Minimal fine limits on trees (Q1765965)

The authors define the minimal filter corresponding to each boundary point of a tree and prove a tree version of the Fatou-Naim-Doob limit theorem. Also, it is given an example of a tree for which minimal fine limits do not imply nontangential limits, even for positive superharmonic functions. Motivated by work on potential theory on halfspaces and Brelot spaces, it is defined the harmonic fine filter corresponding to each boundary point of the tree. In contrast to the classical setting the harmonic fine filter is the same as the minimal fine filter.