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Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces. (English) Zbl 1460.46008
Summary: We study non-linear functionals, including quasi-linear functionals, \(p\)-conic quasi-linear functionals, \(d\)-functionals, \(r\)-functionals, and their relationships to deficient topological measures and topological measures on locally compact spaces. We prove representation theorems and show, in particular, that there is an order-preserving, conic-linear bijection between the class of finite deficient topological measures and the class of bounded \(p\)-conic quasi-linear functionals. Our results imply known representation theorems for finite topological measures and deficient topological measures. When the space is compact we obtain four equivalent definitions of a quasi-linear functional and four equivalent definitions of functionals corresponding to deficient topological measures.
MSC:
46B10 Duality and reflexivity in normed linear and Banach spaces
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28A25 Integration with respect to measures and other set functions
46E27 Spaces of measures
46G99 Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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