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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
Keywords:
quasi-measures; quasi-integrals
Full Text:
References:
 [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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