\cap G) \leq 10g_\varepsilon(t)$, by the definition of Hausdorff $\varepsilon$-content. Letting $\varepsilon \rightarrow 0$ and applying Fatou’s lemma gives $$\int_{[0,1]}^* \mathcal{H}^1(\overline{u^{-1}(t)} \cap G)\, dt \leq 10 \int_0^1 \liminf_{\varepsilon \to 0} g_\varepsilon(t)\, dt \\
            \leq 10 \liminf_{\varepsilon \rightarrow 0} \int_0^1 g_\varepsilon(t) \, dt.$$ Combining estimates, we obtain $$\int_{[0,1]}^*\mathcal{H}^1(\overline{u^{-1}(t)} \cap G)\, dt \leq \frac{4 \cdot 2000}{\pi} \int_E \rho\,d\mathcal{H}^2.$$

We turn our attention next to the set $F = E \setminus G$. We claim that $$\label{equ:bad_points}
            \int_{[0,1]}^* \mathcal{H}^1(\overline{u^{-1}(t)} \cap F)\,dt = 0.$$ By the definition of $F$, for all $x \in F$ there exists $\varepsilon_x = 10^{-k_x}$ (for some integer $k_x \geq 1$) such that $$\label{equ:bad_iteration}
            \int_{B(x,10^{-j})} \rho\,d\mathcal{H}^2  \leq 200^{-1} \int_{B(x,10^{-j+1})} \rho\,d\mathcal{H}^2 \leq \cdots \leq 200^{-(j-k_x)} \int_{B(x,\varepsilon_x)} \rho\,d\mathcal{H}^2$$ for all $j \geq k_x$. For all $k \in \mathbb{N}$, let $F_k = \{x \in F: k_x \leq k\}$. Observe that $F = \bigcup_k F_k$.

Now, fix $k \in \mathbb{N}$ and let $j \geq k$. By definition of the (spherical) Hausdorff measure, there exists a countable collection of balls $B_m=B(x_m, r_m)$ which cover $F_k$, such that $x_m \in F_k$, $r_m \leq \min\{10^{-j}, d(x_m,Q \setminus E),\operatorname{diam}\zeta_1/4, \operatorname{diam}\zeta_3/4\}$, and $\sum 4r_m^2 \leq 4\mathcal{H}^2(F_k)+4/j$. For the last requirement, recall that the spherical Hausdorff 2-measure is at most 4 