[proofplan] The layer-cake representation $u = \int_0^\infty \mathbb{1}_{\{u > t\}} \, d\mathcal{L}^1(t) - \int_{-\infty}^0 (1 - \mathbb{1}_{\{u > t\}}) \, d\mathcal{L}^1(t)$ decomposes $u$ into its superlevel sets. Distributional differentiation commutes with the $d\mathcal{L}^1(t)$ integral by Fubini, giving $Du = \int D\mathbb{1}_{\{u > t\}} \, d\mathcal{L}^1(t)$. The triangle inequality for measures gives $|Du|(U) \leq \int |D\mathbb{1}_{\{u > t\}}|(U) \, d\mathcal{L}^1(t)$. The reverse inequality follows from the [Dual Characterisation](/theorems/591) and Fubini. [/proofplan] [step:Differentiate the layer-cake representation to obtain $Du = \int D\mathbb{1}_{\{u>t\}} \, d\mathcal{L}^1(t)$] For $u \in L^1(U)$, the layer-cake identity holds pointwise a.e.: \begin{align*} u(x) = \int_0^\infty \mathbb{1}_{\{u > t\}}(x) \, d\mathcal{L}^1(t) - \int_{-\infty}^0 (1 - \mathbb{1}_{\{u > t\}}(x)) \, d\mathcal{L}^1(t). \end{align*} Since $|Du|$ is finite, distributional differentiation commutes with the $d\mathcal{L}^1(t)$ integral by Fubini: \begin{align*} Du = \int_{-\infty}^\infty D\mathbb{1}_{\{u > t\}} \, d\mathcal{L}^1(t) \end{align*} as measures. [/step] [step:Prove the upper bound $|Du|(U) \leq \int |D\mathbb{1}_{\{u>t\}}|(U) \, d\mathcal{L}^1(t)$] Taking total variations in the measure identity and applying the triangle inequality for measures: \begin{align*} |Du|(U) \leq \int_{-\infty}^\infty |D\mathbb{1}_{\{u > t\}}|(U) \, d\mathcal{L}^1(t). \end{align*} [/step] [step:Prove the lower bound via the dual characterisation and Fubini] For any $\Phi \in C_c^\infty(U; \mathbb{R}^n)$ with $|\Phi| \leq 1$, the [Dual Characterisation](/theorems/591) applied to each $\mathbb{1}_{\{u > t\}}$ gives: \begin{align*} \int_U \mathbb{1}_{\{u > t\}}\,\operatorname{div}\Phi \, d\mathcal{L}^n \leq |D\mathbb{1}_{\{u > t\}}|(U). \end{align*} Integrating over $t$ and applying Fubini to interchange the $d\mathcal{L}^1(t)$ and $d\mathcal{L}^n(x)$ integrals: \begin{align*} \int_U u\,\operatorname{div}\Phi \, d\mathcal{L}^n = \int_{-\infty}^\infty \int_U \mathbb{1}_{\{u > t\}}\,\operatorname{div}\Phi \, d\mathcal{L}^n \, d\mathcal{L}^1(t) \leq \int_{-\infty}^\infty |D\mathbb{1}_{\{u > t\}}|(U) \, d\mathcal{L}^1(t). \end{align*} Taking the supremum over $\Phi$: \begin{align*} |Du|(U) \leq \int_{-\infty}^\infty |D\mathbb{1}_{\{u > t\}}|(U) \, d\mathcal{L}^1(t). \end{align*} Combined with the upper bound, equality holds. [/step]