P(E_n, \Omega) \rightarrow m.$$

Take $M \leq |\Omega|$ be the supremum in the second problem. Take a sequence o sets $E_n$ such that $|E_n| \rightarrow M$. Then on some subsequence $\chi_{E_{n_k}} \rightarrow \chi_E$, and thus

$$M \geq |E| \geq |E_n| - |E_n \triangle E| = |E_n| - \|\chi_{E_n} - \chi_E \|_1 \rightarrow M - 0.$$

Then we can show existence of a set $T$ of full measure such that for every $t \in T$ we have $\partial E_t \cap \partial \Omega \subset g^{-1}(t)$ and for every $t,s \in T$, $s < t$ the inclusion $E_t \subset \subset E_s$ holds. It enables us to treat $E_t$ as superlevel sets of a certain function; we define it by the following formula:

$$u(x) = \sup \{t \in T: x \in \overline{E_t \cap \Omega} \}.$$

It turns out that $u \in C(\overline{\Omega}) \cap BV(\Omega)$ and $u$ is a solution to LGP for $g$. Moreover $| \{ u \geq t \} \triangle (\overline{E_t \cap \Omega})| = 0$ for a.e. $t$. Uniqueness proof is based on a maximum principle.

In the existence proof in chapter $4$ we are going to use a particularly simple case of the construction. Suppose $\Omega \subset \mathbb{R}^2$ and that $f \in 