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Probability Measures on Product Spaces with Uniform Metrics

Author

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  • Martin Hellwig

    (Max Planck Institute for Research on Collective Goods)

Abstract

The paper provides mathematical foundations for a homeomorphism theorem à la Mertens and Zamir (1985) when the space of belief hierarchies of an agent has the uniform topology rather than the product topology. The Borel σ-algebra for the uniform topology being unsuitable, the theorem relies on the product σ-algebra but defines the topology of weak convergence on the space of measures on this σ-algebra with reference to the uniform topology on the underlying space. For a countable product of complete separable metric spaces, the paper shows that this topology on the space of measures on the product σ-algebra is metrizable by the Prohorov metric. The projection mapping from such measures to sequences of measures on the first ℓ factors, ℓ=1,2,..., is a homeomorphism if the range of this mapping is also given a uniform metric.

Suggested Citation

  • Martin Hellwig, 2017. "Probability Measures on Product Spaces with Uniform Metrics," Discussion Paper Series of the Max Planck Institute for Research on Collective Goods 2017_06, Max Planck Institute for Research on Collective Goods, revised May 2023.
    • Handle: RePEc:mpg:wpaper:2017_06
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    File URL: http://www.coll.mpg.de/pdf_dat/2017_06online.pdf
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    References listed on IDEAS

    as
    1. Yi-Chun Chen & Alfredo Di Tillio & Eduardo Faingold & Siyang Xiong, 2012. "The Strategic Impact of Higher-Order Beliefs," Cowles Foundation Discussion Papers 1875, Cowles Foundation for Research in Economics, Yale University.
    2. Jonathan Weinstein & Muhamet Yildiz, 2007. "A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements," Econometrica, Econometric Society, vol. 75(2), pages 365-400, March.
    3. MERTENS, Jean-François & ZAMIR, Shmuel, 1985. "Formulation of Bayesian analysis for games with incomplete information," LIDAM Reprints CORE 608, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Gizatulina, Alia & Hellwig, Martin, 2017. "The generic possibility of full surplus extraction in models with large type spaces," Journal of Economic Theory, Elsevier, vol. 170(C), pages 385-416.
    5. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-391, June.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Product spaces with uniform metrics; weak convergence of non-Borel measures; σ-algebras generated by the open balls; quasi-separable measures; Prohorov metric;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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