Small sets and balayage in potential theory. (English) Zbl 0636.31005

Kučera, Petr; Netuka, Ivan

Stud. Cercet. Mat. 39, No. 1, 39-41 (1987).

Let S be a standard H-cone of functions on a set X, \(\Phi\) be a family of subsets of X. For a function \(f: X\to [0,\infty]\) and a set \(E\subset X\) define \(^{\Phi}K\) \(E_ f=\{v\in S:\) \(E\cap \{v<f\}\in \Phi \}\) and \(^{\Phi}T\quad E_ f=\inf^{\Phi}K\quad E_ f.\) A typical result of the note reads as follows: Let \(\Phi\) be a \(\sigma\)-ideal of subsets of X. Then the following conditions are equivalent: (1) Every semi-polar set belongs to \(\Phi\). (2) If \(f: X\to [0,\infty]\), \(E\subset X\) and \(^{\Phi}K\) \(E_ f\neq \emptyset\), then \(^{\Phi}T\) \(E_ f\in S\) and \(E\cap \{^{\Phi}T\) \(E_ f<f\}\in \Phi.\)
Therefore (1) is a necessary and sufficient condition for the behaviour of the balayage as known from classical potential theory, semi-classical potential theory and abstract potential theory where the family of polar, Lebesgue measure zero and semi-polar sets is considered for \(\Phi\), respectively.

Reviewer: P.Kučera

MSC:

31D05

Axiomatic potential theory

Keywords:

exceptional sets; polar sets; standard H-cone; semi-polar set; balayage