IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-03506933.html
   My bibliography  Save this paper

Fatou's Lemma for Unbounded Gelfand Integrable Mappings

Author

Listed:
  • Bernard Cornet

    (Department of economics, University of Kansas - KU - University of Kansas [Lawrence])

  • V. Filipe Martins-Da-Rocha

    (LEDa - Laboratoire d'Economie de Dauphine - IRD - Institut de Recherche pour le Développement - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

Abstract

The objective of this paper is to provide Fatou-type results for sequences of Gelfand integrable mappings defined on a measure space with values in the topological dual of a separable Banach space E. We introduce the assumption of mean weak boundedness that encompasses stronger conditions considered previously. The proof of our Fatou-type results consists in applying successively the scalar version of Komlós' theorem. Since Schmeidler (1970), Fatou's lemma is one of the essential tools in proving the existence of Walrasian equilibria in economic models with a measure space of economic agents as initiated by Aumann (1966). With infinitely many commodities, with commodity space being the topological dual of a separable Banach space, the version of Fatou's lemma with the Gelfand integral applies directly to models of spacial economies (Cornet and Médecin 2000) and models with differentiated commodities (Ostroy and Zame 1994 and Martins-da-Rocha 2003).

Suggested Citation

  • Bernard Cornet & V. Filipe Martins-Da-Rocha, 2021. "Fatou's Lemma for Unbounded Gelfand Integrable Mappings," Post-Print hal-03506933, HAL.
    • Handle: RePEc:hal:journl:hal-03506933
    Note: View the original document on HAL open archive server: https://hal.science/hal-03506933
    as

