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Integral representation of continuous comonotonically additive functionals. (English) Zbl 0905.28006
It is shown that for any quasi integral \(I\) on a Stone lattice \(L\) with \(I(1)= 1\) there exists a unique upper-continuous capacity \(\mu\) on the collection \(\Sigma\) of all upper contours of all functions belonging to \(L\) satisfying \(I(a)= \int_X ad\mu\), \(a\in L\). Moreover, \(I\) introduced by \(I(a)= \int_X ad\mu\), \(a\in L\), for some upper-continuous capacity \(\mu\) on \(\Sigma\) is a quasi integral on \(L\). Here \(\int_X ad\mu\) is defined as the Choquet integral \[ \int^\infty_0 \mu(a\geq t)dt+ \int^0_{-\infty} (\mu(a\geq t)- 1)dt,\quad a\in L. \] As an application it is shown that the set consisting of all upper continuous capacities on a compact space equipped with the weak topology is a compact Hausdorff space, which is in addition metrizable, if the underlying compact space is metrizable.

MSC:
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28A12 Contents, measures, outer measures, capacities
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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[1] G. Choquet, Theory of Capacities, Ann. Inst. Fourier (Grenoble) 5 (1955), 131-295. · Zbl 0064.35101
[2] C. Dellacherie, Quelques commentaires sur les prolongements de capacités, Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969-1970), Springer, Berlin, 1971, pp. 77 – 81. Lecture Notes in Math., Vol. 191 (French).
[3] Dieter Denneberg, Non-additive measure and integral, Theory and Decision Library. Series B: Mathematical and Statistical Methods, vol. 27, Kluwer Academic Publishers Group, Dordrecht, 1994. · Zbl 0826.28002
[4] L. Epstein and T. Wang, A “Type” Space for Games of Incomplete Information with Non-Bayesian Players, Econometrica 64 (1996), 1343-1373.
[5] Gabriele H. Greco, On the representation of functionals by means of integrals, Rend. Sem. Mat. Univ. Padova 66 (1982), 21 – 42 (Italian).
[6] George L. O’Brien and Wim Vervaat, Capacities, large deviations and loglog laws, Stable processes and related topics (Ithaca, NY, 1990) Progr. Probab., vol. 25, Birkhäuser Boston, Boston, MA, 1991, pp. 43 – 83. · Zbl 0724.60033
[7] David Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986), no. 2, 255 – 261. · Zbl 0687.28008
[8] Theodore Artikis, On the distribution of a stochastic integral arising in the theory of finance, Studia Sci. Math. Hungar. 23 (1988), no. 1-2, 209 – 213. · Zbl 0663.60013
[9] L. Zhou, Subjective Probability Theory with Alternative Feasible Act Spaces, Journal of Mathematical Economics, forthcoming.
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