Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob 11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see 10, Theorem 3.1] or 11,1.XII.23] – and the second author 15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov 18], Evans 12], Rudin 20], Bagemihland Seidel 6], Schneider 21], Berman 7], and Armitage andNelson 4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans 12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l–, +]: for each neighbourhood N of l in –,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.