Geometry of spaces of capacities on metrizable compacta and components of multiplication of the capacity monad and two its submonads

O.R.Nykyforchyn

It is proved that the space of cap-capacities (also called necessity measures) and the space of cup-capacities (also known as possibility measures or sup-measures) on an infinite metrizable compactum are homeomorphic to the Hilbert cube. It is also proved that the multiplication mappings on a metrizable compactum with $\ge 2$ points for the capacity monad, the monad of cap-capacities and the monad of cup-capacities satisfy the Fibrewise Disjoint Approximation property (in the two latter cases we consider the restriction of the multiplication mapping to multiple points). The multiplication mapping of the capacity monad on a nondegenerate metrizable compactum is a trivial $Q$-fibration, but for the monad of cap-capacities and the monad the restrictions of the multiplication mappings to multiple points are locally trivial $Q$-fibrations only for a two-point compactum.

1. G.~Choquet, Theory of Capacity, Ann. Inst. Fourier (Grenoble), 5 (1953-1954) 131--295.

2. J.~Eichberger, D.~Kelsey, Non-additive beliefs and strategic equilibria, Games Econom. Behav. 30 (2000) 183--215.

3. L.~Epstein, T.~Wang, ``Beliefs about beliefs'' without probabilities, Econometrica, 64 (1996) №5, 1343--1373.

4. V.V.~Fedorchuk, V.V.~Filippov General topology: principal constructions, MGU Press, Moscow, 1988.

5. I.D.~Hlushak, O.R.~Nykyforchyn, Submonads of the capacity monad, Carpathian J. Math. 24 (2008) №1, 56--67.

6. Zhou Lin, Integral representation of continuous comonotonically additive functionals, Trans. Amer. Math. Soc. 350 (1998) №5, 1811--1822.

7. S.~Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, N.Y., 1998

8. O.R.~Nykyforchyn, On the axiom of continuity, openness and bicommutativity, Methods Funct. Anal. Topology. 4 (1998) 82--85.

9. O.R.~Nykyforchyn, Capacities with values in compact Hausdorff lattices, Appl. Categ. Structures. 15 (2007) №3, 243--257.

10. O.R.~Nykyforchyn, A monad for the inclusion hyperspace functor is unique, Mat. Stud. 27 (2007) 3--18.

11. O.R.~Nykyforchyn, Uniqueness of monad for the capacity functor and its subfunctors, preprint, 2007.

12. G.L.~O'Brien, W.~Verwaat, How subsadditive are subadditive capacities? Comment. Math. Univ. Carolin. 35 (1994) №2, 311--324.

13. D.~Schmeidler, Subjective Probability and Expected Utility without Additivity, Econometrica. 57 (1989) 571--587.

14. I.~Gilboa, D.~Schmeidler, Updating Ambiguous Beliefs, J. Econom. Theory. 59 (1993) 33--49.

15. M.M.~Zarichnyi, O.R.~Nykyforchyn, Capacity functor in the category of compacta, Mat. Sb. 199 (2008) №2, 3--26.

	Geometry of spaces of capacities on metrizable compacta and components of multiplication of the capacity monad and two its submonads
Author	O.R.Nykyforchyn nick@pu.if.ua Vasyl' Stefanyk Precarpathian National University, Department of Mathematics and Computer Science,Shevchenka 57, Ivano-Frankivsk, Ukraine
Abstract	It is proved that the space of cap-capacities (also called necessity measures) and the space of cup-capacities (also known as possibility measures or sup-measures) on an infinite metrizable compactum are homeomorphic to the Hilbert cube. It is also proved that the multiplication mappings on a metrizable compactum with $\ge 2$ points for the capacity monad, the monad of cap-capacities and the monad of cup-capacities satisfy the Fibrewise Disjoint Approximation property (in the two latter cases we consider the restriction of the multiplication mapping to multiple points). The multiplication mapping of the capacity monad on a nondegenerate metrizable compactum is a trivial $Q$-fibration, but for the monad of cap-capacities and the monad the restrictions of the multiplication mappings to multiple points are locally trivial $Q$-fibrations only for a two-point compactum.
Keywords	capacity monad, submonad, metrizable compacta
DOI	doi:10.30970/ms.31.2.122-134
Reference	1. G.~Choquet, Theory of Capacity, Ann. Inst. Fourier (Grenoble), 5 (1953-1954) 131--295. 2. J.~Eichberger, D.~Kelsey, Non-additive beliefs and strategic equilibria, Games Econom. Behav. 30 (2000) 183--215. 3. L.~Epstein, T.~Wang, ``Beliefs about beliefs'' without probabilities, Econometrica, 64 (1996) №5, 1343--1373. 4. V.V.~Fedorchuk, V.V.~Filippov General topology: principal constructions, MGU Press, Moscow, 1988. 5. I.D.~Hlushak, O.R.~Nykyforchyn, Submonads of the capacity monad, Carpathian J. Math. 24 (2008) №1, 56--67. 6. Zhou Lin, Integral representation of continuous comonotonically additive functionals, Trans. Amer. Math. Soc. 350 (1998) №5, 1811--1822. 7. S.~Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, N.Y., 1998 8. O.R.~Nykyforchyn, On the axiom of continuity, openness and bicommutativity, Methods Funct. Anal. Topology. 4 (1998) 82--85. 9. O.R.~Nykyforchyn, Capacities with values in compact Hausdorff lattices, Appl. Categ. Structures. 15 (2007) №3, 243--257. 10. O.R.~Nykyforchyn, A monad for the inclusion hyperspace functor is unique, Mat. Stud. 27 (2007) 3--18. 11. O.R.~Nykyforchyn, Uniqueness of monad for the capacity functor and its subfunctors, preprint, 2007. 12. G.L.~O'Brien, W.~Verwaat, How subsadditive are subadditive capacities? Comment. Math. Univ. Carolin. 35 (1994) №2, 311--324. 13. D.~Schmeidler, Subjective Probability and Expected Utility without Additivity, Econometrica. 57 (1989) 571--587. 14. I.~Gilboa, D.~Schmeidler, Updating Ambiguous Beliefs, J. Econom. Theory. 59 (1993) 33--49. 15. M.M.~Zarichnyi, O.R.~Nykyforchyn, Capacity functor in the category of compacta, Mat. Sb. 199 (2008) №2, 3--26.
Pages	122-134
Volume	31
Issue	2
Year	2009
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html