The fine topology on ℝn (n[ges ]2) is
the coarsest topology for which all superharmonic functions on ℝn
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 ℝ3,
then there is a set E⊆ℝ2×{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 ℝn by
X=(X′ x), where
X′∈ℝn−1 and x∈ℝ. Let
π:ℝn→ℝn−1×{0}
denote the projection map given by π(X′, x) = (X′, 0).
For any function f:ℝn→[−∞, +∞]
and point X we define the vertical and fine cluster sets
of f at X respectively by
formula here
and
formula here
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.
