On the mean value property of finely harmonic and finely hyperharmonic functions (Q921187)

Let U be a bounded finely open set in \({\mathbb{R}}^ 2\) and let \(\tilde U\) denote its fine closure. It is shown that, if f is finely lower semicontinuous on \(\tilde U\) and finely hyperharmonic on U, then f(x)\(\geq \int_{*}f d\epsilon_ x^{CV}\) for any finely open set V with \(\tilde V\subseteq U\). (Here \(\epsilon_ x^{CV}\) denotes fine harmonic measure for V). An example is given to demonstrate that this result fails in \({\mathbb{R}}^ n\) when \(n\geq 3\).