[proofplan] The strategy is a two-step approximation, both legs of which are standard. First, we mollify $u$ to obtain a smooth $W^{1,1}$ function $w := u_{\delta_0}$ that approximates $u$ in $L^1$ and whose gradient $L^1$-norm approximates the total variation $|Du|(\mathbb{R}^n)$; this is the strict-convergence theorem for the mollification of a $BV$ function. Second, we apply the $C^1$-approximation theorem for $W^{1,1}$ functions (theorem 3092) to $w$ with tolerance $\varepsilon/3$, producing $g \in C^1(\mathbb{R}^n)$ with $\|w - g\|_{W^{1,1}} < \varepsilon/3$. The triangle inequality in $L^1$ and the reverse triangle inequality $\bigl| \|a\|_{L^1} - \|b\|_{L^1} \bigr| \le \|a - b\|_{L^1}$ then assemble the two legs into the conclusion. No singular-part cancellation is required: the conclusion is purely about the $L^1$-distance of the functions and the closeness of the scalars $|Du|(\mathbb{R}^n)$ and $\|\nabla g\|_{L^1}$, which is the natural notion of strict convergence in $BV$. [/proofplan] [step:Mollify $u$ in $C^\infty \cap W^{1,1}$] Let $u \in BV(\mathbb{R}^n)$ and $\varepsilon > 0$ be as in the statement. Recall that $u \in BV(\mathbb{R}^n)$ means $u \in L^1(\mathbb{R}^n)$ and the distributional gradient $Du$ is a finite $\mathbb{R}^n$-valued Radon measure on $\mathbb{R}^n$ with total variation $|Du|(\mathbb{R}^n) < \infty$. Fix a [standard mollifier](/page/Standard%20Mollifier) $\eta \in C_c^\infty(\mathbb{R}^n)$ with $\eta \ge 0$, $\operatorname{supp} \eta \subseteq B(0, 1)$, and $\int_{\mathbb{R}^n} \eta \, d\mathcal{L}^n = 1$. For $\delta > 0$ define the rescaled mollifier $\eta_\delta(x) := \delta^{-n} \eta(x/\delta)$, so $\operatorname{supp} \eta_\delta \subseteq B(0, \delta)$ and $\int_{\mathbb{R}^n} \eta_\delta \, d\mathcal{L}^n = 1$. Define the mollification \begin{align*} u_\delta: \mathbb{R}^n &\to \mathbb{R} \\ x &\mapsto (\eta_\delta * u)(x) = \int_{\mathbb{R}^n} \eta_\delta(x - y) \, u(y) \, d\mathcal{L}^n(y). \end{align*} Standard properties of mollification (differentiation under the integral) give $u_\delta \in C^\infty(\mathbb{R}^n)$ with classical gradient \begin{align*} \nabla u_\delta(x) = \int_{\mathbb{R}^n} \nabla_x \eta_\delta(x - y) \, u(y) \, d\mathcal{L}^n(y). \end{align*} [claim:$\nabla u_\delta = \eta_\delta * Du$ as $\mathbb{R}^n$-valued Radon measures] [proof] For each fixed $x \in \mathbb{R}^n$, the map $y \mapsto \eta_\delta(x - y)$ lies in $C_c^\infty(\mathbb{R}^n)$ (compactly supported in the ball $\overline{B}(x, \delta)$). Hence we may pair it against the distributional gradient $Du$. Using $\nabla_x \eta_\delta(x - y) = -\nabla_y \eta_\delta(x - y)$ and the definition of the distributional derivative $D_i u$ as a Radon measure paired against test functions in $C_c^\infty(\mathbb{R}^n)$, \begin{align*} \partial_{x_i} u_\delta(x) &= \int_{\mathbb{R}^n} \partial_{x_i} \eta_\delta(x - y) \, u(y) \, d\mathcal{L}^n(y) \\ &= -\int_{\mathbb{R}^n} \partial_{y_i} \bigl(\eta_\delta(x - y)\bigr) \, u(y) \, d\mathcal{L}^n(y) \\ &= \int_{\mathbb{R}^n} \eta_\delta(x - y) \, dD_i u(y). \end{align*} Hence $\nabla u_\delta(x) = \int_{\mathbb{R}^n} \eta_\delta(x - y) \, dDu(y)$, i.e. $\nabla u_\delta$ is the convolution of the scalar function $\eta_\delta$ with the vector-valued Radon measure $Du$. [/proof] [/claim] We now bound $\|\nabla u_\delta\|_{L^1}$ in terms of $|Du|(\mathbb{R}^n)$ via Jensen's inequality and Fubini-Tonelli: \begin{align*} \int_{\mathbb{R}^n} |\nabla u_\delta(x)| \, d\mathcal{L}^n(x) &= \int_{\mathbb{R}^n} \Bigl| \int_{\mathbb{R}^n} \eta_\delta(x - y) \, dDu(y) \Bigr| \, d\mathcal{L}^n(x) \\ &\le \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \eta_\delta(x - y) \, d|Du|(y) \, d\mathcal{L}^n(x) \\ &= \int_{\mathbb{R}^n} \Bigl( \int_{\mathbb{R}^n} \eta_\delta(x - y) \, d\mathcal{L}^n(x) \Bigr) \, d|Du|(y) \\ &= \int_{\mathbb{R}^n} 1 \, d|Du|(y) = |Du|(\mathbb{R}^n). \end{align*} The first inequality is the triangle inequality for vector-valued integrals against the non-negative kernel $\eta_\delta$ (a Jensen-type bound). The swap of integrals is justified by Fubini-Tonelli: the integrand is non-negative and both measures $\mathcal{L}^n$ and $|Du|$ are $\sigma$-finite. The inner $\mathcal{L}^n$-integral evaluates to $1$ by the unit-mass property of $\eta_\delta$ (translation-invariance of $\mathcal{L}^n$). Hence \begin{align*} \|\nabla u_\delta\|_{L^1(\mathbb{R}^n)} \le |Du|(\mathbb{R}^n) < \infty, \end{align*} so $u_\delta \in W^{1,1}(\mathbb{R}^n)$ for every $\delta > 0$. We now invoke two black-box convergence results as $\delta \to 0^+$: (a) [$L^1$ continuity of translation and mollification](/theorems/l1-continuity-of-mollification) (Evans-Gariepy *Measure Theory and Fine Properties of Functions*, revised edition, Section 4.2 Theorem 1; Ambrosio-Fusco-Pallara *Functions of Bounded Variation*, Proposition 3.7): for $u \in L^1(\mathbb{R}^n)$, \begin{align*} \|u_\delta - u\|_{L^1(\mathbb{R}^n)} \to 0 \quad \text{as } \delta \to 0^+. \end{align*} The hypothesis $u \in L^1(\mathbb{R}^n)$ holds because $BV(\mathbb{R}^n) \subseteq L^1(\mathbb{R}^n)$ by definition. (b) [Strict convergence of mollification in $BV$](/theorems/strict-convergence-of-mollification) (Evans-Gariepy Section 5.3 Theorem 2; Ambrosio-Fusco-Pallara Theorem 3.9): for $u \in BV(\mathbb{R}^n)$, \begin{align*} \|\nabla u_\delta\|_{L^1(\mathbb{R}^n)} \to |Du|(\mathbb{R}^n) \quad \text{as } \delta \to 0^+. \end{align*} The hypothesis $u \in BV(\mathbb{R}^n)$ is the standing assumption. Combining (a) and (b), choose $\delta_0 > 0$ small enough that \begin{align*} \|u_{\delta_0} - u\|_{L^1(\mathbb{R}^n)} < \varepsilon/3 \qquad \text{and} \qquad \Bigl| \|\nabla u_{\delta_0}\|_{L^1(\mathbb{R}^n)} - |Du|(\mathbb{R}^n) \Bigr| < \varepsilon/3. \end{align*} Define $w := u_{\delta_0}$. Then $w \in C^\infty(\mathbb{R}^n) \cap