are considered as subsets of $Q$.

Consider the set $$G = \left\{x \in E: \forall \varepsilon>0, \exists r<\varepsilon, \int_{B(x,10r)} \rho\,d\mathcal{H}^2 \leq 200 \int_{B(x,r)} \rho \,d\mathcal{H}^2\right\}.$$ Fix $\varepsilon>0$. We apply the basic covering theorem ([@Hei:01 Thm. 1.2]) to choose a countable collection of pairwise disjoint balls $B_j = B(x_j,r_j)$ such that $x_j \in G$ and $10r_j \leq \min\{\varepsilon, d(x_j, Q\setminus E)\}$ for each $j$, the collection $\{5B_j\}$ covers $G$, and $$\int_{10B_j} \rho\,d\mathcal{H}^2 \leq 200 \int_{B_j} \rho \,d\mathcal{H}^2$$ for each $j$. We also require that $20r_j < \min\{\operatorname{diam}\zeta_1, \operatorname{diam}\zeta_3\}$ for our application of Lemma \[lemm:oscillation\]. We have $$\sum_j \int_{10B_j} \rho\, d\mathcal{H}^2 \leq \sum_j 200\int_{B_j} \rho\, d\mathcal{H}^2 \leq 200\int_{E} \rho\,d\mathcal{H}^2,$$ where the last inequality follows since by our choice the balls $B_j$ are pairwise disjoint subsets of the open set $E$. For each $j$ fix a measurable set $A_j \supset u(5B_j)$ such that $\mathcal{H}^1(A_j)=\mathcal{H}^1(u(5B_j))$. Moreover, define $g_\varepsilon: [0,1] \rightarrow \mathbb{R}$ by $$g_\varepsilon(t) = \sum_j r_j \chi_{A_j}(t).$$ Integrating and applying Lemma \[lemm:oscillation\] gives $$\int_0^1 g_\varepsilon(t)\,dt = \sum_j r_j \mathcal{H}^1(u(5B_j)) \leq \frac{4}{\pi}\sum_j \int_{10B_j} \rho\, d\mathcal{H}^2.$$ We observe that if $x \in \overline{u^{-1}(t)} \cap G$ for a given $t \in [0,1]$, with $j_x$ such that $x \in 5B_{j_x}$, then of necessity $t \in u(5B_{j_x})$. Hence $\mathcal{H}_\varepsilon^1(\overline{u^{-1}(t)} 