Finely harmonic functions and the domination axiom (Q1261224)

This paper analyzes the relation of the axiom of domination in a harmonic space and the sheaf property of finely hyperharmonic functions in finely open sets. In fact, it is known that the sheaf property of positive finely hyperharmonic functions is equivalent with the axiom of domination [see \textit{J. Lukeš}, \textit{J. Malý} and \textit{L. Zajíček}, Fine topology methods in real analysis and potential theory, Lect. Notes Math. 1189 (Springer 1986; Zbl 0607.31001)]. In the present article it is shown, in a \({\mathcal P}\)-harmonic space \(\Omega\) with countable base, that the sheaf property of positive finely hyperharmonic functions is equivalent with the sheaf property of finely harmonic functions dominated by a given finely harmonic function (or by a locally bounded potential on \(\Omega)\).