On the behavior at infinity of logarithmic potentials (Q801188)

Let F be a closed set in \(R^ 2\). A positive measure \(\mu\) supported by F with total mass 1 is called the equilibrium measure on F when the logarithmic potential of \(\mu\) is constant on F except for a set with logarithmic capacity zero. It has been shown by \textit{N. Ninomiya} [Osaka J. Math. 20, 205-216 (1983; Zbl 0519.31002)] that F is thin at infinity if F supports the equilibrium measure. The author shows that the converse of this result is not valid. He proves that F has the equilibrium measure if and only if it satisfies a certain additional condition which is more stringent than the one of thinness.