[proofplan]
The weighted formula extends the [BV Coarea Formula](/theorems/598) by replacing the "constant weight $1$" against $|Du|$ with a general bounded Borel weight $g$. The strategy is a standard layer-by-layer extension: the local coarea formula $|Du|(B) = \int_{-\infty}^\infty P(\{u > t\}, B)\, d\mathcal{L}^1(t)$ for Borel $B \subseteq \Omega$ identifies the integration of $\mathbb{1}_B$ against $|Du|$ with the integration of $\mathbb{1}_B$ against the parameterised perimeter measure. Linearity extends this to non-negative simple functions $g = \sum c_j \mathbb{1}_{B_j}$, and density of simple functions in $L^\infty$ via a monotone or bounded approximation argument produces the formula for general bounded Borel $g$. The two technical pieces are: (i) Fubini–Tonelli applied to the parameterised family of measures $|D\mathbb{1}_{E_t}|$, justifying the exchange of orders of integration; and (ii) the Lebesgue dominated convergence theorem, which carries the simple-function identity into a $g$-pointwise identity in the limit.
[/proofplan]
[step:Establish the formula for indicator functions $g = \mathbb{1}_B$ via the local coarea formula]
Let $B \subseteq \Omega$ be Borel. The [BV Coarea Formula](/theorems/598) asserts
\begin{align*}
|Du|(B) = \int_{-\infty}^\infty P(\{u > t\}, B) \, d\mathcal{L}^1(t) = \int_{-\infty}^\infty |D\mathbb{1}_{\{u > t\}}|(B) \, d\mathcal{L}^1(t).
\end{align*}
Rewriting both sides with the indicator function $\mathbb{1}_B$:
\begin{align*}
\int_\Omega \mathbb{1}_B \, d|Du| = \int_{-\infty}^\infty \int_\Omega \mathbb{1}_B \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
This establishes the weighted coarea formula for the special case $g = \mathbb{1}_B$.
[guided]
We start with the simplest possible weight: an indicator function. The local coarea formula already provides the result for this case.
The local coarea formula in the [BV Coarea Formula](/theorems/598) states: for $u \in BV(\Omega)$ and any Borel set $B \subseteq \Omega$,
\begin{align*}
|Du|(B) = \int_{-\infty}^\infty P(\{u > t\}, B) \, d\mathcal{L}^1(t).
\end{align*}
The notation $P(\{u > t\}, B)$ is by definition $|D\mathbb{1}_{\{u > t\}}|(B)$, the perimeter measure of the level set $\{u > t\}$ evaluated on $B$.
Now, for any Radon measure $\mu$ on $\Omega$ and Borel set $B \subseteq \Omega$,
\begin{align*}
\mu(B) = \int_\Omega \mathbb{1}_B(x) \, d\mu(x).
\end{align*}
This is the basic relationship between a measure and its action on indicator functions. Applying this to $\mu = |Du|$ on the left and to $\mu = |D\mathbb{1}_{\{u > t\}}|$ inside the integral on the right:
\begin{align*}
\int_\Omega \mathbb{1}_B \, d|Du| = |Du|(B) = \int_{-\infty}^\infty |D\mathbb{1}_{\{u > t\}}|(B) \, d\mathcal{L}^1(t) = \int_{-\infty}^\infty \int_\Omega \mathbb{1}_B \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
This is the desired formula with $g = \mathbb{1}_B$. Both sides are well-defined and equal.
The hypothesis met for invoking [BV Coarea Formula](/theorems/598): $u \in BV(\Omega)$, which is given. The local form (over Borel $B$) is the stronger version of the coarea formula stated in part (ii) of the theorem.
