\lim_{j \to \infty} \frac{3200}{\pi}\cdot 200^k \cdot2^{-j}  \left(\int_Q \rho\,d\mathcal{H}^2 \right)\left(\mathcal{H}^2(F_k)+1/j \right)=0.$$ Combining with Fatou’s lemma and shows that $ \mathcal{H}^1(\overline{u^{-1}(t)} \cap F_k)=0$ for almost every $t$. Since this is true for all $k$, follows.

With Proposition \[prop:coarea\_u\] in hand, the proof of Theorem \[thm:main\] is now simple.

First, observe from Proposition \[prop:coarea\_u\] that $\mathcal{H}^1(\overline{u^{-1}(t)}) < \infty$ for almost every $t \in [0,1]$. Also, as shown in Proposition \[prop:connect\], $\overline{u^{-1}(t)}$ is connected for all $t$ and connects $\zeta_2$ and $\zeta_4$. By Proposition \[prop:existence\_paths\], for almost every $t \in [0,1]$, $\overline{u^{-1}(t)}$ contains a simple rectifiable curve $\gamma_t$ joining $\zeta_2$ and $\zeta_4$ in $Q$. Let $g: Q \rightarrow [0,\infty]$ be an admissible function for $\Gamma_2$. Then $$\label{nakka}
        1 \leq \int_{\gamma_t} g \, ds \leq \int_{\overline{u^{-1}(t)}} g\,d\mathcal{H}^1$$ for almost every $0 \leq t \leq 1$. Combining with Proposition \[prop:coarea\_u\] yields $$1 \leq \int^*_{[0,1]}\int_{\overline{u^{-1}(t)}} g\,d\mathcal{H}^1\,dt \leq  \frac{4 \cdot 2000}{\pi} \int_Q g\rho\, d\mathcal{H}^2.$$ By Hölder’s inequality, $$\int_Q g\rho\,d\mathcal{H}^2 \leq \left( \int_Q g^2\,d\mathcal{H}^2 \right)^{1/2} \left( \int_Q \rho^2\, d\mathcal{H}^2 \right)^{1/2} = \left( \int_Q g^2\,d\mathcal{H}^2 \right)^{1/2} (\operatorname{mod}\Gamma_1)^{1/2}.$$ Infimizing over all admissible $g$, we obtain $$\frac{1}{2000^2\cdot (4/\pi)^2} \leq \operatorname{mod}\Gamma_1 \cdot \operatorname{mod}\Gamma_2.$$

We can improve the value of $\kappa$ as follows. For $\delta>0$, a version of the basic covering theorem yields a 