\[prop:coarea\_u\], like Proposition 15.7 in [@Raj:16], is based on standard arguments such as that in [@AmbTil:04 Prop. 3.1.5].

\[prop:coarea\_u\] Let $u$ and $\rho$ be as above. For all Borel functions $g: Q \rightarrow [0,\infty]$, $$\int_{[0,1]}^* \int_{\overline{u^{-1}(t)}} g\,d\mathcal{H}^1\,dt \leq 2000 \int_Q g\rho\, d\mathcal{H}^2.$$ Here $\int^*_A a(t)\, dt $ is the upper Lebesgue integral of $a$ over $A$ (see [@Fed:69 Sec. 2.4.2]).

It suffices to consider the case where $g$ is a characteristic function, that is, $g = \chi_E$ for some Borel set $E \subset Q$. Moreover, we may assume that $E$ is open in $Q$. Indeed, for a Borel set $E$ we find open sets $U_j \supset E$, $U_{j+1} \subset U_j$, such that $\mathcal{H}^2(U_j) \to \mathcal{H}^2(E)$. Assuming the proposition for $g=\chi_{U_j}$, we have $$\begin{aligned}
            \int_{[0,1]}^* \int_{\overline{u^{-1}(t)}} \chi_E \,d\mathcal{H}^1\,dt  &\leq &\int_{[0,1]}^* \int_{\overline{u^{-1}(t)}} \chi_{U_j} \,d\mathcal{H}^1\,dt \leq 2000 \int_Q \chi_{U_j}\rho\, d\mathcal{H}^2 \\ 
            &\longrightarrow& 2000 \int_Q \chi_{E}\rho\, d\mathcal{H}^2. 
        \end{aligned}$$ So we want to show that $$\label{equ:set_E} 
        \int_{[0,1]}^* \mathcal{H}^1(\overline{u^{-1}(t)} \cap E) \,dt \leq 2000 \int_E \rho\,d\mathcal{H}^2$$ whenever $E$ is open in $Q$. The proof is divided into two steps, the first dealing with the subset of “good” points of $E$ and the second dealing with the subset of “bad” points. Throughout this proof, all metric balls 