    Download full text from publisher

    File URL: https://hal.science/hal-03506933/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Konard Podczeck, 1997. "Markets with infinitely many commodities and a continuum of agents with non-convex preferences (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(3), pages 385-426.
    2. Bernard Cornet & Jean-Philippe Medecin, 2000. "Existence of General Equilibrium for Spatial Economies," Econometric Society World Congress 2000 Contributed Papers 1107, Econometric Society.
    3. Artstein, Zvi, 1979. "A note on fatou's lemma in several dimensions," Journal of Mathematical Economics, Elsevier, vol. 6(3), pages 277-282, December.
    4. SCHMEIDLER, David, 1970. "Fatou's lemma in several dimensions," LIDAM Reprints CORE 46, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Filipe Martins-da-Rocha, V., 2003. "Equilibria in large economies with a separable Banach commodity space and non-ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 39(8), pages 863-889, November.
    6. Ostroy, Joseph M & Zame, William R, 1994. "Nonatomic Economies and the Boundaries of Perfect Competition," Econometrica, Econometric Society, vol. 62(3), pages 593-633, May.
    7. Hildenbrand, Werner, 1970. "Existence of Equilibria for Economies with Production and a Measure Space of Consumers," Econometrica, Econometric Society, vol. 38(5), pages 608-623, September.
    8. Erik J. Balder & Christian Hess, 1995. "Fatou's Lemma for Multifunctions with Unbounded Values," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 175-188, February.
    9. 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.
    10. Lee, Sangjik, 2013. "Competitive Equilibrium With An Atomless Measure Space Of Agents And Infinite Dimensional Commodity Spaces Without Convex And Complete Preferences," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 54(2), pages 221-230, December.
    11. HILDENBRAND, Werner, 1970. "Existence of equilibria for economies with production and a measure space of consumers," LIDAM Reprints CORE 79, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    12. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    13. Noguchi, Mitsunori, 1997. "Economies with a continuum of consumers, a continuum of suppliers and an infinite dimensional commodity space," Journal of Mathematical Economics, Elsevier, vol. 27(1), pages 1-21, February.
    14. BEWLEY, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," LIDAM Reprints CORE 122, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. Greinecker, Michael & Podczeck, Konrad, 2017. "Core equivalence with differentiated commodities," Journal of Mathematical Economics, Elsevier, vol. 73(C), pages 54-67.
    16. Cornet, Bernard & Topuzu, Mihaela & Yildiz, Aysegul, 2003. "Equilibrium theory with a measure space of possibly satiated consumers," Journal of Mathematical Economics, Elsevier, vol. 39(3-4), pages 175-196, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bernard Cornet & V. F. Martins-Da-Rocha, 2005. "Fatou¡¯S Lemma For Unbounded Gelfand Integrable Mappings," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 200503, University of Kansas, Department of Economics, revised Feb 2005.
    2. Motoki Otsuka, 2024. "The existence of Walrasian equilibrium: infinitely many commodities, measure space of agents, and discontinuous preferences," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(2), pages 119-140, December.
    3. Jang, Hyo Seok & Lee, Sangjik, 2020. "Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 57-64.
    4. 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.
    5. Martins-da-Rocha, Victor Filipe & Angeloni, Laura, 2008. "Large economies with differential information but without free disposal," FGV EPGE Economics Working Papers (Ensaios Economicos da EPGE) 671, EPGE Brazilian School of Economics and Finance - FGV EPGE (Brazil).
    6. Niccolò Urbinati, 2023. "The Walrasian objection mechanism and Mas-Colell’s bargaining set in economies with many commodities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(1), pages 45-68, July.
    7. Angeloni, Laura & Martins-da-Rocha, Victor Filipe, 2007. "Contract enforcement and incentive compatibility in large economies with differential information: the role of exact feasibility," FGV EPGE Economics Working Papers (Ensaios Economicos da EPGE) 647, EPGE Brazilian School of Economics and Finance - FGV EPGE (Brazil).
    8. Niccolò Urbinati, 2020. "Walrasian objection mechanism and Mas Colell's bargaining set in economies with many commodities," Working Papers 07, Venice School of Management - Department of Management, Università Ca' Foscari Venezia.
    9. Cea-Echenique, Sebastián & Fuentes, Matías, 2024. "On the continuity of the Walras correspondence in distributional economies with an infinite-dimensional commodity space," Mathematical Social Sciences, Elsevier, vol. 129(C), pages 61-69.
    10. Filipe Martins-da-Rocha, V. & Topuzu, Mihaela, 2008. "Cournot-Nash equilibria in continuum games with non-ordered preferences," Journal of Economic Theory, Elsevier, vol. 140(1), pages 314-327, May.
    11. Jiuqiang Liu, 2022. "Equivalence of Competitive Equilibria, Fuzzy Cores, and Fuzzy Bargaining Sets in Finite Production Economies," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    12. Jiuqiang Liu & Huihui Zhang, 2016. "Coincidence of the Mas-Colell bargaining set and the set of competitive equilibria in a continuum coalition production economy," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 1095-1109, November.
    13. He, Wei & Sun, Yeneng, 2022. "Conditional expectation of Banach valued correspondences and economic applications," Journal of Mathematical Economics, Elsevier, vol. 101(C).
    14. Jiuqiang Liu, 2017. "Existence of competitive equilibrium in coalition production economies with a continuum of agents," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 941-955, November.
    15. repec:dau:papers:123456789/6273 is not listed on IDEAS
    16. repec:dau:papers:123456789/2344 is not listed on IDEAS
    17. Noguchi, Mitsunori, 1997. "Economies with a continuum of agents with the commodity-price pairing (l[infin], l1)," Journal of Mathematical Economics, Elsevier, vol. 28(3), pages 265-287, October.
    18. Ghosal, Sayantan & Tonin, Simone, 2014. "Non-Cooperative Asymptotic Oligopoly in Economies with Infinitely Many Commodities," 2007 Annual Meeting, July 29-August 1, 2007, Portland, Oregon TN 2015-23, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    19. Hyo Seok Jang & Sangjik Lee, 2019. "Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences," Papers 1904.07444, arXiv.org, revised Feb 2020.
    20. Monteiro, P. K. & Araújo, Aloísio Pessoa de & Martins-da-Rocha, Victor Filipe, 2003. "Equilibria in security markets with a continuum of agents," FGV EPGE Economics Working Papers (Ensaios Economicos da EPGE) 513, EPGE Brazilian School of Economics and Finance - FGV EPGE (Brazil).
    21. Bernard Cornet & Mihaela Topuzu, 2005. "Existence of equilibria for economies with externalities and a measure space of consumers," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(2), pages 397-421, August.
    22. Laura Angeloni & V. Martins-da-Rocha, 2009. "Large economies with differential information and without free disposal," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 263-286, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:hal-03506933. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.