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On the convexity and compactness of the integral of a Banach space valued correspondence

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  • Podczeck, Konrad
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    Abstract

    We characterize the class of finite measure spaces which guarantee that for a correspondence [phi] from to a general Banach space the Bochner integral of [phi] is convex. In addition, it is shown that if [phi] has weakly compact values and is integrably bounded, then, for this class of measure spaces, the Bochner integral of [phi] is weakly compact, too. Analogous results are provided with regard to the Gelfand integral of correspondences taking values in the dual of a separable Banach space, with "weakly compact" replaced by "weak*-compact." The crucial condition on the measure space concerns its measure algebra and is consistent with having T=[0,1] and [mu] to be an extension of Lebesgue measure.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Mathematical Economics.

    Volume (Year): 44 (2008)
    Issue (Month): 7-8 (July)
    Pages: 836-852

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    Handle: RePEc:eee:mateco:v:44:y:2008:i:7-8:p:836-852

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    Web page: http://www.elsevier.com/locate/jmateco

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    Cited by:
    1. M. Ali Khan & Kali P. Rath & Yeneng Sun & Haomiao Yu, 2012. "Large Games with a Bio-Social Typology," Working Papers, Ryerson University, Department of Economics 035, Ryerson University, Department of Economics.
    2. Haomiao Yu, 2014. "Rationalizability in large games," Economic Theory, Springer, Springer, vol. 55(2), pages 457-479, February.
    3. Noguchi, Mitsunori, 2009. "Existence of Nash equilibria in large games," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 168-184, January.
    4. Barlo, Mehmet & Carmona, Guilherme, 2011. "Strategic behavior in non-atomic games," MPRA Paper 35549, University Library of Munich, Germany.
    5. Jianwei Wang & Yongchao Zhang, 2012. "Purification, saturation and the exact law of large numbers," Economic Theory, Springer, Springer, vol. 50(3), pages 527-545, August.
    6. Sun, Yeneng & Zhang, Yongchao, 2008. "Individual Risk and Lebesgue Extension without Aggregate Uncertainty," MPRA Paper 7448, University Library of Munich, Germany.
    7. Michael Greinecker & Konrad Podczeck, 2013. "Liapounoff's vector measure theorem in Banach spaces," Working Papers 2013-20, Faculty of Economics and Statistics, University of Innsbruck.
    8. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
    9. Sun, Yeneng & Yannelis, Nicholas C., 2008. "Saturation and the integration of Banach valued correspondences," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 861-865, July.

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