Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on R n ( n ⩾2) is the coarsest topology for which all superharmonic functions on R n are continuous. We refer to Doob [ 11 , 1.XI] for its basic properties and its relationship to the notion of thinness. This paper presents several theorems relating the fine topology to limits of functions along parallel lines. (Results of this nature for the minimal fine topology have 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 results of Lusin and Privalov [ 18 ], Evans [ 12 ], Rudin [ 20 ], Bagemihl and Seidel [ 6 ], Schneider [ 21 ], Berman [ 7 ], and Armitage and Nelson [ 4 ], and will also solve a problem posed by the latter authors. An early version of our first result is due to Evans [ 12 , p. 234], who proved that, if u is a superharmonic function on R 3 , then there is a set E ⊆R 2 ×{0}, of two‐dimensional measure 0, such that u ( x , y ,·) is continuous on R whenever ( x , y , 0)∉ E . We denote a typical point of R n by X =( X ′ x ), where X ′∈R n −1 and x ∈R. Let π:R n →R n −1 ×{0} denote the projection map given by π( X ′, x ) = ( X ′, 0). For any function f :R n →[−∞, +∞] and point X we define the vertical and fine cluster sets of f at X respectively by C V ( f ; X )={ l ∈[−∞, +∞]: there is a sequence ( t m ) of numbers in ℝ∖{ x } such that t m → x 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 finely open , and functions which are continuous with respect to the fine topology will be called finely continuous . Corollary 1(ii) below is an improvement of Evans' result.