Separately finely superharmonic functions. (Q1417880)

The author extends classical results about separately superharmonic functions obtained by \textit{V. Avanissian} [Semin. Probab. I, Univ. Strasbourg 1966/67, 3--17 (1967; Zbl 0153.15402)] to the fine potential theory, as developed by \textit{B. Fuglede} [Finely harmonic functions. Lecture Notes in Mathematics. 289. Berlin: Springer (1972; Zbl 0248.31010)]. Let \(\tau \) denote the product of the fine topologies on \(\mathbb{R}^ {n} \times \mathbb{R}^ {m}\). The fine topology on \(\mathbb{R}^ {n + m}\) is strictly finer than \(\tau \). The main result states that any \(\tau \)-locally lower bounded, separately finely superharmonic function on a \(\tau \)-open set is finely superharmonic.