Removing the boundary before applying scalar coarea

The boundary is Borel: in the half-space charts its complement is open. Each boundary-chart image of $\mathrm{\partial }R$ lies in a Lebesgue-null coordinate plane by [F4] and [F6]. Its volume is therefore zero by the chart formula [F5]; a finite chart cover of the compact region gives $V_k(\mathrm{\partial }R)=0$. By [F2] the Lipschitz function $u$ is continuous in the manifold topology. Since the boundary constants lie outside $I$, the closed level $E_t=\{u=t\}$ misses the boundary for $t\in I$, and [F10] makes it compact.

Let $C$ be any interior component. Every curve confined to $C$ is a curve in $R$, so $d_R(x,y)\le d_C(x,y)$ for $x,y\in C$. If $L$ is the given Lipschitz constant, then $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le L{d}_R(x,y)\le L{d}_C(x,y)$. Thus $u{\mathrm{\mid }}_C$ satisfies the global Lipschitz hypothesis of the boundaryless theorem; completeness of $C$ is not required by that theorem.

Fix an interior point $p\in C$ and choose coordinates centered at $p$ with the closed Euclidean ball ${\bar{B}}_r$ inside the chart and inside $C$. Use [F4] on this compact set and choose $0<\epsilon <r$ with $2b\epsilon <a(r-\epsilon )$. For any $y,z$ with coordinates in $B_{\epsilon }$, the coordinate segment between them has metric length at most $2b\epsilon$. Any curve in $R$ from $y$ to $z$ that leaves the coordinate ball $B_r$ has an initial segment reaching its sphere; its length is at least $a(r-\epsilon )$, since Euclidean curve length bounds endpoint displacement. Hence such curves cannot lower the infimum below the segment bound, and $d_R(y,z)$ equals the infimum over curves confined to $B_r$. As these curves are also confined to $C$, step 1.2 yields $d_R(y,z)\le d_C(y,z)\le d_R(y,z)$. Thus the two distances agree on the neighborhood $B_{\epsilon }$. This argument uses infima and does not assume existence of minimizing curves.

Fix a Borel interval $J\subset I$ and put $A=\{x\in R:u(x)\in J\}$. It is Borel by continuity, and step 1.1 gives $A\subset R^{\circ }$. For each interior component $C$, apply [F9] to $X=C$, $Y=\mathbb{R}$ with its Euclidean metric, $F=u{\mathrm{\mid }}_C$, $n=2$, $k=1$, and the Borel set $A\cap C$. Its Lipschitz hypothesis is step 1.2. The source normalization identifies full-dimensional Hausdorff measure with $d{V}_k$ and target measure with $dt$. By [F1] and [F3] its Jacobian is $\mathrm{\mid }{\mathrm{\nabla }}_{k}u\mathrm{\mid }$ almost everywhere. The resulting identity is ${\int }_{A\cap C}\mathrm{\mid }{\mathrm{\nabla }}_{k}u\mathrm{\mid }d{V}_k={\int }_J{\mathcal{H}}_{d_C}^2(C\cap E_t)dt$, with nonnegative extended integrals.

For a subset of one of the neighborhoods in step 2.1, intersect every Hausdorff covering set with that neighborhood and discard empty intersections. This does not increase diameters, so Hausdorff outer measure equals the one computed within the neighborhood. The restricted distances agree there; therefore the two normalized Hausdorff outer measures agree on every such subset. Second countability supplies a countable cover $(U_j)$ by these neighborhoods: take a countable basis refinement and choose a containing neighborhood for each selected basis element, using [A1]. Partition $C$ into the Borel sets $P_j=U_j\setminus {\bigcup }_{i<j}{U}_i$. For a Borel $E\subset C$, [F7] gives equality of the two measures of $E$ by summing their equal values on $E\cap P_j$. The Borel domains agree by [F2]. Interior components are open, because coordinate balls are connected; they are countably many, since disjoint nonempty open sets contain different members of a countable basis. Consequently the $d_R$-area of any Borel interior set is the sum of its $d_C$-areas over those components.

Sum the identities in step 2.2 over the countably many interior components. The coarea input [F9] supplies measurable fiber-area functions, and [F8] interchanges their nonnegative sum with the $t$-integral. Step 3.1 identifies the sum of component areas with ${\mathcal{H}}_{d_R}^2(E_t)$ for $t\in J$. Thus ${\int }_A\mathrm{\mid }{\mathrm{\nabla }}_{k}u\mathrm{\mid }d{V}_k={\int }_J{\mathcal{H}}_{d_R}^2(E_t)dt$. On the volume side the density measure on the interior is the restriction of the measure on $R$ by [F5], and the removed boundary is null by step 1.1; on the level side $E_t\subset R^{\circ }$ and step 3.1 gives agreement of the Hausdorff areas. This proves the claimed absence of boundary contributions.