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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 3, Pages 45–64 (Mi izv2785)  

This article is cited in 8 scientific papers (total in 8 papers)

Spaces and maps of idempotent measures

M. M. Zarichnyi

Ivan Franko National University of L'viv

Abstract: We prove that the weak* topologization of the set of all idempotent measures (Maslov measures) on compact Hausdorff spaces defines a functor on the category $\operatorname{\mathbf{Comp}}$ of compact Hausdorff spaces, and this functor is normal in the sense of E. V. Shchepin; in particular, it has many properties in common with the probability measure functor and the hyperspace functor. Moreover, we establish that this functor defines a monad in the category $\operatorname{\mathbf{Comp}}$, and prove that the idempotent measure monad contains the hyperspace monad as a submonad. For the space of idempotent measures there is an analogue of the Milyutin map (that is, of a continuous map of compact Hausdorff spaces which admits a regular averaging operator for spaces of continuous functions). Using the assertion of the existence of Milyutin maps for idempotent measures, we prove that the idempotent measure functor is open, that is, it preserves the class of open surjective maps. We also prove that, in contrast to the case of probability measure spaces, the correspondence assigning to any pair of idempotent measures the set of measures on their product which have the given marginals is not continuous.

Keywords: idempotent measure (Maslov measure), compact Hausdorff space, open map, Milyutin map, monad.

DOI: https://doi.org/10.4213/im2785

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English version:
Izvestiya: Mathematics, 2010, 74:3, 481–499

Bibliographic databases:

UDC: 515.122.5+512.582.2
MSC: Primary 18B30; Secondary 12K10, 16Y60, 54B20, 60B05
Received: 01.04.2008

Citation: M. M. Zarichnyi, “Spaces and maps of idempotent measures”, Izv. RAN. Ser. Mat., 74:3 (2010), 45–64; Izv. Math., 74:3 (2010), 481–499

Citation in format AMSBIB
\by M.~M.~Zarichnyi
\paper Spaces and maps of idempotent measures
\jour Izv. RAN. Ser. Mat.
\yr 2010
\vol 74
\issue 3
\pages 45--64
\jour Izv. Math.
\yr 2010
\vol 74
\issue 3
\pages 481--499

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  • https://doi.org/10.4213/im2785
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    This publication is cited in the following articles:
    1. U. A. Rozikov, M. M. Karimov, “Dynamics of linear maps of idempotent measures”, Lobachevskii J Math, 34:1 (2013), 20  crossref  mathscinet  zmath  elib  scopus
    2. Cencelj M., Repovs D., Zarichnyi M., “Max-Min Measures on Ultrametric Spaces”, Topology Appl., 160:5 (2013), 673–681  crossref  mathscinet  zmath  isi  elib  scopus
    3. Mazurenko N., Zarichnyi M., “Invariant Idempotent Measures”, Carpathian Math. Publ., 10:1 (2018), 172–178  crossref  zmath  isi
    4. A. A. Zaitov, A. Ya. Ishmetov, “Homotopy Properties of the Space $I_f(X)$ of Idempotent Probability Measures”, Math. Notes, 106:4 (2019), 562–571  mathnet  crossref  crossref  mathscinet  isi  elib
    5. Radul T., “On the Openness of the Idempotent Barycenter Map”, Topology Appl., 265 (2019), UNSP 106809  crossref  mathscinet  isi
    6. Brydun V., Savchenko A., Zarichnyi M., “Fuzzy Metrization of the Spaces of Idempotent Measures”, Eur. J. Math., 6:1, SI (2020), 98–109  crossref  isi
    7. Zaitov A.A., “On a Metric on the Space of Idempotent Probability Measures”, Appl. Gen. Topol., 21:1 (2020), 35–51  crossref  mathscinet  isi
    8. Radul T., “Idempotent Measures: Absolute Retracts and Soft Maps”, Topol. Methods Nonlinear Anal., 56:1 (2020), 161–172  crossref  mathscinet  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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