for Lipschitz functions on metric spaces, found in [@AmbTil:04 Proposition 3.1.5].

\[prop:coarea\] Let $A \subset X$ be Borel measurable. If $m\colon X \rightarrow \mathbb{R}$ is $L$-Lipschitz and $g\colon A \rightarrow [0, \infty]$ is Borel measurable, then $$\int_{\mathbb{R}} \int_{A \cap m^{-1}(t)} g(s)\, d\mathcal{H}^1(s)\, dt \leq \frac{4L}{\pi} \int_A g(x)\, d\mathcal{H}^2(x).$$

We also need a topological lemma, cf. [@Moo:62 IV Theorem 26].

\[lemm:separating\_continuum\] Let $A,B \subset Q$ be non-empty sets, and let $K \subset Q$ be a compact set such that $A$ and $B$ belong to different components of $Q \setminus K$. Then there is a continuum $F \subset K$ such that $A$ and $B$ belong to different components of $Q \setminus F$. Moreover, if $\mathcal{H}^1(K) < \infty$ and the component of $Q \setminus K$ containing $A$ is contained in the interior of $Q$, then $F$ may be taken to be the image of an injective Lipschitz mapping $\gamma: \mathbb{S}^1 \rightarrow K$.

Level sets of $u$ {#sec:level_sets}
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In this section, we prove a number of topological properties for the level sets of the harmonic function $u$, or, more precisely, for the closure of these level sets. This section can be viewed as an extension of Section 4 in [@Raj:16], which also studies 