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

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  • Podczeck, Konrad

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.

Suggested Citation

  • Podczeck, Konrad, 2008. "On the convexity and compactness of the integral of a Banach space valued correspondence," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 836-852, July.
  • Handle: RePEc:eee:mateco:v:44:y:2008:i:7-8:p:836-852
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    Cited by:

    1. 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.
    2. Barlo, Mehmet & Carmona, Guilherme, 2015. "Strategic behavior in non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 134-144.
    3. repec:spr:etbull:v:1:y:2013:i:2:d:10.1007_s40505-013-0018-0 is not listed on IDEAS
    4. Haomiao Yu, 2014. "Rationalizability in large games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 55(2), pages 457-479, February.
    5. Khan, M. Ali & Rath, Kali P. & Sun, Yeneng & Yu, Haomiao, 2013. "Large games with a bio-social typology," Journal of Economic Theory, Elsevier, vol. 148(3), pages 1122-1149.
    6. Jianwei Wang & Yongchao Zhang, 2012. "Purification, saturation and the exact law of large numbers," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 527-545, August.
    7. 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.
    8. Sun, Yeneng & Zhang, Yongchao, 2009. "Individual risk and Lebesgue extension without aggregate uncertainty," Journal of Economic Theory, Elsevier, vol. 144(1), pages 432-443, January.
    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.
    10. Noguchi, Mitsunori, 2009. "Existence of Nash equilibria in large games," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 168-184, January.
    11. Xiang Sun & Yongchao Zhang, 2015. "Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(1), pages 161-182, January.
    12. Khan, M. Ali & Sagara, Nobusumi, 2016. "Relaxed large economies with infinite-dimensional commodity spaces: The existence of Walrasian equilibria," Journal of Mathematical Economics, Elsevier, vol. 67(C), pages 95-107.
    13. Takashi Suzuki, 2016. "A coalitional production economy with infinitely many indivisible commodities," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 35-52, April.

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