Maximum Sets of Semicontinuous Functions

Abstract

This paper presents a description of G_δ sets. It also presents how this description can be used in potential theory. For a set E of type G_δ which is polar in Rⁿ we give a new construction of subharmonic function u such that

$$E=\{z\in\mathbb{R}^{n}:u(z)=-\infty\}.\vspace{-1pt}$$

We also give some tools which can be used to obtain similar results for pluripolar sets in Cⁿ. In particular, if E⊂F₁×⋅⋅⋅×F_n, where E is of type G_δ and F_i is a polar set in C, then we give a construction of plurisubharmonic function u such that E={z∈Cⁿ:u(z)=−∞}.