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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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##### References:
 [1] Aarnes, J.F., Quasi-states and quasi-measures, Adv. math., 86, 41-67, (1991) · Zbl 0744.46052 [2] Aarnes, J.F., Pure quasi-states and extremal quasi-measures, Math. ann., 295, 575-588, (1993) · Zbl 0791.46028 [3] Aarnes, J.F., Construction of non-subadditive measures and discretization of Borel measures, Fund. math., 147, 213-237, (1995) · Zbl 0842.28004 [4] Aarnes, J.F.; Rustad, A.B., Probability and quasi-measures—a new interpretation, Math. scand., 85, 278-284, (1999) · Zbl 0967.28014 [5] Aarnes, J.F., Quasi-states on $$C^\ast$$-algebras, Trans. amer. math. soc., 149, 601-625, (1970) · Zbl 0212.15403 [6] J.F. Aarnes, Private communication · Zbl 1207.68077 [7] D.J. Grubb, Private communication [8] Grubb, D.J.; LaBerge, T., Additivity of quasi-measures, Proc. amer. math. soc., 126, 3007-3012, (1998) · Zbl 0907.28007 [9] Knudsen, F.F., Topology and the construction of extreme quasi-measures, Adv. math., 120, 302-321, (1996) · Zbl 0914.28010 [10] Rustad, A.B., Quasi-measures with image transformations as generalized variables, J. math. anal. appl., 271, 16-30, (2002) · Zbl 1078.28010 [11] Wheeler, R.F., Quasi-measures and dimension theory, Topology appl., 66, 75-92, (1995) · Zbl 0842.28005
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