[/guided]
[/step]
[step:Extend by linearity to non-negative simple Borel functions]
Let $g = \sum_{j=1}^N c_j \mathbb{1}_{B_j}$ be a non-negative simple function with $c_j \ge 0$ and $B_j \subseteq \Omega$ Borel. By Step 1 applied to each $\mathbb{1}_{B_j}$ and linearity of the integral,
\begin{align*}
\int_\Omega g \, d|Du| = \sum_{j=1}^N c_j \int_\Omega \mathbb{1}_{B_j} \, d|Du| = \sum_{j=1}^N c_j \int_{-\infty}^\infty \int_\Omega \mathbb{1}_{B_j} \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
Exchanging summation and integration (a finite sum, hence justified) and using linearity of integration against $|D\mathbb{1}_{\{u > t\}}|$,
\begin{align*}
\int_\Omega g \, d|Du| = \int_{-\infty}^\infty \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
The exchange of $\sum_{j}$ and $\int_t$ is also a finite sum and is justified by Fubini for finite sums.
[guided]
For a non-negative simple function $g = \sum_{j=1}^N c_j \mathbb{1}_{B_j}$ with $c_j \ge 0$ and Borel $B_j$, the formula extends by linearity. We must check both:
- Linearity of the LHS: $\int_\Omega g \, d|Du| = \sum_{j=1}^N c_j \int_\Omega \mathbb{1}_{B_j} \, d|Du| = \sum_{j=1}^N c_j |Du|(B_j)$.
- Linearity of the RHS: $\int_{-\infty}^\infty \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t) = \sum_{j=1}^N c_j \int_{-\infty}^\infty |D\mathbb{1}_{\{u > t\}}|(B_j) \, d\mathcal{L}^1(t)$.
Both linearity statements are standard properties of integration with respect to a positive measure (the integral of a finite linear combination of indicator functions is the same linear combination of measures of sets). The exchange of the finite sum $\sum_{j=1}^N$ with the $t$-integral is automatic for finite sums (Fubini for the product of the finite counting space $\{1, \dots, N\}$ with $(\mathbb{R}, \mathcal{L}^1)$).
Applying Step 1 to each $\mathbb{1}_{B_j}$:
\begin{align*}
\int_\Omega \mathbb{1}_{B_j} \, d|Du| = \int_{-\infty}^\infty \int_\Omega \mathbb{1}_{B_j} \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
Multiplying by $c_j$ and summing over $j = 1, \dots, N$, and using both linearity statements:
\begin{align*}
\int_\Omega g \, d|Du| = \sum_{j=1}^N c_j \int_\Omega \mathbb{1}_{B_j} \, d|Du| = \sum_{j=1}^N c_j \int_{-\infty}^\infty \int_\Omega \mathbb{1}_{B_j} \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
Pulling the sum inside the $t$-integral (finite sum, automatic):
\begin{align*}
\int_\Omega g \, d|Du| = \int_{-\infty}^\infty \sum_{j=1}^N c_j \int_\Omega \mathbb{1}_{B_j} \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t) = \int_{-\infty}^\infty \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
The formula holds for non-negative simple Borel functions.
[/guided]
[/step]
[step:Pass from simple functions to general bounded Borel $g$ via monotone convergence]
Let $g: \Omega \to [0, \infty)$ be bounded Borel-measurable with $0 \le g \le M$. By the standard approximation of non-negative measurable functions, there exists an increasing sequence of non-negative simple Borel functions $g_k$ with $g_k \nearrow g$ pointwise as $k \to \infty$, and $0 \le g_k \le g \le M$ for all $k$.
For each $k$, Step 2 gives
\begin{align*}
\int_\Omega g_k \, d|Du| = \int_{-\infty}^\infty \int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
We pass to the limit on both sides. On the LHS, by the monotone convergence theorem applied to the measure $|Du|$,
\begin{align*}
\int_\Omega g_k \, d|Du| \nearrow \int_\Omega g \, d|Du|.
