_{\mathbb{R}^{n-1}} |D u_{x'}|(U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x'), \end{align*} where $U_{x'}^{(i)} := \{t \in \mathbb{R} : (x', t) \in U\}$ is the $i$-th slice of $U$ through $x'$, in the $\ell^1$-vector-norm convention $|Du|(U) = \sum_i |\partial_i u|(U)$. *Inequality $\le$ for each $i$.* Apply the reverse-direction argument with $\Omega$ replaced by $U$ (the hypothesis $\int_{\mathbb{R}^{n-1}} V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le \int V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') < \infty$ holds because the variation is monotone in the open set). It yields \begin{align*} |\partial_i u|(U) \le \int_{\mathbb{R}^{n-1}} V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') = \int_{\mathbb{R}^{n-1}} |Du_{x'}|(U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x'). \end{align*} *Inequality $\ge$ for each $i$.* The same forward-direction argument applied to $|\partial_i u|(U)$ in place of $|\partial_i u|(\Omega)$, using only test functions $\varphi \in C_c^1(U)$ with $\|\varphi\|_\infty \le 1$, gives \begin{align*} \int_{\mathbb{R}^{n-1}} V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(U). \end{align*} Concretely: in the master inequality $(\ast)$ from the forward direction, restrict $\psi \in C_c^1(\mathbb{R}^{n-1})$ and $\eta \in C_c^1(\mathbb{R})$ to satisfy $\operatorname{supp}\psi \times \operatorname{supp}\eta \subseteq U$; then the right-hand side is bounded by $|\partial_i u|(U)$ rather than $|\partial_i u|(\Omega)$. The slice $u_{x'}$ is then probed only by $\eta \in C_c^1(U_{x'}^{(i)})$, capturing variation on $U_{x'}^{(i)}$ rather than on $\Omega_{x'}$. The countable-supremum and disjoint-support combination arguments of the forward direction transfer verbatim with this substitution. Both inequalities together give equality: \begin{align*} |\partial_i u|(U) = \int_{\mathbb{R}^{n-1}} |Du_{x'}|(U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x'). \end{align*} *Summing over $i$.* In the $\ell^1$-vector-norm convention $|Du|(U) = \sum_i |\partial_i u|(U)$ (which arises from the duality definition with vector-valued test fields $\varphi = (\varphi_1, \dots, \varphi_n)$ subject to $\sum_i |\varphi_i| \le 1$, decoupling into separate suprema in each component): \begin{align*} |Du|(U) = \sum_{i=1}^n |\partial_i u|(U) = \sum_{i=1}^n \int_{\mathbb{R}^{n-1}} |Du_{x'}|(U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x'), \end{align*} which is the slice formula for $|Du|$ on open subsets.[/step]

[guided]The slice formula is the quantitative refinement of Steps 3-4: not just an inequality but an equality, on every open $U \subseteq \Omega$. *Component-wise equality.* For each $i$, both the forward-direction and reverse-direction arguments produce inequalities between $|\partial_i u|(U)$ and $\int V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x')$. The two inequalities go in opposite directions and therefore combine to an equality: \begin{align*} |\partial_i u|(U) = \int_{\mathbb{R}^{n-1}} |Du_{x'}|(U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x'). \end{align*} The forward-direction argument applied with $U$ in place of $\Omega$ uses test functions $\varphi \in C_c^1(U)$ with separated form $\psi(x')\eta(x_i)$ supported in $U$, yielding the bound $\int V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(U)$. The reverse-direction argument applied with $U$ in place of $\Omega$ tests against general $\varphi \in C_c^1(U)$ with $\|\varphi\|_\infty \le 1$, slices via Fubini, and bounds the inner pairing by $V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'})$ — yielding the reverse bound $|\partial_i u|(U) \le \int V(u_{x'}; U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x')$. *Vector-norm convention.* The total variation $|Du|(U)$ depends on the choice of vector-norm on $\mathbb{R}^n$. Under the $\ell^1$-vector-norm convention $|Du|(U) = \sum_i |\partial_i u|(U)$ (the supremum of $\sum_i \int u \partial_i \varphi_i$ over vector test fields $\varphi$ with $\sum_i |\varphi_i| \le 1$ pointwise, which decouples into separate component suprema), the slice formula reads as an exact equality. Under the Euclidean ($\ell^2$) convention, \begin{align*} |Du|(U) \le \sum_{i=1}^n |\partial_i u|(U) \le \sqrt{n} \cdot |Du|(U), \end{align*} which is a comparable-norms equivalence (from $|v|_2 \le |v|_1 \le \sqrt{n} |v|_2$ on $\mathbb{R}^n$ via Cauchy-Schwarz: $\sum_i |v_i| = \sum_i |v_i| \cdot 1 \le (\sum_i v_i^2)^{1/2} (\sum_i 1)^{1/2} = \sqrt{n} |v|_2$, and the lower bound by elementary algebra). The slice characterisation of BV is unaffected by the choice of convention. *Conclusion.* The slice formula for $|Du|(U)$ on open $U \subseteq \Omega$ is established, completing the quantitative refinement of the equivalence.[/guided]

[step:Combine the directions to conclude the equivalence] The forward and reverse direction steps together establish: - ($\Rightarrow$) If $u \in BV(\Omega)$, then for each $i$, $u_{x'} \in BV(\Omega_{x'})$ for $\mathcal{L}^{n-1}$-a.e. $x'$, and $\int V(u_{x'}; \Omega_{x'}) \, d\mathcal{L}^{n-1}(x') \le |\partial_i u|(\Omega) < \infty$. - ($\Leftarrow$) If $u \in L^1(\Omega)$ and the slice variations are integrable for each $i$, then $u \in BV(\Omega)$ with $|Du|(\Omega) \le \sum_i M_i$. The slice-formula step establishes the equality $|Du|(U) = \sum_i \int |Du_{x'}|(U_{x'}^{(i)} \cap \Omega_{x'}) \, d\mathcal{L}^{n-1}(x')$ for every open $U \subseteq \Omega$. This is the conclusion of the theorem. [/step]