We construct a connected, compact set \(K \subset \mathbb {C}^2\) with the following property: there exist points \(p \in \hat{K} {\setminus } K\) such that there does not exist a sequence \(\{A_\nu \}\) of analytic sets \(A_\nu \subset \subset \mathbb {C}^2\) with boundary satisfying \(p \in A_\nu \) for every \(\nu \in \mathbb {N}\) and \(\lim _{\nu \rightarrow \infty } bA_\nu \subset K\) . For every point in \(\hat{K} {\setminus } K\) , we explicitly construct a sequence of Poletsky discs, and we compute the weak limit of the pushforwards of the Green current under these discs.
