Suppose I have a continuous, compactly supported function $f : \mathbb{R}^2 \to \mathbb{R}_{+}$ and I define the set $S := f^{-1}([a,\infty)) \subset \mathbb{R}^2$ for some $a > 0$. It is a compact set and therefore its logarithmic capacity is well defined. Let's denote $C(E)$ the logarithmic capacity of a capacitable set $E \subset \mathbb{R}^{2}$.

What conditions on $f$ ensure the existence of a sequence of capacitable sets with regular (say Lipshitz) border $S_{n} \subset S$, $n \in \mathbb{N}$, such that $\lim_{n \to \infty} C(S_{n}) = C(S)$? In particular, is continuity a sufficient condition? Does more regularity help?

Suppose I have a continuous, compactly supported function $f : \mathbb{R}^2 \to \mathbb{R}_{+}$ and I define the set $S := f^{-1}([a,\infty)) \subset \mathbb{R}^2$ for some $a > 0$. It is a compact set and therefore its logarithmic capacity is well defined. Let's denote $C(E)$ the logarithmic capacity of a capacitable set $E \subset \mathbb{R}^{2}$.

What conditions on $f$ ensure the existence of a sequence of capacitable sets with regular (say Lipschitz) border $S_{n} \subset S$, $n \in \mathbb{N}$, such that $\lim_{n \to \infty} C(S_{n}) = C(S)$? In particular, is continuity a sufficient condition? Does more regularity help?