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Unbounded quasi-integrals. (English) Zbl 0957.28004
The aim of this paper is to extend the theory of quasi-measures in [J. F. Aarnes, Adv. Math. 86, No. 1, 41-67 (1991; Zbl 0744.46052)] to locally compact Hausdorff spaces \(X\). For this purpose, the author introduces the notion of a quasi-measure in \(X\) and the notion of a quasi-integral on \(C_0(X)\) (resp. \(C_c(X)\)). A real-valued function \(\rho\) on \(C_0(X)\) (resp. \(C_c(X)\)) is called a quasi-integral on \(C_0(X)\) (resp. \(C_c(X)\)) if the following conditions are satisfied: (1) \(b\geq 0\Rightarrow \rho(b)\geq 0\) whenever \(b\in C_0(X)\) (resp. \(C_c(X)\)). (2) \(\rho\) is linear on \(A_0(a)\) (the smallest uniformly closed subalgebra of \(C_0(X)\) (resp. \(C_c(X)\)) containing \(a)\) for each \(a\in C_0(X)\) (resp. \(C_c(X)\)). A quasi-integral \(\rho\) is said to be bounded if \(\sup\{\rho(a): 0\leq a\leq 1, a\in C_c(X)\}<+ \infty\). Then it is shown that all quasi-integrals on \(C_0(X)\) are bounded, a representation of quasi-integrals on \(C_c(X)\) in terms of quasi-measures (a generalization of the Riesz representation theorem), and unique extensions of quasi-integrals on \(C_c(X)\) to \(C_0(X)\).

MSC:
28A25 Integration with respect to measures and other set functions
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[1] Johan F. Aarnes, Quasi-states and quasi-measures, Adv. Math. 86 (1991), no. 1, 41 – 67. · Zbl 0744.46052
[2] J. F. Aarnes: “Image transformations and attractors,” Dept. of Math., University of Trondheim, Preprint no. 2 (1994).
[3] J. F. Aarnes: “Quasi-measures in locally compact spaces,” Norwegian University of Science and Technology, Preprint no. 1 (1996).
[4] J. F. Aarnes and A. B. Rustad: “Probability and Quasi-measures -a new interpretation,” To appear in Math. Scand. · Zbl 0967.28014
[5] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005
[6] A. B. Rustad: “The multidimensional median as a quasi-measure,” Norwegian University of Science and Technology, Preprint no. 5 (1998).
[7] G. Taraldsen: “Image-transformations and quasi-measures in locally compact Hausdorff spaces,” University of Trondheim, Preprint (1995).
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