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\(q\)-functions and extreme topological measures. (English) Zbl 1074.28007
Summary: \(q\)-functions provide a method for constructing topological measures. We give necessary and sufficient conditions for a composition of a \(q\)-function and a topological measure to be a topological measure. Regular and extreme step \(q\)-functions are characterized by certain regions in \(\mathbb R^n\). Then extreme \(q\)-functions are used to study extreme topological measures. For example, we prove (under some assumptions on the underlying set) that given \(n\), there are different types of extreme topological measures with values \(0,1/n,\dots ,1\). In contrast, in the case of measures the only extreme points are \(\{0,1\}\)-valued, i.e., point masses.

MSC:
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28A10 Real- or complex-valued set functions
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
46J05 General theory of commutative topological algebras
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