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Integral representation of continuous comonotonically additive functionals
Author(s):
Lin
Zhou
Journal:
Trans. Amer. Math. Soc.
350
(1998),
1811-1822.
MSC (1991):
Primary 28A12, 28C05, 28C15;
Secondary 60A05, 60A15, 90A05
MathSciNet review:
1373649
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Abstract:
In this paper, I first prove an integral representation theorem: Every quasi-integral on a Stone lattice can be represented by a unique upper-continuous capacity. I then apply this representation theorem to study the topological structure of the space of all upper-continuous capacities on a compact space, and to prove the existence of an upper-continuous capacity on the product space of infinitely many compact Hausdorff spaces with a collection of consistent finite marginals.
References:
- 1.
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- 2.
- C. Dellacherie, Quelques commentaires sur les prolongements de capacités, Seminaire Probabilities V, Strasbourg, Lecture Notes in Math., Vol. 191, Springer-Verlag, Berlin, 1970. MR 52:3568
- 3.
- D. Denneberg, Non-Additive Measure and Integral, Kluwer, Dordrecht, 1994. MR 96c:28017
- 4.
- L. Epstein and T. Wang, A ``Type'' Space for Games of Incomplete Information with Non-Bayesian Players, Econometrica 64 (1996), 1343-1373.
- 5.
- G. Greco, Sulla rappresentazione di funzionali mediante integrali, Rend. Sem. Math. Univ. Padova 66 (1982), 21-42. MR 83m:28008
- 6.
- G. L. O'Brien, and W. Vervaat, Capacities, Large Deviations and LogLog Laws, in Stable Processes, S. Cambahis, G Samorodnitsky, and M. S. Taquu (eds.), Birkhäuser, 1991. MR 92k:60007
- 7.
- D. Schmeidler, Integral Representation without Additivity, Proc. Amer. Math. Soc. 97 (1986), 253-261. MR 87f:28014
- 8.
- D. Schmeidler, Subjective Probability and Expected Utility without Additivity, Econometrica 57 (1989), 571-587. MR 90c:90019
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Additional Information:
Lin
Zhou
Affiliation:
Department of Economics, Duke University, Box 90097, Durham, North Carolina 27708-0097
Email:
linzhou@econ.duke.edu
DOI:
10.1090/S0002-9947-98-01735-8
PII:
S 0002-9947(98)01735-8
Keywords:
Upper-continuous capacities,
regular capacities,
Choquet integrals,
Stone lattices,
comonotonically additive functionals,
monotonic functionals,
continuous functionals,
the weak topology,
Kolmogorov's theorem,
consistent marginals
Received by editor(s):
August 16, 1995
Received by editor(s) in revised form:
October 30, 1995
Additional Notes:
I want to thank L. Epstein, D. Schmeidler, and in particular, M. Marinacci, as well as an anonymous referee, for their helpful comments. The revision was done while I was visiting the Economics Department of the Hong Kong University of Science and Technology, whose hospitality I deeply appreciated
Copyright of article:
Copyright
1998,
American Mathematical Society
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