W^{1,1}(\mathbb{R}^n)$. [/step] [step:Apply $C^1$ approximation of Sobolev functions] We apply the [$C^1$ Approximation of Sobolev Functions](/theorems/c1-approximation-of-sobolev-functions) (theorem 3092) to $w$ with exponent $p = 1$ and tolerance $\varepsilon/3$. We verify the hypotheses of theorem 3092: it requires $p \in [1, \infty)$ and an input function in $W^{1,p}(\mathbb{R}^n)$. Here $p = 1 \in [1, \infty)$, and $w \in W^{1,1}(\mathbb{R}^n)$ by Step 1. Theorem 3092 produces $g \in C^1(\mathbb{R}^n)$ such that \begin{align*} \mathcal{L}^n(\{w \ne g\}) < \varepsilon/3 \qquad \text{and} \qquad \|w - g\|_{W^{1,1}(\mathbb{R}^n)} < \varepsilon/3. \end{align*} We will not use the bound on $\mathcal{L}^n(\{w \ne g\})$. From the $W^{1,1}$ bound, recalling $\|h\|_{W^{1,1}} = \|h\|_{L^1} + \|\nabla h\|_{L^1}$, we extract the two component bounds \begin{align*} \|w - g\|_{L^1(\mathbb{R}^n)} < \varepsilon/3 \qquad \text{and} \qquad \|\nabla w - \nabla g\|_{L^1(\mathbb{R}^n)} < \varepsilon/3. \end{align*} [/step] [step:Aggregate via triangle inequality] We now verify the two conclusions of the theorem for the function $g \in C^1(\mathbb{R}^n)$ produced in Step 2. For the $L^1$-distance, the triangle inequality gives \begin{align*} \|u - g\|_{L^1(\mathbb{R}^n)} &\le \|u - w\|_{L^1(\mathbb{R}^n)} + \|w - g\|_{L^1(\mathbb{R}^n)} \\ &< \varepsilon/3 + \varepsilon/3 = 2\varepsilon/3 < \varepsilon, \end{align*} using the bound on $\|u - w\|_{L^1}$ from Step 1 and the bound on $\|w - g\|_{L^1}$ from Step 2. For the gradient-mass discrepancy, we combine the strict-convergence bound from Step 1 with the reverse triangle inequality \begin{align*} \bigl| \|\nabla w\|_{L^1(\mathbb{R}^n)} - \|\nabla g\|_{L^1(\mathbb{R}^n)} \bigr| \le \|\nabla w - \nabla g\|_{L^1(\mathbb{R}^n)}, \end{align*} which follows from the triangle inequality $\|\nabla w\|_{L^1} \le \|\nabla w - \nabla g\|_{L^1} + \|\nabla g\|_{L^1}$ and the symmetric estimate. Then \begin{align*} \Bigl| |Du|(\mathbb{R}^n) - \int_{\mathbb{R}^n} |\nabla g| \, d\mathcal{L}^n \Bigr| &= \bigl| |Du|(\mathbb{R}^n) - \|\nabla g\|_{L^1(\mathbb{R}^n)} \bigr| \\ &\le \bigl| |Du|(\mathbb{R}^n) - \|\nabla w\|_{L^1(\mathbb{R}^n)} \bigr| + \bigl| \|\nabla w\|_{L^1(\mathbb{R}^n)} - \|\nabla g\|_{L^1(\mathbb{R}^n)} \bigr| \\ &\le \bigl| |Du|(\mathbb{R}^n) - \|\nabla w\|_{L^1(\mathbb{R}^n)} \bigr| + \|\nabla w - \nabla g\|_{L^1(\mathbb{R}^n)} \\ &< \varepsilon/3 + \varepsilon/3 = 2\varepsilon/3 < \varepsilon, \end{align*} using the bound on $\bigl| \|\nabla w\|_{L^1} - |Du|(\mathbb{R}^n) \bigr|$ from Step 1 and the bound on $\|\nabla w - \nabla g\|_{L^1}$ from Step 2. Thus $g \in C^1(\mathbb{R}^n)$ satisfies both \begin{align*} \|u - g\|_{L^1(\mathbb{R}^n)} < \varepsilon \qquad \text{and} \qquad \Bigl| |Du|(\mathbb{R}^n) - \int_{\mathbb{R}^n} |\nabla g| \, d\mathcal{L}^n \Bigr| < \varepsilon, \end{align*} which is the conclusion of the theorem. [/step]