Fine topology

in potential theory

The weakest topology in which all superharmonic functions on $ \mathbf R ^ {n} $ are continuous. Objects related to the fine topology are described as "fine" , "finely" , etc.

The notion of fine topology is closely connected with that of a thin set (cf. also Thinness of a set). The fine topology is stronger that the usual Euclidean topology on $ \mathbf R ^ {n} $, i.e. every Euclidean-open set is finely open. A fine neighbourhood of a point $ x _ {0} \in \mathbf R ^ {n} $ is a set $ V ( x _ {0} ) $ such that $ x _ {0} \in V ( x _ {0} ) $ and such that the complement $ {C V ( x _ {0} ) } $ is a thin set at $ x $. The finely-open sets are unions of pre-images under mappings by superharmonic functions of the extended real line $ \overline{\mathbf R} $ and of intervals of the form $ ( a, + \infty ] $, $ [- \infty , b) $, $ ( a, b) $, $ - \infty < a < b < + \infty $. Every superharmonic function on an open set $ E \subset \mathbf R ^ {n} $ is finely continuous on $ E $. A set $ E \subset \mathbf R ^ {n} $ is thin at a point $ x _ {0} \in E $ if and only if $ x _ {0} $ is a finely-isolated point of $ E $.

Let $ x _ {0} $ be a fine-limit point of $ E $, that is, $ E $ is not thin at $ x _ {0} $, and let $ f $ be a function defined on $ E $. The number $ \lambda $ is called the fine limit of $ f $ at $ x _ {0} $ if for every neighbourhood $ U ( \lambda ) $ of $ \lambda $ in $ \overline{\mathbf R} $ there exists a fine neighbourhood $ V ( x _ {0} ) $ of $ x _ {0} $ such that

$$ x \in E \cap V ( x _ {0} ) \Rightarrow f ( x) \in U ( \lambda ). $$

If $ \lambda $ is the fine limit of $ f $ at $ x _ {0} $, then there exists a fine neighbourhood $ V ( x _ {0} ) $ such that $ \lambda $ is an ordinary limit at $ x _ {0} $ of the restriction $ f \mid _ {E \cap V ( x _ {0} ) } $ (Cartan's theorem).

Let $ E $ be a closed set, thin at a point $ x _ {0} $, and let $ f > 0 $ be a superharmonic function defined on $ C E $ in a neighbourhood of $ x _ {0} $. Then $ f $ has a fine limit $ \lambda $ at $ x _ {0} $.

A fine topology has also been constructed in axiomatic potential theory (cf. [3]).

References

[1]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[2]	N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[3]	M. Brélot, "Lectures on potential theory" , Tata Inst. (1960)

Comments

A potential theory for finely-harmonic and finely-hyperharmonic functions is developed in [a1]. See also [a2].

References

[a1]	B. Fuglede, "Finely harmonic functions" , Springer (1972)
[a2]	J. Lukeš, J. Malý, L. Zajíček, "Fine topology methods in real analysis and potential theory" , Springer (1986)