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