Review on: Capacity functor in the category of compacta

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Abstract

The paper is a contribution to the theory of capacities in the sense of Choquet. Given a compact Hausdorff space X the collection of all upper-semicontinuous capacities is denoted by M(X). By identifying capacities with their Choquet integral, M(X) is turned into a compact Hausdorff space, via its embedding in the product RC+(X), where C+(X) stands for the collection of all non-negative continuous realvalued maps. The space of capacities contains as a subspace, the space of probability measures on X, endowed with the weak ?-topology. In case of X being a compact metric space, natural metrics for the space M(X) are introduced. Special attention goes to the capacity endofunctor M : Comp -> Comp on compact Hausdorff spaces. It is shown that this functor preserves the class of open surjective maps, thus generalising the analogue result of Ditor-Eifler for the functor of probability measures. Natural transformations \eta and \mu are introduced such that the triple (M, \eta, \mu) turns into a monad on Comp. Among other results, this monad is compared to some monads (introduced previously by other authors) such as the monad of probability measures, the monad of order preserving functionals and the monad of inclusion hyperspaces.
Original languageEnglish
JournalSt. Petersburg mathematical journal
Issue numberEuropean Mathematical Society
Publication statusPublished - 2009

Keywords

  • Capacity

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