\end{align*}
On the RHS, for each fixed $t$, the inner integral $\int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}|$ converges to $\int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}|$ by the monotone convergence theorem. Moreover, the inner integrals are dominated uniformly in $k$:
\begin{align*}
\int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}| \le M \cdot |D\mathbb{1}_{\{u > t\}}|(\Omega) = M \cdot P(\{u > t\}, \Omega),
\end{align*}
and $t \mapsto P(\{u > t\}, \Omega)$ is in $L^1(\mathcal{L}^1)$ by the [BV Coarea Formula](/theorems/598) with total mass $|Du|(\Omega) < \infty$. So the dominated convergence theorem on the outer $t$-integral applies:
\begin{align*}
\int_{-\infty}^\infty \int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t) \to \int_{-\infty}^\infty \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
Taking the limit $k \to \infty$ in the displayed identity gives the formula for general bounded Borel $g \ge 0$.
[guided]
We extend the formula from simple functions to bounded Borel $g \ge 0$ by approximation. The approximation tool is the standard simple-function approximation of non-negative measurable functions.
*Approximation.* For any Borel $g: \Omega \to [0, M]$ with $0 \le g \le M$, define
\begin{align*}
g_k(x) := \sum_{j=0}^{k 2^k - 1} \frac{j}{2^k} \mathbb{1}_{\{j 2^{-k} \le g < (j+1) 2^{-k}\}}(x) + k \mathbb{1}_{\{g \ge k\}}(x).
\end{align*}
This is the standard dyadic simple-function approximation: $g_k$ is non-negative, simple, Borel, and $0 \le g_k \le g$ pointwise, with $g_k(x) \nearrow g(x)$ for every $x$ as $k \to \infty$. The functions $g_k$ are bounded by $M$ for $k \ge M$.
*Convergence on the LHS.* The monotone convergence theorem (applied with respect to the positive Radon measure $|Du|$ on $\Omega$, which is $\sigma$-finite since $|Du|(\Omega) < \infty$): if $g_k \nearrow g$ pointwise, then $\int g_k \, d|Du| \nearrow \int g \, d|Du|$. Hence
\begin{align*}
\lim_{k \to \infty} \int_\Omega g_k \, d|Du| = \int_\Omega g \, d|Du|.
\end{align*}
*Convergence on the RHS.* For each fixed $t \in \mathbb{R}$, the monotone convergence theorem with respect to $|D\mathbb{1}_{\{u > t\}}|$ (a positive Radon measure on $\Omega$, $\sigma$-finite for $\mathcal{L}^1$-a.e. $t$):
\begin{align*}
\lim_{k \to \infty} \int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}| = \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}|.
\end{align*}
We need to interchange this $k$-limit with the $t$-integral. The dominated convergence theorem applies:
\begin{align*}
\int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}| \le \int_\Omega M \, d|D\mathbb{1}_{\{u > t\}}| = M \cdot |D\mathbb{1}_{\{u > t\}}|(\Omega) = M \cdot P(\{u > t\}, \Omega).
\end{align*}
The dominating function $t \mapsto M \cdot P(\{u > t\}, \Omega)$ is in $L^1(\mathbb{R}, \mathcal{L}^1)$:
\begin{align*}
\int_{-\infty}^\infty M \cdot P(\{u > t\}, \Omega) \, d\mathcal{L}^1(t) = M \cdot |Du|(\Omega) < \infty,
\end{align*}
where we have used the [BV Coarea Formula](/theorems/598) and the hypothesis $u \in BV(\Omega)$.
By dominated convergence on $(\mathbb{R}, \mathcal{L}^1)$:
\begin{align*}
\lim_{k \to \infty} \int_{-\infty}^\infty \int_\Omega g_k \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t) = \int_{-\infty}^\infty \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
*Combining.* Taking $k \to \infty$ in the Step 2 identity for $g_k$:
\begin{align*}
\int_\Omega g \, d|Du| = \int_{-\infty}^\infty \int_\Omega g \, d|D\mathbb{1}_{\{u > t\}}| \, d\mathcal{L}^1(t).
\end{align*}
This is the weighted coarea formula for non-negative bounded Borel $g$. Since the theorem statement assumes $g \ge 0$ (codomain $[0, \infty)$), the proof is complete. The formula extends to bounded real-valued Borel $g$ by splitting $g = g^+ - g^-$, but the statement of the theorem only requires the non-negative case, so we stop here.
[/guided]
[/step]
