Maximum sets of semicontinuous functions (Q1781891)

Let \(E\) be a \(G_\delta\)-set in the metric space \((X, \rho)\). It is shown that there exist sequences \((D_i)\) and \((T_i)\) of closed sets such that \[ X \setminus E =\bigcup^\infty_{i=1} D_i,\quad D_i\subset D_{i+1}\text{ for }i\in \mathbb N, \] \[ \rho((T_j,D))>0\text{ for }j\in\mathbb N, \] \[ E=\bigcap^\infty_{j=1}\bigcup^\infty_{i=j}T_i. \] If \(\sigma\) is a Borel probability measure on \(X\) and \((t_i)\) a sequence of positive numbers, then the sequences \((D_i)\) and \((T_i)\) can be chosen such that \(\sigma(X\setminus (D_i\cup E)) < t_i\) for all \(i\in \mathbb N\). Various applications of this result in potential theory are presented. In particular, it is shown how to construct a plurisubharmonic function \(u\) such that \(E = \{z \in\mathbb C^n : u(z) = -\infty\}\) if \(E\subset F_1\times \cdots \times F_n\), \(E\) is a \(G_\delta\)-set and the \(F_i\) are polar sets in \(\mathbb C\).